diff --git "a/test/test.json" "b/test/test.json"
new file mode 100644--- /dev/null
+++ "b/test/test.json"
@@ -0,0 +1 @@
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\u03b1\nx y : \u03b1\n\u22a2 span {y} \u2264 span {x} \u2227 span {x} \u2264 span {y} \u2194 x \u2223 y \u2227 y \u2223 x"}, {"tactic": "apply and_congr <;> rw [span_singleton_le_span_singleton]", "annotated_tactic": ["apply <a>and_congr</a> <;> rw [<a>span_singleton_le_span_singleton</a>]", [{"full_name": "and_congr", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [43, 9], "def_end_pos": [43, 18]}, {"full_name": "Ideal.span_singleton_le_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [511, 9], "def_end_pos": [511, 41]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\na b : \u03b1\u271d\ninst\u271d\u00b2 : CommSemiring \u03b1\u271d\nI : Ideal \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : IsDomain \u03b1\nx y : \u03b1\n\u22a2 span {y} \u2264 span {x} \u2227 span {x} \u2264 span {y} \u2194 x \u2223 y \u2227 y \u2223 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean", "full_name": "MeasureTheory.OuterMeasure.comap_ofFunction", "start": [172, 1], "end": [184, 88], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type ?u.25201\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 (fun s => m (f '' s)) \u2205 = 0", "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type ?u.25201\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 m \u2205 = 0"}, {"tactic": "simp [m_empty]", "annotated_tactic": ["simp [m_empty]", []], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type ?u.25201\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 m \u2205 = 0", "state_after": "no goals"}, {"tactic": "refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>le_ofFunction</a>.2 fun s => ?_) fun s => ?_", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.OuterMeasure.le_ofFunction", "def_path": "Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean", "def_pos": [116, 9], "def_end_pos": [116, 22]}]], "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 (comap f) (OuterMeasure.ofFunction m m_empty) = OuterMeasure.ofFunction (fun s => m (f '' s)) \u22ef", "state_after": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 ((comap f) (OuterMeasure.ofFunction m m_empty)) s \u2264 m (f '' s)\n\ncase refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 (OuterMeasure.ofFunction (fun s => m (f '' s)) \u22ef) s \u2264 ((comap f) (OuterMeasure.ofFunction m m_empty)) s"}, {"tactic": "rw [comap_apply]", "annotated_tactic": ["rw [<a>comap_apply</a>]", [{"full_name": "MeasureTheory.OuterMeasure.comap_apply", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Operations.lean", "def_pos": [323, 9], "def_end_pos": [323, 20]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 ((comap f) (OuterMeasure.ofFunction m m_empty)) s \u2264 m (f '' s)", "state_after": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 (OuterMeasure.ofFunction m m_empty) (f '' s) \u2264 m (f '' s)"}, {"tactic": "apply ofFunction_le", "annotated_tactic": ["apply <a>ofFunction_le</a>", [{"full_name": "MeasureTheory.OuterMeasure.ofFunction_le", "def_path": "Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean", "def_pos": [99, 9], "def_end_pos": [99, 22]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 (OuterMeasure.ofFunction m m_empty) (f '' s) \u2264 m (f '' s)", "state_after": "no goals"}, {"tactic": "rw [comap_apply, ofFunction_apply, ofFunction_apply]", "annotated_tactic": ["rw [<a>comap_apply</a>, <a>ofFunction_apply</a>, <a>ofFunction_apply</a>]", [{"full_name": "MeasureTheory.OuterMeasure.comap_apply", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Operations.lean", "def_pos": [323, 9], "def_end_pos": [323, 20]}, {"full_name": "MeasureTheory.OuterMeasure.ofFunction_apply", "def_path": "Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean", "def_pos": [92, 9], "def_end_pos": [92, 25]}, {"full_name": "MeasureTheory.OuterMeasure.ofFunction_apply", "def_path": "Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean", "def_pos": [92, 9], "def_end_pos": [92, 25]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 (OuterMeasure.ofFunction (fun s => m (f '' s)) \u22ef) s \u2264 ((comap f) (OuterMeasure.ofFunction m m_empty)) s", "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f '' t n) \u2264 \u2a05 t, \u2a05 (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)"}, {"tactic": "refine iInf_mono' fun t => \u27e8fun k => f \u207b\u00b9' t k, ?_\u27e9", "annotated_tactic": ["refine <a>iInf_mono'</a> fun t => \u27e8fun k => f \u207b\u00b9' t k, ?_\u27e9", [{"full_name": "iInf_mono'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [808, 9], "def_end_pos": [808, 19]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f '' t n) \u2264 \u2a05 t, \u2a05 (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)", "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264\n    \u2a05 (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)"}, {"tactic": "refine iInf_mono' fun ht => ?_", "annotated_tactic": ["refine <a>iInf_mono'</a> fun ht => ?_", [{"full_name": "iInf_mono'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [808, 9], "def_end_pos": [808, 19]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264\n    \u2a05 (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)", "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f '' s \u2286 iUnion t\n\u22a2 \u2203 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)"}, {"tactic": "rw [Set.image_subset_iff, preimage_iUnion] at ht", "annotated_tactic": ["rw [<a>Set.image_subset_iff</a>, <a>preimage_iUnion</a>] at ht", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [481, 9], "def_end_pos": [481, 25]}, {"full_name": "Set.preimage_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1713, 9], "def_end_pos": [1713, 24]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f '' s \u2286 iUnion t\n\u22a2 \u2203 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)", "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2203 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)"}, {"tactic": "refine \u27e8ht, ENNReal.tsum_le_tsum fun n => ?_\u27e9", "annotated_tactic": ["refine \u27e8ht, <a>ENNReal.tsum_le_tsum</a> fun n => ?_\u27e9", [{"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [854, 19], "def_end_pos": [854, 31]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\n\u22a2 \u2203 (_ : s \u2286 \u22c3 k, f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)", "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)"}, {"tactic": "cases' h with hl hr", "annotated_tactic": ["cases' h with hl hr", []], "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)", "state_after": "case refine_2.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhl : Monotone m\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)\n\ncase refine_2.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhr : Surjective f\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)"}, {"tactic": "exacts [hl (image_preimage_subset _ _), (congr_arg m (hr.image_preimage (t n))).le]", "annotated_tactic": ["exacts [hl (<a>image_preimage_subset</a> _ _), (<a>congr_arg</a> m (hr.image_preimage (t n))).<a>le</a>]", [{"full_name": "Set.image_preimage_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [485, 9], "def_end_pos": [485, 30]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [154, 7], "def_end_pos": [154, 12]}]], "state_before": "case refine_2.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhl : Monotone m\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)\n\ncase refine_2.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 i, f \u207b\u00b9' t i\nn : \u2115\nhr : Surjective f\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsGroupHom.inv_ker_one'", "start": [367, 1], "end": [370, 43], "traced_tactics": [{"tactic": "have : (f a)\u207b\u00b9 * f b = 1 := by rw [h, mul_left_inv]", "annotated_tactic": ["have : (f a)\u207b\u00b9 * f b = 1 := by rw [h, <a>mul_left_inv</a>]", [{"full_name": "mul_left_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1222, 9], "def_end_pos": [1222, 21]}]], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 f (a\u207b\u00b9 * b) = 1", "state_after": "G : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\nthis : (f a)\u207b\u00b9 * f b = 1\n\u22a2 f (a\u207b\u00b9 * b) = 1"}, {"tactic": "rwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this", "annotated_tactic": ["rwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this", []], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\nthis : (f a)\u207b\u00b9 * f b = 1\n\u22a2 f (a\u207b\u00b9 * b) = 1", "state_after": "no goals"}, {"tactic": "rw [h, mul_left_inv]", "annotated_tactic": ["rw [h, <a>mul_left_inv</a>]", [{"full_name": "mul_left_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1222, 9], "def_end_pos": [1222, 21]}]], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 (f a)\u207b\u00b9 * f b = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Normed.lean", "full_name": "Convex.thickening", "start": [70, 1], "end": [72, 33], "traced_tactics": [{"tactic": "rw [\u2190 add_ball_zero]", "annotated_tactic": ["rw [\u2190 <a>add_ball_zero</a>]", [{"full_name": "add_ball_zero", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [221, 3], "def_end_pos": [221, 14]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\ns t : Set E\nhs : Convex \u211d s\n\u03b4 : \u211d\n\u22a2 Convex \u211d (Metric.thickening \u03b4 s)", "state_after": "\u03b9 : Type u_1\nE : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\ns t : Set E\nhs : Convex \u211d s\n\u03b4 : \u211d\n\u22a2 Convex \u211d (s + ball 0 \u03b4)"}, {"tactic": "exact hs.add (convex_ball 0 _)", "annotated_tactic": ["exact hs.add (<a>convex_ball</a> 0 _)", [{"full_name": "convex_ball", "def_path": "Mathlib/Analysis/Convex/Normed.lean", "def_pos": [62, 9], "def_end_pos": [62, 20]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\ns t : Set E\nhs : Convex \u211d s\n\u03b4 : \u211d\n\u22a2 Convex \u211d (s + ball 0 \u03b4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Roots.lean", "full_name": "Polynomial.ne_zero_of_mem_roots", "start": [118, 1], "end": [119, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/Basic.lean", "full_name": "SetTheory.PGame.zero_lf_inv'", "start": [975, 1], "end": [978, 8], "traced_tactics": [{"tactic": "convert lf_mk _ _ InvTy.zero", "annotated_tactic": ["convert <a>lf_mk</a> _ _ <a>InvTy.zero</a>", [{"full_name": "SetTheory.PGame.lf_mk", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [607, 9], "def_end_pos": [607, 14]}, {"full_name": "SetTheory.PGame.InvTy.zero", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [919, 5], "def_end_pos": [919, 9]}]], "state_before": "xl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 0 \u29cf (mk xl xr xL xR).inv'", "state_after": "case h.e'_1\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 0 = invVal (fun i => xL \u2191i) xR (fun i => (xL \u2191i).inv') (fun i => (xR i).inv') InvTy.zero"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_1\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 0 = invVal (fun i => xL \u2191i) xR (fun i => (xL \u2191i).inv') (fun i => (xR i).inv') InvTy.zero", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNReal/Basic.lean", "full_name": "NNReal.le_inv_iff_mul_le", "start": [923, 1], "end": [924, 75], "traced_tactics": [{"tactic": "rw [\u2190 mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]", "annotated_tactic": ["rw [\u2190 <a>mul_le_mul_left</a> (<a>pos_iff_ne_zero</a>.2 h), <a>mul_inv_cancel</a> h, <a>mul_comm</a>]", [{"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [265, 9], "def_end_pos": [265, 24]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [230, 3], "def_end_pos": [230, 14]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [223, 15], "def_end_pos": [223, 29]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "r p : \u211d\u22650\nh : p \u2260 0\n\u22a2 r \u2264 p\u207b\u00b9 \u2194 r * p \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integrable_of_intervalIntegral_norm_bounded", "start": [541, 1], "end": [548, 58], "traced_tactics": [{"tactic": "have h\u03c6 : AECover \u03bc l _ := aecover_Ioc ha hb", "annotated_tactic": ["have h\u03c6 : <a>AECover</a> \u03bc l _ := <a>aecover_Ioc</a> ha hb", [{"full_name": "MeasureTheory.AECover", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [100, 11], "def_end_pos": [100, 18]}, {"full_name": "MeasureTheory.aecover_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [191, 9], "def_end_pos": [191, 20]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\n\u22a2 Integrable f \u03bc", "state_after": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\n\u22a2 Integrable f \u03bc"}, {"tactic": "refine h\u03c6.integrable_of_integral_norm_bounded I hfi (h.mp ?_)", "annotated_tactic": ["refine h\u03c6.integrable_of_integral_norm_bounded I hfi (h.mp ?_)", []], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\n\u22a2 Integrable f \u03bc", "state_after": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u222b (x : \u211d) in a x..b x, \u2016f x\u2016 \u2202\u03bc \u2264 I \u2192 \u222b (x : \u211d) in Ioc (a x) (b x), \u2016f x\u2016 \u2202\u03bc \u2264 I"}, {"tactic": "filter_upwards [ha.eventually (eventually_le_atBot 0),\n  hb.eventually (eventually_ge_atTop 0)] with i hai hbi ht", "annotated_tactic": ["filter_upwards [ha.eventually (<a>eventually_le_atBot</a> 0),\n    hb.eventually (<a>eventually_ge_atTop</a> 0)] with i hai hbi ht", [{"full_name": "Filter.eventually_le_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}, {"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}]], "state_before": "\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u222b (x : \u211d) in a x..b x, \u2016f x\u2016 \u2202\u03bc \u2264 I \u2192 \u222b (x : \u211d) in Ioc (a x) (b x), \u2016f x\u2016 \u2202\u03bc \u2264 I", "state_after": "case h\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\ni : \u03b9\nhai : a i \u2264 0\nhbi : 0 \u2264 b i\nht : \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2202\u03bc \u2264 I"}, {"tactic": "rwa [\u2190 intervalIntegral.integral_of_le (hai.trans hbi)]", "annotated_tactic": ["rwa [\u2190 <a>intervalIntegral.integral_of_le</a> (hai.trans hbi)]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [487, 9], "def_end_pos": [487, 23]}]], "state_before": "case h\n\u03b9 : Type u_1\nE : Type u_2\n\u03bc : Measure \u211d\nl : Filter \u03b9\ninst\u271d\u00b2 : l.NeBot\ninst\u271d\u00b9 : l.IsCountablyGenerated\ninst\u271d : NormedAddCommGroup E\na b : \u03b9 \u2192 \u211d\nf : \u211d \u2192 E\nI : \u211d\nhfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) \u03bc\nha : Tendsto a l atBot\nhb : Tendsto b l atTop\nh : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\nh\u03c6 : AECover \u03bc l fun i => Ioc (a i) (b i)\ni : \u03b9\nhai : a i \u2264 0\nhbi : 0 \u2264 b i\nht : \u222b (x : \u211d) in a i..b i, \u2016f x\u2016 \u2202\u03bc \u2264 I\n\u22a2 \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2202\u03bc \u2264 I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean", "full_name": "Nat.casesAuxOn_succ", "start": [40, 1], "end": [42, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "mul_right_cancel", "start": [249, 1], "end": [250, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "full_name": "CauSeq.const_zero", "start": [287, 1], "end": [288, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/SeparableDegree.lean", "full_name": "IntermediateField.separable_of_mem_isSeparable", "start": [741, 1], "end": [743, 72], "traced_tactics": [{"tactic": "simpa only [minpoly_eq] using IsSeparable.separable F (K := L) \u27e8x, h\u27e9", "annotated_tactic": ["simpa only [<a>minpoly_eq</a>] using <a>IsSeparable.separable</a> F (K := L) \u27e8x, h\u27e9", [{"full_name": "IntermediateField.minpoly_eq", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [831, 9], "def_end_pos": [831, 19]}, {"full_name": "IsSeparable.separable", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [572, 9], "def_end_pos": [572, 30]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F E\nK : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra F K\nL : IntermediateField F E\ninst\u271d : IsSeparable F \u21a5L\nx : E\nh : x \u2208 L\n\u22a2 (minpoly F x).Separable", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on", "start": [1067, 1], "end": [1071, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.untrop_mul", "start": [392, 1], "end": [393, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "CochainComplex.mk'_d_1_0", "start": [1085, 1], "end": [1087, 36], "traced_tactics": [{"tactic": "change ite (1 = 0 + 1) (d\u2080 \u226b \ud835\udfd9 X\u2081) 0 = d\u2080", "annotated_tactic": ["change <a>ite</a> (1 = 0 + 1) (d\u2080 \u226b \ud835\udfd9 X\u2081) 0 = d\u2080", [{"full_name": "ite", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [970, 21], "def_end_pos": [970, 24]}]], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\nX\u2080 X\u2081 X\u2082 : V\nd\u2080 : X\u2080 \u27f6 X\u2081\nd\u2081 : X\u2081 \u27f6 X\u2082\ns : d\u2080 \u226b d\u2081 = 0\nsucc : (S : ShortComplex V) \u2192 (X\u2084 : V) \u00d7' (d\u2082 : S.X\u2083 \u27f6 X\u2084) \u00d7' S.g \u226b d\u2082 = 0\nsucc' : {X\u2080 X\u2081 : V} \u2192 (f : X\u2080 \u27f6 X\u2081) \u2192 (X\u2082 : V) \u00d7' (d : X\u2081 \u27f6 X\u2082) \u00d7' f \u226b d = 0\n\u22a2 (mk' X\u2080 X\u2081 d\u2080 fun {X\u2080 X\u2081} => succ').d 0 1 = d\u2080", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\nX\u2080 X\u2081 X\u2082 : V\nd\u2080 : X\u2080 \u27f6 X\u2081\nd\u2081 : X\u2081 \u27f6 X\u2082\ns : d\u2080 \u226b d\u2081 = 0\nsucc : (S : ShortComplex V) \u2192 (X\u2084 : V) \u00d7' (d\u2082 : S.X\u2083 \u27f6 X\u2084) \u00d7' S.g \u226b d\u2082 = 0\nsucc' : {X\u2080 X\u2081 : V} \u2192 (f : X\u2080 \u27f6 X\u2081) \u2192 (X\u2082 : V) \u00d7' (d : X\u2081 \u27f6 X\u2082) \u00d7' f \u226b d = 0\n\u22a2 (if 1 = 0 + 1 then d\u2080 \u226b \ud835\udfd9 X\u2081 else 0) = d\u2080"}, {"tactic": "rw [if_pos rfl, Category.comp_id]", "annotated_tactic": ["rw [<a>if_pos</a> <a>rfl</a>, <a>Category.comp_id</a>]", [{"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}]], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\nX\u2080 X\u2081 X\u2082 : V\nd\u2080 : X\u2080 \u27f6 X\u2081\nd\u2081 : X\u2081 \u27f6 X\u2082\ns : d\u2080 \u226b d\u2081 = 0\nsucc : (S : ShortComplex V) \u2192 (X\u2084 : V) \u00d7' (d\u2082 : S.X\u2083 \u27f6 X\u2084) \u00d7' S.g \u226b d\u2082 = 0\nsucc' : {X\u2080 X\u2081 : V} \u2192 (f : X\u2080 \u27f6 X\u2081) \u2192 (X\u2082 : V) \u00d7' (d : X\u2081 \u27f6 X\u2082) \u00d7' f \u226b d = 0\n\u22a2 (if 1 = 0 + 1 then d\u2080 \u226b \ud835\udfd9 X\u2081 else 0) = d\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Equiv.Perm.prod_comp'", "start": [495, 1], "end": [498, 30], "traced_tactics": [{"tactic": "convert \u03c3.prod_comp s (fun x => f x (\u03c3.symm x)) hs", "annotated_tactic": ["convert \u03c3.prod_comp s (fun x => f x (\u03c3.symm x)) hs", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\n\u03c3 : Equiv.Perm \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\nhs : {a | \u03c3 a \u2260 a} \u2286 \u2191s\n\u22a2 \u220f x \u2208 s, f (\u03c3 x) x = \u220f x \u2208 s, f x ((Equiv.symm \u03c3) x)", "state_after": "case h.e'_2.a.h.e'_2\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\n\u03c3 : Equiv.Perm \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\nhs : {a | \u03c3 a \u2260 a} \u2286 \u2191s\nx\u271d : \u03b1\na\u271d : x\u271d \u2208 s\n\u22a2 x\u271d = (Equiv.symm \u03c3) (\u03c3 x\u271d)"}, {"tactic": "rw [Equiv.symm_apply_apply]", "annotated_tactic": ["rw [<a>Equiv.symm_apply_apply</a>]", [{"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [282, 17], "def_end_pos": [282, 33]}]], "state_before": "case h.e'_2.a.h.e'_2\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\n\u03c3 : Equiv.Perm \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\nhs : {a | \u03c3 a \u2260 a} \u2286 \u2191s\nx\u271d : \u03b1\na\u271d : x\u271d \u2208 s\n\u22a2 x\u271d = (Equiv.symm \u03c3) (\u03c3 x\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.erase_toFinmap", "start": [419, 1], "end": [420, 15], "traced_tactics": [{"tactic": "simp [erase]", "annotated_tactic": ["simp [<a>erase</a>]", [{"full_name": "Finmap.erase", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [414, 5], "def_end_pos": [414, 10]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 erase a \u27e6s\u27e7 = \u27e6AList.erase a s\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_add_atTop_nat", "start": [1787, 1], "end": [1788, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.sUnion_subset_iff", "start": [1023, 1], "end": [1024, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.coe_inv", "start": [76, 1], "end": [77, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.subset_biUnion_of_mem", "start": [830, 1], "end": [833, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.to_hasBasis'", "start": [340, 1], "end": [345, 33], "traced_tactics": [{"tactic": "refine \u27e8fun t => \u27e8fun ht => ?_, fun \u27e8i', hi', ht\u27e9 => mem_of_superset (h' i' hi') ht\u27e9\u27e9", "annotated_tactic": ["refine \u27e8fun t => \u27e8fun ht => ?_, fun \u27e8i', hi', ht\u27e9 => <a>mem_of_superset</a> (h' i' hi') ht\u27e9\u27e9", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\n\u22a2 l.HasBasis p' s'", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht : t \u2208 l\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t"}, {"tactic": "rcases hl.mem_iff.1 ht with \u27e8i, hi, ht\u27e9", "annotated_tactic": ["rcases hl.mem_iff.1 ht with \u27e8i, hi, ht\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht : t \u2208 l\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t"}, {"tactic": "rcases h i hi with \u27e8i', hi', hs's\u27e9", "annotated_tactic": ["rcases h i hi with \u27e8i', hi', hs's\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni'\u271d : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\ni' : \u03b9'\nhi' : p' i'\nhs's : s' i' \u2286 s i\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t"}, {"tactic": "exact \u27e8i', hi', hs's.trans ht\u27e9", "annotated_tactic": ["exact \u27e8i', hi', hs's.trans ht\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni'\u271d : \u03b9'\nhl : l.HasBasis p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\ni' : \u03b9'\nhi' : p' i'\nhs's : s' i' \u2286 s i\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean", "full_name": "FreeAlgebra.toTensor_\u03b9", "start": [352, 1], "end": [353, 18], "traced_tactics": [{"tactic": "simp [toTensor]", "annotated_tactic": ["simp [<a>toTensor</a>]", [{"full_name": "FreeAlgebra.toTensor", "def_path": "Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean", "def_pos": [347, 5], "def_end_pos": [347, 13]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nM : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : M\n\u22a2 toTensor (\u03b9 R m) = (TensorAlgebra.\u03b9 R) m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "String.revPosOfAux_eq", "start": [267, 1], "end": [267, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "Filter.EventuallyEq.isExtrFilter_iff", "start": [676, 1], "end": [678, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_sdiff_eq_self", "start": [2413, 1], "end": [2414, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/NormTrace.lean", "full_name": "Algebra.map_leftMulMatrix_localization", "start": [50, 1], "end": [56, 90], "traced_tactics": [{"tactic": "ext i j", "annotated_tactic": ["ext i j", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\na : S\n\u22a2 (algebraMap R R\u2098).mapMatrix ((leftMulMatrix b) a) =\n    (leftMulMatrix (Basis.localizationLocalization R\u2098 M S\u2098 b)) ((algebraMap S S\u2098) a)", "state_after": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\na : S\ni j : \u03b9\n\u22a2 (algebraMap R R\u2098).mapMatrix ((leftMulMatrix b) a) i j =\n    (leftMulMatrix (Basis.localizationLocalization R\u2098 M S\u2098 b)) ((algebraMap S S\u2098) a) i j"}, {"tactic": "simp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, \u2190 map_mul,\n  Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_algebraMap]", "annotated_tactic": ["simp only [<a>Matrix.map_apply</a>, <a>RingHom.mapMatrix_apply</a>, <a>leftMulMatrix_eq_repr_mul</a>, \u2190 <a>map_mul</a>,\n    <a>Basis.localizationLocalization_apply</a>, <a>Basis.localizationLocalization_repr_algebraMap</a>]", [{"full_name": "Matrix.map_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "RingHom.mapMatrix_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1551, 3], "def_end_pos": [1551, 8]}, {"full_name": "Algebra.leftMulMatrix_eq_repr_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [927, 9], "def_end_pos": [927, 34]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "Basis.localizationLocalization_apply", "def_path": "Mathlib/RingTheory/Localization/Module.lean", "def_pos": [153, 9], "def_end_pos": [153, 45]}, {"full_name": "Basis.localizationLocalization_repr_algebraMap", "def_path": "Mathlib/RingTheory/Localization/Module.lean", "def_pos": [159, 9], "def_end_pos": [159, 55]}]], "state_before": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\na : S\ni j : \u03b9\n\u22a2 (algebraMap R R\u2098).mapMatrix ((leftMulMatrix b) a) i j =\n    (leftMulMatrix (Basis.localizationLocalization R\u2098 M S\u2098 b)) ((algebraMap S S\u2098) a) i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Monic.lean", "full_name": "Polynomial.Monic.degree_mul_comm", "start": [182, 1], "end": [187, 57], "traced_tactics": [{"tactic": "by_cases h : q = 0", "annotated_tactic": ["by_cases h : q = 0", []], "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\n\u22a2 (p * q).degree = (q * p).degree", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : q = 0\n\u22a2 (p * q).degree = (q * p).degree\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 (p * q).degree = (q * p).degree"}, {"tactic": "rw [degree_mul', hp.degree_mul]", "annotated_tactic": ["rw [<a>degree_mul'</a>, hp.degree_mul]", [{"full_name": "Polynomial.degree_mul'", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [995, 9], "def_end_pos": [995, 20]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 (p * q).degree = (q * p).degree", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 p.degree + q.degree = q.degree + p.degree\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 p.leadingCoeff * q.leadingCoeff \u2260 0"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : q = 0\n\u22a2 (p * q).degree = (q * p).degree", "state_after": "no goals"}, {"tactic": "exact add_comm _ _", "annotated_tactic": ["exact <a>add_comm</a> _ _", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 p.degree + q.degree = q.degree + p.degree", "state_after": "no goals"}, {"tactic": "rwa [hp.leadingCoeff, one_mul, leadingCoeff_ne_zero]", "annotated_tactic": ["rwa [hp.leadingCoeff, <a>one_mul</a>, <a>leadingCoeff_ne_zero</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Polynomial.leadingCoeff_ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [695, 9], "def_end_pos": [695, 29]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q\u271d r : R[X]\nhp : p.Monic\nq : R[X]\nh : \u00acq = 0\n\u22a2 p.leadingCoeff * q.leadingCoeff \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.head_cyclicPermutations", "start": [612, 1], "end": [615, 72], "traced_tactics": [{"tactic": "have h : 0 < length (cyclicPermutations l) := length_pos_of_ne_nil (cyclicPermutations_ne_nil _)", "annotated_tactic": ["have h : 0 < <a>length</a> (<a>cyclicPermutations</a> l) := <a>length_pos_of_ne_nil</a> (<a>cyclicPermutations_ne_nil</a> _)", [{"full_name": "List.length", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2316, 5], "def_end_pos": [2316, 16]}, {"full_name": "List.cyclicPermutations", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [562, 5], "def_end_pos": [562, 23]}, {"full_name": "List.length_pos_of_ne_nil", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [176, 30], "def_end_pos": [176, 50]}, {"full_name": "List.cyclicPermutations_ne_nil", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [593, 9], "def_end_pos": [593, 34]}]], "state_before": "\u03b1 : Type u\nl\u271d l' l : List \u03b1\n\u22a2 l.cyclicPermutations.head \u22ef = l", "state_after": "\u03b1 : Type u\nl\u271d l' l : List \u03b1\nh : 0 < l.cyclicPermutations.length\n\u22a2 l.cyclicPermutations.head \u22ef = l"}, {"tactic": "rw [\u2190 get_mk_zero h, get_cyclicPermutations, Fin.val_mk, rotate_zero]", "annotated_tactic": ["rw [\u2190 <a>get_mk_zero</a> h, <a>get_cyclicPermutations</a>, <a>Fin.val_mk</a>, <a>rotate_zero</a>]", [{"full_name": "List.get_mk_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [236, 9], "def_end_pos": [236, 20]}, {"full_name": "List.get_cyclicPermutations", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [606, 9], "def_end_pos": [606, 31]}, {"full_name": "Fin.val_mk", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}, {"full_name": "List.rotate_zero", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}]], "state_before": "\u03b1 : Type u\nl\u271d l' l : List \u03b1\nh : 0 < l.cyclicPermutations.length\n\u22a2 l.cyclicPermutations.head \u22ef = l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Nonneg/Floor.lean", "full_name": "Nonneg.nat_ceil_coe", "start": [44, 1], "end": [46, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_natCast", "start": [1307, 1], "end": [1307, 94], "traced_tactics": [{"tactic": "induction n <;> simp [*]", "annotated_tactic": ["induction n <;> simp [*]", []], "state_before": "\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 lift.{u, v} \u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Trace.lean", "full_name": "Algebra.trace_prod", "start": [179, 1], "end": [181, 64], "traced_tactics": [{"tactic": "rw [coprod_apply, trace_prod_apply]", "annotated_tactic": ["rw [<a>coprod_apply</a>, <a>trace_prod_apply</a>]", [{"full_name": "LinearMap.coprod_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [223, 9], "def_end_pos": [223, 21]}, {"full_name": "Algebra.trace_prod_apply", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [169, 9], "def_end_pos": [169, 25]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\np : S \u00d7 T\n\u22a2 (trace R (S \u00d7 T)) p = ((trace R S).coprod (trace R T)) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_congr_Prop", "start": [168, 1], "end": [170, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.NeBot.of_vsub_right", "start": [1154, 1], "end": [1155, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Ici_subset_Ioi", "start": [427, 1], "end": [428, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.typevecCasesCons\u2082_appendFun", "start": [371, 1], "end": [376, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "full_name": "GeneralizedContinuedFraction.of_s_tail", "start": [319, 1], "end": [320, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.pred_eq_iff_covBy", "start": [880, 1], "end": [883, 39], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred b = a \u2192 a \u22d6 b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\nb : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred b \u22d6 b"}, {"tactic": "exact pred_covBy _", "annotated_tactic": ["exact <a>pred_covBy</a> _", [{"full_name": "Order.pred_covBy", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [756, 9], "def_end_pos": [756, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\nb : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred b \u22d6 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WellFoundedSet.lean", "full_name": "Set.wellFoundedOn_iff", "start": [77, 1], "end": [89, 59], "traced_tactics": [{"tactic": "have f : RelEmbedding (fun (a : s) (b : s) => r a b) fun a b : \u03b1 => r a b \u2227 a \u2208 s \u2227 b \u2208 s :=\n  \u27e8\u27e8(\u2191), Subtype.coe_injective\u27e9, by simp\u27e9", "annotated_tactic": ["have f : <a>RelEmbedding</a> (fun (a : s) (b : s) => r a b) fun a b : \u03b1 => r a b \u2227 a \u2208 s \u2227 b \u2208 s :=\n    \u27e8\u27e8(\u2191), <a>Subtype.coe_injective</a>\u27e9, by simp\u27e9", [{"full_name": "RelEmbedding", "def_path": "Mathlib/Order/RelIso/Basic.lean", "def_pos": [197, 11], "def_end_pos": [197, 23]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\n\u22a2 s.WellFoundedOn r \u2194 WellFounded fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\n\u22a2 s.WellFoundedOn r \u2194 WellFounded fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s"}, {"tactic": "refine \u27e8fun h => ?_, f.wellFounded\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, f.wellFounded\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\n\u22a2 s.WellFoundedOn r \u2194 WellFounded fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\n\u22a2 WellFounded fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s"}, {"tactic": "rw [WellFounded.wellFounded_iff_has_min]", "annotated_tactic": ["rw [<a>WellFounded.wellFounded_iff_has_min</a>]", [{"full_name": "WellFounded.wellFounded_iff_has_min", "def_path": "Mathlib/Order/WellFounded.lean", "def_pos": [82, 9], "def_end_pos": [82, 32]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\n\u22a2 WellFounded fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\n\u22a2 \u2200 (s_1 : Set \u03b1), s_1.Nonempty \u2192 \u2203 m \u2208 s_1, \u2200 x \u2208 s_1, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\n\u22a2 \u2200 (s_1 : Set \u03b1), s_1.Nonempty \u2192 \u2203 m \u2208 s_1, \u2200 x \u2208 s_1, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "by_cases hst : (s \u2229 t).Nonempty", "annotated_tactic": ["by_cases hst : (s \u2229 t).<a>Nonempty</a>", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Init/Set.lean", "def_pos": [218, 15], "def_end_pos": [218, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (s \u2229 t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : \u00ac(s \u2229 t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns t : Set \u03b1\nx y : \u03b1\n\u22a2 \u2200 {a b : \u2191s},\n    r ({ toFun := Subtype.val, inj' := \u22ef } a) ({ toFun := Subtype.val, inj' := \u22ef } b) \u2227\n        { toFun := Subtype.val, inj' := \u22ef } a \u2208 s \u2227 { toFun := Subtype.val, inj' := \u22ef } b \u2208 s \u2194\n      r \u2191a \u2191b", "state_after": "no goals"}, {"tactic": "rw [\u2190 Subtype.preimage_coe_nonempty] at hst", "annotated_tactic": ["rw [\u2190 <a>Subtype.preimage_coe_nonempty</a>] at hst", [{"full_name": "Subtype.preimage_coe_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1458, 9], "def_end_pos": [1458, 30]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (s \u2229 t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (Subtype.val \u207b\u00b9' t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "rcases h.has_min (Subtype.val \u207b\u00b9' t) hst with \u27e8\u27e8m, ms\u27e9, mt, hm\u27e9", "annotated_tactic": ["rcases h.has_min (<a>Subtype.val</a> \u207b\u00b9' t) hst with \u27e8\u27e8m, ms\u27e9, mt, hm\u27e9", [{"full_name": "Subtype.val", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [587, 3], "def_end_pos": [587, 6]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (Subtype.val \u207b\u00b9' t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "case pos.intro.mk.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (Subtype.val \u207b\u00b9' t).Nonempty\nm : \u03b1\nms : m \u2208 s\nmt : \u27e8m, ms\u27e9 \u2208 Subtype.val \u207b\u00b9' t\nhm : \u2200 x \u2208 Subtype.val \u207b\u00b9' t, \u00acr \u2191x \u2191\u27e8m, ms\u27e9\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "exact \u27e8m, mt, fun x xt \u27e8xm, xs, _\u27e9 => hm \u27e8x, xs\u27e9 xt xm\u27e9", "annotated_tactic": ["exact \u27e8m, mt, fun x xt \u27e8xm, xs, _\u27e9 => hm \u27e8x, xs\u27e9 xt xm\u27e9", []], "state_before": "case pos.intro.mk.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : (Subtype.val \u207b\u00b9' t).Nonempty\nm : \u03b1\nms : m \u2208 s\nmt : \u27e8m, ms\u27e9 \u2208 Subtype.val \u207b\u00b9' t\nhm : \u2200 x \u2208 Subtype.val \u207b\u00b9' t, \u00acr \u2191x \u2191\u27e8m, ms\u27e9\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "no goals"}, {"tactic": "rcases ht with \u27e8m, mt\u27e9", "annotated_tactic": ["rcases ht with \u27e8m, mt\u27e9", []], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nht : t.Nonempty\nhst : \u00ac(s \u2229 t).Nonempty\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "case neg.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nhst : \u00ac(s \u2229 t).Nonempty\nm : \u03b1\nmt : m \u2208 t\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)"}, {"tactic": "exact \u27e8m, mt, fun x _ \u27e8_, _, ms\u27e9 => hst \u27e8m, \u27e8ms, mt\u27e9\u27e9\u27e9", "annotated_tactic": ["exact \u27e8m, mt, fun x _ \u27e8_, _, ms\u27e9 => hst \u27e8m, \u27e8ms, mt\u27e9\u27e9\u27e9", []], "state_before": "case neg.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr r' : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d : \u03b2 \u2192 \u03b1\ns t\u271d : Set \u03b1\nx y : \u03b1\nf : (fun a b => r \u2191a \u2191b) \u21aar fun a b => r a b \u2227 a \u2208 s \u2227 b \u2208 s\nh : s.WellFoundedOn r\nt : Set \u03b1\nhst : \u00ac(s \u2229 t).Nonempty\nm : \u03b1\nmt : m \u2208 t\n\u22a2 \u2203 m \u2208 t, \u2200 x \u2208 t, \u00ac(r x m \u2227 x \u2208 s \u2227 m \u2208 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean", "full_name": "ExteriorAlgebra.\u03b9_range_disjoint_one", "start": [260, 1], "end": [266, 30], "traced_tactics": [{"tactic": "rw [Submodule.disjoint_def]", "annotated_tactic": ["rw [<a>Submodule.disjoint_def</a>]", [{"full_name": "Submodule.disjoint_def", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [374, 9], "def_end_pos": [374, 21]}]], "state_before": "R : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 Disjoint (LinearMap.range (\u03b9 R)) 1", "state_after": "R : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 \u2200 x \u2208 LinearMap.range (\u03b9 R), x \u2208 1 \u2192 x = 0"}, {"tactic": "rintro _ \u27e8x, hx\u27e9 \u27e8r, rfl : algebraMap R (ExteriorAlgebra R M) r = _\u27e9", "annotated_tactic": ["rintro _ \u27e8x, hx\u27e9 \u27e8r, rfl : <a>algebraMap</a> R (<a>ExteriorAlgebra</a> R M) r = _\u27e9", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "ExteriorAlgebra", "def_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean", "def_pos": [57, 8], "def_end_pos": [57, 23]}]], "state_before": "R : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 \u2200 x \u2208 LinearMap.range (\u03b9 R), x \u2208 1 \u2192 x = 0", "state_after": "case intro.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nhx : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\n\u22a2 (algebraMap R (ExteriorAlgebra R M)) r = 0"}, {"tactic": "rw [\u03b9_eq_algebraMap_iff x] at hx", "annotated_tactic": ["rw [<a>\u03b9_eq_algebraMap_iff</a> x] at hx", [{"full_name": "ExteriorAlgebra.\u03b9_eq_algebraMap_iff", "def_path": "Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 28]}]], "state_before": "case intro.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nhx : (\u03b9 R) x = (algebraMap R (ExteriorAlgebra R M)) r\n\u22a2 (algebraMap R (ExteriorAlgebra R M)) r = 0", "state_after": "case intro.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nhx : x = 0 \u2227 r = 0\n\u22a2 (algebraMap R (ExteriorAlgebra R M)) r = 0"}, {"tactic": "rw [hx.2, RingHom.map_zero]", "annotated_tactic": ["rw [hx.2, <a>RingHom.map_zero</a>]", [{"full_name": "RingHom.map_zero", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [546, 19], "def_end_pos": [546, 27]}]], "state_before": "case intro.intro\nR : Type u1\ninst\u271d\u2074 : CommRing R\nM : Type u2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : M\nr : R\nhx : x = 0 \u2227 r = 0\n\u22a2 (algebraMap R (ExteriorAlgebra R M)) r = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "full_name": "MonoidHom.map_zpow'", "start": [1026, 11], "end": [1028, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Basic.lean", "full_name": "lt_trans'", "start": [76, 1], "end": [77, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "MulOpposite.unop_list_prod", "start": [848, 1], "end": [850, 26], "traced_tactics": [{"tactic": "rw [\u2190 op_inj, op_unop, MulOpposite.op_list_prod, map_reverse, map_map, reverse_reverse,\n  op_comp_unop, map_id]", "annotated_tactic": ["rw [\u2190 <a>op_inj</a>, <a>op_unop</a>, <a>MulOpposite.op_list_prod</a>, <a>map_reverse</a>, <a>map_map</a>, <a>reverse_reverse</a>,\n    <a>op_comp_unop</a>, <a>map_id</a>]", [{"full_name": "MulOpposite.op_inj", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [167, 9], "def_end_pos": [167, 15]}, {"full_name": "MulOpposite.op_unop", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}, {"full_name": "MulOpposite.op_list_prod", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [841, 7], "def_end_pos": [841, 19]}, {"full_name": "List.map_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1480, 17], "def_end_pos": [1480, 28]}, {"full_name": "List.map_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [842, 17], "def_end_pos": [842, 24]}, {"full_name": "List.reverse_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1469, 17], "def_end_pos": [1469, 32]}, {"full_name": "MulOpposite.op_comp_unop", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [102, 9], "def_end_pos": [102, 21]}, {"full_name": "List.map_id", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [716, 17], "def_end_pos": [716, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d : Monoid M\nl : List M\u1d50\u1d52\u1d56\n\u22a2 unop l.prod = (map unop l).reverse.prod", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_add_iff_integrable_left", "start": [706, 1], "end": [709, 56], "traced_tactics": [{"tactic": "rw [add_comm, integrable_add_iff_integrable_right hf]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>integrable_add_iff_integrable_right</a> hf]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "MeasureTheory.integrable_add_iff_integrable_right", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [701, 7], "def_end_pos": [701, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf g : \u03b1 \u2192 \u03b2\nhf : Integrable f \u03bc\n\u22a2 Integrable (g + f) \u03bc \u2194 Integrable g \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/LightProfinite/Limits.lean", "full_name": "LightProfinite.finiteCoproduct.\u03b9_desc", "start": [164, 1], "end": [166, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Inverse.lean", "full_name": "PowerSeries.ker_coeff_eq_max_ideal", "start": [378, 1], "end": [380, 86], "traced_tactics": [{"tactic": "rw [RingHom.mem_ker, maximalIdeal_eq_span_X, Ideal.mem_span_singleton, X_dvd_iff]", "annotated_tactic": ["rw [<a>RingHom.mem_ker</a>, <a>maximalIdeal_eq_span_X</a>, <a>Ideal.mem_span_singleton</a>, <a>X_dvd_iff</a>]", [{"full_name": "RingHom.mem_ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [605, 9], "def_end_pos": [605, 16]}, {"full_name": "PowerSeries.maximalIdeal_eq_span_X", "def_path": "Mathlib/RingTheory/PowerSeries/Inverse.lean", "def_pos": [335, 9], "def_end_pos": [335, 31]}, {"full_name": "Ideal.mem_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 27]}, {"full_name": "PowerSeries.X_dvd_iff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [571, 9], "def_end_pos": [571, 18]}]], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nx\u271d : k\u27e6X\u27e7\n\u22a2 x\u271d \u2208 RingHom.ker (constantCoeff k) \u2194 x\u271d \u2208 maximalIdeal k\u27e6X\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.card_classes_ker_le", "start": [78, 1], "end": [81, 93], "traced_tactics": [{"tactic": "classical exact\n    le_trans (Set.card_le_card (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)", "annotated_tactic": ["classical exact\n      <a>le_trans</a> (<a>Set.card_le_card</a> (<a>classes_ker_subset_fiber_set</a> f)) (<a>Fintype.card_range_le</a> _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Set.card_le_card", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1299, 9], "def_end_pos": [1299, 21]}, {"full_name": "Setoid.classes_ker_subset_fiber_set", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [67, 9], "def_end_pos": [67, 37]}, {"full_name": "Fintype.card_range_le", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [509, 9], "def_end_pos": [509, 22]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u2191(ker f).classes\n\u22a2 Fintype.card \u2191(ker f).classes \u2264 Fintype.card \u03b2", "state_after": "no goals"}, {"tactic": "exact\nle_trans (Set.card_le_card (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)", "annotated_tactic": ["exact\n      <a>le_trans</a> (<a>Set.card_le_card</a> (<a>classes_ker_subset_fiber_set</a> f)) (<a>Fintype.card_range_le</a> _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Set.card_le_card", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1299, 9], "def_end_pos": [1299, 21]}, {"full_name": "Setoid.classes_ker_subset_fiber_set", "def_path": "Mathlib/Data/Setoid/Partition.lean", "def_pos": [67, 9], "def_end_pos": [67, 37]}, {"full_name": "Fintype.card_range_le", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [509, 9], "def_end_pos": [509, 22]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u2191(ker f).classes\n\u22a2 Fintype.card \u2191(ker f).classes \u2264 Fintype.card \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.smul_kronecker", "start": [312, 1], "end": [314, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Basic.lean", "full_name": "FiberPrebundle.mem_pretrivializationAt_source", "start": [832, 1], "end": [835, 41], "traced_tactics": [{"tactic": "simp only [(a.pretrivializationAt b).source_eq, mem_preimage, TotalSpace.proj]", "annotated_tactic": ["simp only [(a.pretrivializationAt b).<a>source_eq</a>, <a>mem_preimage</a>, <a>TotalSpace.proj</a>]", [{"full_name": "Pretrivialization.source_eq", "def_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "def_pos": [68, 3], "def_end_pos": [68, 12]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "Bundle.TotalSpace.proj", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [55, 3], "def_end_pos": [55, 7]}]], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n\u22a2 { proj := b, snd := x } \u2208 (a.pretrivializationAt b).source", "state_after": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n\u22a2 b \u2208 (a.pretrivializationAt b).baseSet"}, {"tactic": "exact a.mem_base_pretrivializationAt b", "annotated_tactic": ["exact a.mem_base_pretrivializationAt b", []], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n\u22a2 b \u2208 (a.pretrivializationAt b).baseSet", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_singleton", "start": [347, 1], "end": [352, 23], "traced_tactics": [{"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\na : \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2191{a}).Pairwise fun a b => Commute (f a) (f b)", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\na : \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 {a}.Pairwise fun a b => Commute (f a) (f b)"}, {"tactic": "exact Set.pairwise_singleton _ _", "annotated_tactic": ["exact <a>Set.pairwise_singleton</a> _ _", [{"full_name": "Set.pairwise_singleton", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 27]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\na : \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 {a}.Pairwise fun a b => Commute (f a) (f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/NFA.lean", "full_name": "NFA.toDFA_correct", "start": [119, 1], "end": [122, 57], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\n\u22a2 M.toDFA.accepts = M.accepts", "state_after": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 M.toDFA.accepts \u2194 x \u2208 M.accepts"}, {"tactic": "rw [mem_accepts, DFA.mem_accepts]", "annotated_tactic": ["rw [<a>mem_accepts</a>, <a>DFA.mem_accepts</a>]", [{"full_name": "NFA.mem_accepts", "def_path": "Mathlib/Computability/NFA.lean", "def_pos": [107, 9], "def_end_pos": [107, 20]}, {"full_name": "DFA.mem_accepts", "def_path": "Mathlib/Computability/DFA.lean", "def_pos": [97, 9], "def_end_pos": [97, 20]}]], "state_before": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 x \u2208 M.toDFA.accepts \u2194 x \u2208 M.accepts", "state_after": "case h\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 M.toDFA.evalFrom M.toDFA.start x \u2208 M.toDFA.accept \u2194 \u2203 S \u2208 M.accept, S \u2208 M.evalFrom M.start x"}, {"tactic": "exact fun \u27e8w, h2, h3\u27e9 => \u27e8w, h3, h2\u27e9", "annotated_tactic": ["exact fun \u27e8w, h2, h3\u27e9 => \u27e8w, h3, h2\u27e9", []], "state_before": "case h.mpr\n\u03b1 : Type u\n\u03c3 \u03c3' : Type v\nM : NFA \u03b1 \u03c3\nx : List \u03b1\n\u22a2 (\u2203 S \u2208 M.accept, S \u2208 M.evalFrom M.start x) \u2192 M.toDFA.evalFrom M.toDFA.start x \u2208 M.toDFA.accept", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "full_name": "FractionalIdeal.count_mul", "start": [328, 1], "end": [349, 7], "traced_tactics": [{"tactic": "have hv : Irreducible (Associates.mk v.asIdeal) := by apply v.associates_irreducible", "annotated_tactic": ["have hv : <a>Irreducible</a> (<a>Associates.mk</a> v.asIdeal) := by apply v.associates_irreducible", [{"full_name": "Irreducible", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [191, 11], "def_end_pos": [191, 22]}, {"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [834, 18], "def_end_pos": [834, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "obtain \u27e8a, J, ha, haJ\u27e9 := exists_eq_spanSingleton_mul I", "annotated_tactic": ["obtain \u27e8a, J, ha, haJ\u27e9 := <a>exists_eq_spanSingleton_mul</a> I", [{"full_name": "FractionalIdeal.exists_eq_spanSingleton_mul", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [828, 9], "def_end_pos": [828, 36]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "have ha_ne_zero : Associates.mk (Ideal.span {a} : Ideal R) \u2260 0 := by\n  rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha", "annotated_tactic": ["have ha_ne_zero : <a>Associates.mk</a> (<a>Ideal.span</a> {a} : <a>Ideal</a> R) \u2260 0 := by\n    rw [<a>ne_eq</a>, <a>Associates.mk_eq_zero</a>, <a>Ideal.zero_eq_bot</a>, <a>Ideal.span_singleton_eq_bot</a>]; exact ha", [{"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [834, 18], "def_end_pos": [834, 20]}, {"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Associates.mk_eq_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 19]}, {"full_name": "Ideal.zero_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [408, 9], "def_end_pos": [408, 20]}, {"full_name": "Ideal.span_singleton_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}]], "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "have hJ_ne_zero : Associates.mk J \u2260 0 := Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI haJ)", "annotated_tactic": ["have hJ_ne_zero : <a>Associates.mk</a> J \u2260 0 := Associates.mk_ne_zero.mpr (<a>ideal_factor_ne_zero</a> hI haJ)", [{"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [834, 18], "def_end_pos": [834, 20]}, {"full_name": "FractionalIdeal.ideal_factor_ne_zero", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [854, 9], "def_end_pos": [854, 29]}]], "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "obtain \u27e8a', J', ha', haJ'\u27e9 := exists_eq_spanSingleton_mul I'", "annotated_tactic": ["obtain \u27e8a', J', ha', haJ'\u27e9 := <a>exists_eq_spanSingleton_mul</a> I'", [{"full_name": "FractionalIdeal.exists_eq_spanSingleton_mul", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [828, 9], "def_end_pos": [828, 36]}]], "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "have ha'_ne_zero : Associates.mk (Ideal.span {a'} : Ideal R) \u2260 0 := by\n  rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha'", "annotated_tactic": ["have ha'_ne_zero : <a>Associates.mk</a> (<a>Ideal.span</a> {a'} : <a>Ideal</a> R) \u2260 0 := by\n    rw [<a>ne_eq</a>, <a>Associates.mk_eq_zero</a>, <a>Ideal.zero_eq_bot</a>, <a>Ideal.span_singleton_eq_bot</a>]; exact ha'", [{"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [834, 18], "def_end_pos": [834, 20]}, {"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Associates.mk_eq_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 19]}, {"full_name": "Ideal.zero_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [408, 9], "def_end_pos": [408, 20]}, {"full_name": "Ideal.span_singleton_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "have hJ'_ne_zero : Associates.mk J' \u2260 0 :=\n  Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI' haJ')", "annotated_tactic": ["have hJ'_ne_zero : <a>Associates.mk</a> J' \u2260 0 :=\n    Associates.mk_ne_zero.mpr (<a>ideal_factor_ne_zero</a> hI' haJ')", [{"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [834, 18], "def_end_pos": [834, 20]}, {"full_name": "FractionalIdeal.ideal_factor_ne_zero", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [854, 9], "def_end_pos": [854, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "have h_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J') := by\n  rw [haJ, haJ', mul_assoc, mul_comm (J : FractionalIdeal R\u2070 K), mul_assoc, \u2190 mul_assoc,\n    spanSingleton_mul_spanSingleton, coeIdeal_mul, RingHom.map_mul, mul_inv,\n    mul_comm (J : FractionalIdeal R\u2070 K)]", "annotated_tactic": ["have h_prod : I * I' = <a>spanSingleton</a> R\u2070 ((<a>algebraMap</a> R K) (a * a'))\u207b\u00b9 * \u2191(J * J') := by\n    rw [haJ, haJ', <a>mul_assoc</a>, <a>mul_comm</a> (J : <a>FractionalIdeal</a> R\u2070 K), <a>mul_assoc</a>, \u2190 <a>mul_assoc</a>,\n      <a>spanSingleton_mul_spanSingleton</a>, <a>coeIdeal_mul</a>, <a>RingHom.map_mul</a>, <a>mul_inv</a>,\n      <a>mul_comm</a> (J : <a>FractionalIdeal</a> R\u2070 K)]", [{"full_name": "FractionalIdeal.spanSingleton", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [623, 17], "def_end_pos": [623, 30]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "FractionalIdeal.spanSingleton_mul_spanSingleton", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [708, 9], "def_end_pos": [708, 40]}, {"full_name": "FractionalIdeal.coeIdeal_mul", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 21]}, {"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [561, 19], "def_end_pos": [561, 26]}, {"full_name": "mul_inv", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 count K v (I * I') = count K v I + count K v I'"}, {"tactic": "rw [count_well_defined K v hI haJ, count_well_defined K v hI' haJ',\n  count_well_defined K v (mul_ne_zero hI hI') h_prod, \u2190 Associates.mk_mul_mk,\n  Associates.count_mul hJ_ne_zero hJ'_ne_zero hv, \u2190 Ideal.span_singleton_mul_span_singleton,\n  \u2190 Associates.mk_mul_mk, Associates.count_mul ha_ne_zero ha'_ne_zero hv]", "annotated_tactic": ["rw [<a>count_well_defined</a> K v hI haJ, <a>count_well_defined</a> K v hI' haJ',\n    <a>count_well_defined</a> K v (<a>mul_ne_zero</a> hI hI') h_prod, \u2190 <a>Associates.mk_mul_mk</a>,\n    <a>Associates.count_mul</a> hJ_ne_zero hJ'_ne_zero hv, \u2190 <a>Ideal.span_singleton_mul_span_singleton</a>,\n    \u2190 <a>Associates.mk_mul_mk</a>, <a>Associates.count_mul</a> ha_ne_zero ha'_ne_zero hv]", [{"full_name": "FractionalIdeal.count_well_defined", "def_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "def_pos": [293, 9], "def_end_pos": [293, 27]}, {"full_name": "FractionalIdeal.count_well_defined", "def_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "def_pos": [293, 9], "def_end_pos": [293, 27]}, {"full_name": "FractionalIdeal.count_well_defined", "def_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "def_pos": [293, 9], "def_end_pos": [293, 27]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "Associates.mk_mul_mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "Associates.count_mul", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 18]}, {"full_name": "Ideal.span_singleton_mul_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [514, 9], "def_end_pos": [514, 42]}, {"full_name": "Associates.mk_mul_mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [913, 9], "def_end_pos": [913, 18]}, {"full_name": "Associates.count_mul", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 count K v (I * I') = count K v I + count K v I'", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors +\n          (Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n      \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors +\n          (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors) =\n    \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors +\n          (Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n      \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors +\n          (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors) =\n    \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors))", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) +\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors)) =\n    \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\nh_prod : I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')\n\u22a2 \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) +\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors)) =\n    \u2191((Associates.mk v.asIdeal).count (Associates.mk J).factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) +\n      (\u2191((Associates.mk v.asIdeal).count (Associates.mk J').factors) -\n        \u2191((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a'})).factors))", "state_after": "no goals"}, {"tactic": "apply v.associates_irreducible", "annotated_tactic": ["apply v.associates_irreducible", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\n\u22a2 Irreducible (Associates.mk v.asIdeal)", "state_after": "no goals"}, {"tactic": "rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]", "annotated_tactic": ["rw [<a>ne_eq</a>, <a>Associates.mk_eq_zero</a>, <a>Ideal.zero_eq_bot</a>, <a>Ideal.span_singleton_eq_bot</a>]", [{"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Associates.mk_eq_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 19]}, {"full_name": "Ideal.zero_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [408, 9], "def_end_pos": [408, 20]}, {"full_name": "Ideal.span_singleton_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\n\u22a2 Associates.mk (Ideal.span {a}) \u2260 0", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\n\u22a2 \u00aca = 0"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\n\u22a2 \u00aca = 0", "state_after": "no goals"}, {"tactic": "rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]", "annotated_tactic": ["rw [<a>ne_eq</a>, <a>Associates.mk_eq_zero</a>, <a>Ideal.zero_eq_bot</a>, <a>Ideal.span_singleton_eq_bot</a>]", [{"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Associates.mk_eq_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 19]}, {"full_name": "Ideal.zero_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [408, 9], "def_end_pos": [408, 20]}, {"full_name": "Ideal.span_singleton_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\n\u22a2 Associates.mk (Ideal.span {a'}) \u2260 0", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\n\u22a2 \u00aca' = 0"}, {"tactic": "exact ha'", "annotated_tactic": ["exact ha'", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\n\u22a2 \u00aca' = 0", "state_after": "no goals"}, {"tactic": "rw [haJ, haJ', mul_assoc, mul_comm (J : FractionalIdeal R\u2070 K), mul_assoc, \u2190 mul_assoc,\n  spanSingleton_mul_spanSingleton, coeIdeal_mul, RingHom.map_mul, mul_inv,\n  mul_comm (J : FractionalIdeal R\u2070 K)]", "annotated_tactic": ["rw [haJ, haJ', <a>mul_assoc</a>, <a>mul_comm</a> (J : <a>FractionalIdeal</a> R\u2070 K), <a>mul_assoc</a>, \u2190 <a>mul_assoc</a>,\n      <a>spanSingleton_mul_spanSingleton</a>, <a>coeIdeal_mul</a>, <a>RingHom.map_mul</a>, <a>mul_inv</a>,\n      <a>mul_comm</a> (J : <a>FractionalIdeal</a> R\u2070 K)]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "FractionalIdeal.spanSingleton_mul_spanSingleton", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [708, 9], "def_end_pos": [708, 40]}, {"full_name": "FractionalIdeal.coeIdeal_mul", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 21]}, {"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [561, 19], "def_end_pos": [561, 26]}, {"full_name": "mul_inv", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R\u2070 K\nhI : I \u2260 0\nhI' : I' \u2260 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a \u2260 0\nhaJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J\nha_ne_zero : Associates.mk (Ideal.span {a}) \u2260 0\nhJ_ne_zero : Associates.mk J \u2260 0\na' : R\nJ' : Ideal R\nha' : a' \u2260 0\nhaJ' : I' = spanSingleton R\u2070 ((algebraMap R K) a')\u207b\u00b9 * \u2191J'\nha'_ne_zero : Associates.mk (Ideal.span {a'}) \u2260 0\nhJ'_ne_zero : Associates.mk J' \u2260 0\n\u22a2 I * I' = spanSingleton R\u2070 ((algebraMap R K) (a * a'))\u207b\u00b9 * \u2191(J * J')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Booleanisation.lean", "full_name": "Booleanisation.lift_le_lift", "start": [132, 1], "end": [132, 93], "traced_tactics": [{"tactic": "rintro \u27e8_\u27e9", "annotated_tactic": ["rintro \u27e8_\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : Booleanisation \u03b1\na b : \u03b1\n\u22a2 lift a \u2264 lift b \u2192 a \u2264 b", "state_after": "case lift\n\u03b1 : Type u_1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : Booleanisation \u03b1\na b : \u03b1\na\u271d : a \u2264 b\n\u22a2 a \u2264 b"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case lift\n\u03b1 : Type u_1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : Booleanisation \u03b1\na b : \u03b1\na\u271d : a \u2264 b\n\u22a2 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "PowerBasis.equivAdjoinSimple_symm_aeval", "start": [1527, 1], "end": [1529, 91], "traced_tactics": [{"tactic": "rw [equivAdjoinSimple, equivOfMinpoly_symm, equivOfMinpoly_aeval, adjoin.powerBasis_gen]", "annotated_tactic": ["rw [<a>equivAdjoinSimple</a>, <a>equivOfMinpoly_symm</a>, <a>equivOfMinpoly_aeval</a>, <a>adjoin.powerBasis_gen</a>]", [{"full_name": "PowerBasis.equivAdjoinSimple", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [1509, 19], "def_end_pos": [1509, 36]}, {"full_name": "PowerBasis.equivOfMinpoly_symm", "def_path": "Mathlib/RingTheory/PowerBasis.lean", "def_pos": [412, 9], "def_end_pos": [412, 28]}, {"full_name": "PowerBasis.equivOfMinpoly_aeval", "def_path": "Mathlib/RingTheory/PowerBasis.lean", "def_pos": [399, 9], "def_end_pos": [399, 29]}, {"full_name": "IntermediateField.adjoin.powerBasis_gen", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [1131, 3], "def_end_pos": [1131, 8]}]], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\npb : PowerBasis K L\nf : K[X]\n\u22a2 pb.equivAdjoinSimple.symm ((aeval pb.gen) f) = (aeval (AdjoinSimple.gen K pb.gen)) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tsum_lt_tsum", "start": [1320, 1], "end": [1322, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.one_pos'", "start": [678, 1], "end": [678, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.zero_add", "start": [458, 11], "end": [459, 50], "traced_tactics": [{"tactic": "simp [ext_iff, add_def, mod_eq_of_lt (is_lt k)]", "annotated_tactic": ["simp [<a>ext_iff</a>, <a>add_def</a>, <a>mod_eq_of_lt</a> (<a>is_lt</a> k)]", [{"full_name": "Fin.ext_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [42, 9], "def_end_pos": [42, 16]}, {"full_name": "Fin.add_def", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [202, 9], "def_end_pos": [202, 16]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [131, 9], "def_end_pos": [131, 21]}, {"full_name": "Fin.is_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [31, 17], "def_end_pos": [31, 22]}]], "state_before": "n m : \u2115\ninst\u271d : NeZero n\nk : Fin n\n\u22a2 0 + k = k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "full_name": "Function.isFixedPt_id", "start": [44, 1], "end": [45, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactification/OnePoint.lean", "full_name": "OnePoint.nhdsWithin_coe", "start": [301, 1], "end": [302, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Opposite.lean", "full_name": "CategoryTheory.unmop_comp", "start": [125, 1], "end": [126, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.Prepartition.distortion_biUnionTagged", "start": [434, 1], "end": [437, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/PrimeCounting.lean", "full_name": "Nat.monotone_primeCounting'", "start": [58, 1], "end": [59, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "full_name": "ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus", "start": [166, 1], "end": [168, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.get_attach", "start": [3563, 1], "end": [3564, 69], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 L : List \u03b1\ni : Fin L.attach.length\n\u22a2 \u2191(L.attach.get i) = L.get \u27e8\u2191i, \u22ef\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/RingDivision.lean", "full_name": "Polynomial.associated_of_dvd_of_natDegree_le", "start": [644, 1], "end": [647, 79], "traced_tactics": [{"tactic": "rwa [\u2190 leadingCoeff_ne_zero, \u2190 isUnit_iff_ne_zero] at hq", "annotated_tactic": ["rwa [\u2190 <a>leadingCoeff_ne_zero</a>, \u2190 <a>isUnit_iff_ne_zero</a>] at hq", [{"full_name": "Polynomial.leadingCoeff_ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [695, 9], "def_end_pos": [695, 29]}, {"full_name": "isUnit_iff_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 27]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\np\u271d q\u271d : R[X]\nK : Type u_1\ninst\u271d : Field K\np q : K[X]\nhpq : p \u2223 q\nhq : q \u2260 0\nh\u2081 : q.natDegree \u2264 p.natDegree\n\u22a2 IsUnit q.leadingCoeff", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Kleene.lean", "full_name": "kstar_mul_le", "start": [224, 1], "end": [225, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Configuration.lean", "full_name": "Configuration.HasLines.exists_bijective_of_card_eq", "start": [256, 1], "end": [263, 95], "traced_tactics": [{"tactic": "classical\n  obtain \u27e8f, hf1, hf2\u27e9 := Nondegenerate.exists_injective_of_card_le (ge_of_eq h)\n  have hf3 := (Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf1, h.symm\u27e9\n  exact \u27e8f, hf3, fun l \u21a6 (sum_eq_sum_iff_of_le fun l _ \u21a6 pointCount_le_lineCount (hf2 l)).1\n        ((hf3.sum_comp _).trans (sum_lineCount_eq_sum_pointCount P L)).symm _ <| mem_univ _\u27e9", "annotated_tactic": ["classical\n    obtain \u27e8f, hf1, hf2\u27e9 := <a>Nondegenerate.exists_injective_of_card_le</a> (<a>ge_of_eq</a> h)\n    have hf3 := (<a>Fintype.bijective_iff_injective_and_card</a> f).<a>mpr</a> \u27e8hf1, h.symm\u27e9\n    exact \u27e8f, hf3, fun l \u21a6 (<a>sum_eq_sum_iff_of_le</a> fun l _ \u21a6 <a>pointCount_le_lineCount</a> (hf2 l)).1\n          ((hf3.sum_comp _).<a>trans</a> (<a>sum_lineCount_eq_sum_pointCount</a> P L)).<a>symm</a> _ <| <a>mem_univ</a> _\u27e9", [{"full_name": "Configuration.Nondegenerate.exists_injective_of_card_le", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [125, 9], "def_end_pos": [125, 50]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 17]}, {"full_name": "Fintype.bijective_iff_injective_and_card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [693, 9], "def_end_pos": [693, 41]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "Finset.sum_eq_sum_iff_of_le", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [530, 3], "def_end_pos": [530, 14]}, {"full_name": "Configuration.HasLines.pointCount_le_lineCount", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [200, 9], "def_end_pos": [200, 41]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}, {"full_name": "Configuration.sum_lineCount_eq_sum_pointCount", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "P : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)", "state_after": "no goals"}, {"tactic": "obtain \u27e8f, hf1, hf2\u27e9 := Nondegenerate.exists_injective_of_card_le (ge_of_eq h)", "annotated_tactic": ["obtain \u27e8f, hf1, hf2\u27e9 := <a>Nondegenerate.exists_injective_of_card_le</a> (<a>ge_of_eq</a> h)", [{"full_name": "Configuration.Nondegenerate.exists_injective_of_card_le", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [125, 9], "def_end_pos": [125, 50]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 17]}]], "state_before": "P : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)", "state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), f l \u2209 l\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)"}, {"tactic": "have hf3 := (Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf1, h.symm\u27e9", "annotated_tactic": ["have hf3 := (<a>Fintype.bijective_iff_injective_and_card</a> f).<a>mpr</a> \u27e8hf1, h.symm\u27e9", [{"full_name": "Fintype.bijective_iff_injective_and_card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [693, 9], "def_end_pos": [693, 41]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), f l \u2209 l\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)", "state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), f l \u2209 l\nhf3 : Function.Bijective f\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)"}, {"tactic": "exact \u27e8f, hf3, fun l \u21a6 (sum_eq_sum_iff_of_le fun l _ \u21a6 pointCount_le_lineCount (hf2 l)).1\n      ((hf3.sum_comp _).trans (sum_lineCount_eq_sum_pointCount P L)).symm _ <| mem_univ _\u27e9", "annotated_tactic": ["exact \u27e8f, hf3, fun l \u21a6 (<a>sum_eq_sum_iff_of_le</a> fun l _ \u21a6 <a>pointCount_le_lineCount</a> (hf2 l)).1\n          ((hf3.sum_comp _).<a>trans</a> (<a>sum_lineCount_eq_sum_pointCount</a> P L)).<a>symm</a> _ <| <a>mem_univ</a> _\u27e9", [{"full_name": "Finset.sum_eq_sum_iff_of_le", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [530, 3], "def_end_pos": [530, 14]}, {"full_name": "Configuration.HasLines.pointCount_le_lineCount", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [200, 9], "def_end_pos": [200, 41]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}, {"full_name": "Configuration.sum_lineCount_eq_sum_pointCount", "def_path": "Mathlib/Combinatorics/Configuration.lean", "def_pos": [186, 9], "def_end_pos": [186, 40]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), f l \u2209 l\nhf3 : Function.Bijective f\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/NNReal.lean", "full_name": "NNReal.tendsto_coe_atTop", "start": [130, 1], "end": [132, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Icc_subset_Iic_self", "start": [487, 1], "end": [487, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Idempotents/Basic.lean", "full_name": "CategoryTheory.Idempotents.split_iff_of_iso", "start": [143, 1], "end": [154, 8], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 (\u2203 Y i e, i \u226b e = \ud835\udfd9 Y \u2227 e \u226b i = p) \u2194 \u2203 Y' i' e', i' \u226b e' = \ud835\udfd9 Y' \u2227 e' \u226b i' = p'", "state_after": "case mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 (\u2203 Y i e, i \u226b e = \ud835\udfd9 Y \u2227 e \u226b i = p) \u2192 \u2203 Y' i' e', i' \u226b e' = \ud835\udfd9 Y' \u2227 e' \u226b i' = p'\n\ncase mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 (\u2203 Y' i' e', i' \u226b e' = \ud835\udfd9 Y' \u2227 e' \u226b i' = p') \u2192 \u2203 Y i e, i \u226b e = \ud835\udfd9 Y \u2227 e \u226b i = p"}, {"tactic": "exact split_imp_of_iso \u03c6 p p' hpp'", "annotated_tactic": ["exact <a>split_imp_of_iso</a> \u03c6 p p' hpp'", [{"full_name": "CategoryTheory.Idempotents.split_imp_of_iso", "def_path": "Mathlib/CategoryTheory/Idempotents/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 25]}]], "state_before": "case mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 (\u2203 Y i e, i \u226b e = \ud835\udfd9 Y \u2227 e \u226b i = p) \u2192 \u2203 Y' i' e', i' \u226b e' = \ud835\udfd9 Y' \u2227 e' \u226b i' = p'", "state_after": "no goals"}, {"tactic": "apply split_imp_of_iso \u03c6.symm p' p", "annotated_tactic": ["apply <a>split_imp_of_iso</a> \u03c6.symm p' p", [{"full_name": "CategoryTheory.Idempotents.split_imp_of_iso", "def_path": "Mathlib/CategoryTheory/Idempotents/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 25]}]], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 (\u2203 Y' i' e', i' \u226b e' = \ud835\udfd9 Y' \u2227 e' \u226b i' = p') \u2192 \u2203 Y i e, i \u226b e = \ud835\udfd9 Y \u2227 e \u226b i = p", "state_after": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b p"}, {"tactic": "rw [\u2190 comp_id p, \u2190 \u03c6.hom_inv_id]", "annotated_tactic": ["rw [\u2190 <a>comp_id</a> p, \u2190 \u03c6.hom_inv_id]", [{"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}]], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b p", "state_after": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b p \u226b \u03c6.hom \u226b \u03c6.inv"}, {"tactic": "slice_rhs 2 3 => rw [hpp']", "annotated_tactic": ["slice_rhs 2 3 => rw [hpp']", []], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b p \u226b \u03c6.hom \u226b \u03c6.inv", "state_after": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b (\u03c6.hom \u226b p') \u226b \u03c6.inv"}, {"tactic": "slice_rhs 1 2 => erw [\u03c6.inv_hom_id]", "annotated_tactic": ["slice_rhs 1 2 => erw [\u03c6.inv_hom_id]", []], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = \u03c6.symm.hom \u226b (\u03c6.hom \u226b p') \u226b \u03c6.inv", "state_after": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = (\ud835\udfd9 X' \u226b p') \u226b \u03c6.inv"}, {"tactic": "simp only [id_comp]", "annotated_tactic": ["simp only [<a>id_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = (\ud835\udfd9 X' \u226b p') \u226b \u03c6.inv", "state_after": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = p' \u226b \u03c6.inv"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX X' : C\n\u03c6 : X \u2245 X'\np : X \u27f6 X\np' : X' \u27f6 X'\nhpp' : p \u226b \u03c6.hom = \u03c6.hom \u226b p'\n\u22a2 p' \u226b \u03c6.symm.hom = p' \u226b \u03c6.inv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.mk_bounded_subset_le", "start": [1306, 1], "end": [1315, 91], "traced_tactics": [{"tactic": "refine le_trans ?_ (mk_bounded_set_le s c)", "annotated_tactic": ["refine <a>le_trans</a> ?_ (<a>mk_bounded_set_le</a> s c)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Cardinal.mk_bounded_set_le", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 26]}]], "state_before": "\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 #{ t // t \u2286 s \u2227 #\u2191t \u2264 c } \u2264 max #\u2191s \u2135\u2080 ^ c", "state_after": "\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 #{ t // t \u2286 s \u2227 #\u2191t \u2264 c } \u2264 #{ t // #\u2191t \u2264 c }"}, {"tactic": "refine \u27e8Embedding.codRestrict _ ?_ ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Embedding.codRestrict</a> _ ?_ ?_\u27e9", [{"full_name": "Function.Embedding.codRestrict", "def_path": "Mathlib/Logic/Embedding/Set.lean", "def_pos": [63, 5], "def_end_pos": [63, 16]}]], "state_before": "\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 #{ t // t \u2286 s \u2227 #\u2191t \u2264 c } \u2264 #{ t // #\u2191t \u2264 c }", "state_after": "case refine_1\n\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 { t // t \u2286 s \u2227 #\u2191t \u2264 c } \u21aa Set \u2191s\n\ncase refine_2\n\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 \u2200 (a : { t // t \u2286 s \u2227 #\u2191t \u2264 c }), ?refine_1 a \u2208 fun t => Quot.lift ((fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) \u2191t) \u22ef c"}, {"tactic": "rintro \u27e8t, _, h2t\u27e9", "annotated_tactic": ["rintro \u27e8t, _, h2t\u27e9", []], "state_before": "case refine_2\n\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\n\u22a2 \u2200 (a : { t // t \u2286 s \u2227 #\u2191t \u2264 c }),\n    { toFun := fun t => Subtype.val \u207b\u00b9' \u2191t, inj' := \u22ef } a \u2208 fun t => Quot.lift ((fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) \u2191t) \u22ef c", "state_after": "case refine_2.mk.intro\n\u03b1 : Type u\ns : Set \u03b1\nc : Cardinal.{u}\nt : Set \u03b1\nleft\u271d : t \u2286 s\nh2t : #\u2191t \u2264 c\n\u22a2 { toFun := fun t => Subtype.val \u207b\u00b9' \u2191t, inj' := \u22ef } \u27e8t, \u22ef\u27e9 \u2208 fun t => Quot.lift ((fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) \u2191t) \u22ef c"}, {"tactic": "exact (mk_preimage_of_injective _ _ Subtype.val_injective).trans h2t", "annotated_tactic": ["exact (<a>mk_preimage_of_injective</a> _ _ <a>Subtype.val_injective</a>).<a>trans</a> h2t", 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inv_def, transpose_smul, det_transpose, adjugate_transpose]", "annotated_tactic": ["rw [<a>inv_def</a>, <a>inv_def</a>, <a>transpose_smul</a>, <a>det_transpose</a>, <a>adjugate_transpose</a>]", [{"full_name": "Matrix.inv_def", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [217, 9], "def_end_pos": [217, 16]}, {"full_name": "Matrix.inv_def", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [217, 9], "def_end_pos": [217, 16]}, {"full_name": "Matrix.transpose_smul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2149, 9], "def_end_pos": [2149, 23]}, {"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"full_name": "Matrix.adjugate_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [218, 9], "def_end_pos": [218, 27]}]], "state_before": "l : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 A\u207b\u00b9\u1d40 = A\u1d40\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.toLinearEquiv_symm", "start": [587, 1], "end": [588, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.cons_rev", "start": [927, 1], "end": [929, 36], "traced_tactics": [{"tactic": "simpa using insertNth_rev 0 a f i", "annotated_tactic": ["simpa using <a>insertNth_rev</a> 0 a f i", [{"full_name": "Fin.insertNth_rev", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [915, 7], "def_end_pos": [915, 20]}]], "state_before": "m n\u271d : 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"CategoryTheory.MorphismProperty.LeftFraction.map", "def_path": "Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean", "def_pos": [67, 19], "def_end_pos": [67, 22]}]], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} C\ninst\u271d\u00b9 : Category.{u_4, u_2} D\nW : MorphismProperty C\nX Y : C\nf : X \u27f6 Y\nL : C \u2964 D\nhL : W.IsInvertedBy L\ninst\u271d : W.ContainsIdentities\n\u22a2 (ofHom W f).map L hL = L.map f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "FiniteField.isSquare_iff", "start": [648, 1], "end": [657, 38], "traced_tactics": [{"tactic": "apply\n  (iff_congr _ (by simp [Units.ext_iff])).mp (FiniteField.unit_isSquare_iff hF (Units.mk0 a ha))", "annotated_tactic": ["apply\n    (<a>iff_congr</a> _ (by simp [<a>Units.ext_iff</a>])).<a>mp</a> 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: Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 IsSquare (Units.mk0 a ha) \u2194 IsSquare a"}, {"tactic": "simp only [IsSquare, Units.ext_iff, Units.val_mk0, Units.val_mul]", "annotated_tactic": ["simp only [<a>IsSquare</a>, <a>Units.ext_iff</a>, <a>Units.val_mk0</a>, <a>Units.val_mul</a>]", [{"full_name": "IsSquare", "def_path": "Mathlib/Algebra/Group/Even.lean", "def_pos": [48, 5], "def_end_pos": [48, 13]}, {"full_name": "Units.ext_iff", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [160, 9], "def_end_pos": [160, 16]}, {"full_name": "Units.val_mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 16]}, {"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [226, 9], "def_end_pos": [226, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 IsSquare (Units.mk0 a ha) \u2194 IsSquare a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2194 \u2203 r, a = r * r"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2194 \u2203 r, a = r * r", "state_after": "case mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2192 \u2203 r, a = r * r\n\ncase mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = r * r) \u2192 \u2203 r, a = \u2191r * \u2191r"}, {"tactic": "simp [Units.ext_iff]", "annotated_tactic": ["simp [<a>Units.ext_iff</a>]", [{"full_name": "Units.ext_iff", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [160, 9], "def_end_pos": [160, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 Units.mk0 a ha ^ (Fintype.card F / 2) = 1 \u2194 a ^ (Fintype.card F / 2) = 1", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hy\u27e9", []], "state_before": "case mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2192 \u2203 r, a = r * r", "state_after": "case mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\ny : F\u02e3\nhy : a = \u2191y * \u2191y\n\u22a2 \u2203 r, a = r * r"}, {"tactic": "exact \u27e8y, hy\u27e9", "annotated_tactic": ["exact \u27e8y, hy\u27e9", []], "state_before": "case mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\ny : F\u02e3\nhy : a = \u2191y * \u2191y\n\u22a2 \u2203 r, a = r * r", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, rfl\u27e9", []], "state_before": "case mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = r * r) \u2192 \u2203 r, a = \u2191r * \u2191r", "state_after": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r"}, {"tactic": "have hy : y \u2260 0 := by rintro rfl; simp at ha", "annotated_tactic": ["have hy : y \u2260 0 := by rintro rfl; simp at ha", []], "state_before": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r", "state_after": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r"}, {"tactic": "refine \u27e8Units.mk0 y hy, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Units.mk0</a> y hy, ?_\u27e9", [{"full_name": "Units.mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [186, 5], "def_end_pos": [186, 8]}]], "state_before": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r", "state_after": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 y * y = \u2191(Units.mk0 y hy) * \u2191(Units.mk0 y hy)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 y * y = \u2191(Units.mk0 y hy) * \u2191(Units.mk0 y hy)", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\n\u22a2 y \u2260 0", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\nha : 0 * 0 \u2260 0\n\u22a2 False"}, {"tactic": "simp at ha", "annotated_tactic": ["simp at ha", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\nha : 0 * 0 \u2260 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "gcd_dvd_gcd_mul_left", "start": [503, 1], "end": [504, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "full_name": "deriv_neg''", "start": [258, 1], "end": [259, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.ball_finset_sup_eq_iInter", "start": [883, 1], "end": [887, 80], "traced_tactics": [{"tactic": "lift r to NNReal using hr.le", "annotated_tactic": ["lift r to <a>NNReal</a> using hr.le", [{"full_name": "NNReal", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 11]}]], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d : Seminorm \ud835\udd5c E\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\nx : E\nr : \u211d\nhr : 0 < r\n\u22a2 (s.sup p).ball x r = \u22c2 i \u2208 s, (p i).ball x r", "state_after": "case intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d : Seminorm \ud835\udd5c E\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\nx : E\nr : \u211d\u22650\nhr : 0 < \u2191r\n\u22a2 (s.sup p).ball x \u2191r = \u22c2 i \u2208 s, (p i).ball x \u2191r"}, {"tactic": "simp_rw [ball, iInter_setOf, finset_sup_apply, NNReal.coe_lt_coe,\n  Finset.sup_lt_iff (show \u22a5 < r from hr), \u2190 NNReal.coe_lt_coe, NNReal.coe_mk]", "annotated_tactic": ["simp_rw [<a>ball</a>, <a>iInter_setOf</a>, <a>finset_sup_apply</a>, <a>NNReal.coe_lt_coe</a>,\n    <a>Finset.sup_lt_iff</a> (show \u22a5 < r from hr), \u2190 <a>NNReal.coe_lt_coe</a>, <a>NNReal.coe_mk</a>]", [{"full_name": "Seminorm.ball", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [661, 5], "def_end_pos": [661, 9]}, {"full_name": "Set.iInter_setOf", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [398, 9], "def_end_pos": [398, 21]}, {"full_name": "Seminorm.finset_sup_apply", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [382, 9], "def_end_pos": [382, 25]}, {"full_name": "NNReal.coe_lt_coe", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [375, 26], "def_end_pos": [375, 36]}, {"full_name": "Finset.sup_lt_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [736, 19], "def_end_pos": [736, 29]}, {"full_name": "NNReal.coe_lt_coe", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [375, 26], "def_end_pos": [375, 36]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 34]}]], "state_before": "case intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d : Seminorm \ud835\udd5c E\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\nx : E\nr : \u211d\u22650\nhr : 0 < \u2191r\n\u22a2 (s.sup p).ball x \u2191r = \u22c2 i \u2208 s, (p i).ball x \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjointed.lean", "full_name": "iSup_disjointed", "start": [145, 1], "end": [146, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BooleanAlgebra.lean", "full_name": "compl_surjective", "start": [672, 1], "end": [673, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "full_name": "Set.IicExtend_coe", "start": [289, 1], "end": [290, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Ring.lean", "full_name": "Filter.EventuallyLE.mul_le_mul'", "start": [27, 1], "end": [30, 74], "traced_tactics": [{"tactic": "filter_upwards [hf, hg] with x hfx hgx using _root_.mul_le_mul' hfx hgx", "annotated_tactic": ["filter_upwards [hf, hg] with x hfx hgx using <a>_root_.mul_le_mul'</a> hfx hgx", [{"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Mul \u03b2\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass \u03b2 \u03b2 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b2 \u03b2 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : Filter \u03b1\nf\u2081 f\u2082 g\u2081 g\u2082 : \u03b1 \u2192 \u03b2\nhf : f\u2081 \u2264\u1da0[l] f\u2082\nhg : g\u2081 \u2264\u1da0[l] g\u2082\n\u22a2 f\u2081 * g\u2081 \u2264\u1da0[l] f\u2082 * g\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/OneHypercover.lean", "full_name": "CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve\u2081", "start": [242, 1], "end": [247, 33], "traced_tactics": [{"tactic": "ext Y f", "annotated_tactic": ["ext Y f", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nA : Type u_1\ninst\u271d : Category.{?u.31721, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf\u2081 f\u2082 : S.Arrow\nW : C\np\u2081 : W \u27f6 f\u2081.Y\np\u2082 : W \u27f6 f\u2082.Y\nw : p\u2081 \u226b f\u2081.f = p\u2082 \u226b f\u2082.f\n\u22a2 S.preOneHypercover.sieve\u2081 p\u2081 p\u2082 = \u22a4", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nA : Type u_1\ninst\u271d : Category.{?u.31721, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf\u2081 f\u2082 : S.Arrow\nW : C\np\u2081 : W \u27f6 f\u2081.Y\np\u2082 : W \u27f6 f\u2082.Y\nw : p\u2081 \u226b f\u2081.f = p\u2082 \u226b f\u2082.f\nY : C\nf : Y \u27f6 W\n\u22a2 (S.preOneHypercover.sieve\u2081 p\u2081 p\u2082).arrows f \u2194 \u22a4.arrows f"}, {"tactic": "simp only [Sieve.top_apply, iff_true]", "annotated_tactic": ["simp only [<a>Sieve.top_apply</a>, <a>iff_true</a>]", [{"full_name": "CategoryTheory.Sieve.top_apply", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [394, 9], "def_end_pos": [394, 18]}, {"full_name": "iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [127, 17], "def_end_pos": [127, 25]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nA : Type u_1\ninst\u271d : Category.{?u.31721, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf\u2081 f\u2082 : S.Arrow\nW : C\np\u2081 : W \u27f6 f\u2081.Y\np\u2082 : W \u27f6 f\u2082.Y\nw : p\u2081 \u226b f\u2081.f = p\u2082 \u226b f\u2082.f\nY : C\nf : Y \u27f6 W\n\u22a2 (S.preOneHypercover.sieve\u2081 p\u2081 p\u2082).arrows f \u2194 \u22a4.arrows f", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nA : Type u_1\ninst\u271d : Category.{?u.31721, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf\u2081 f\u2082 : S.Arrow\nW : C\np\u2081 : W \u27f6 f\u2081.Y\np\u2082 : W \u27f6 f\u2082.Y\nw : p\u2081 \u226b f\u2081.f = p\u2082 \u226b f\u2082.f\nY : C\nf : Y \u27f6 W\n\u22a2 (S.preOneHypercover.sieve\u2081 p\u2081 p\u2082).arrows f"}, {"tactic": "exact \u27e8{ w := w}, f, rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8{ w := w}, f, <a>rfl</a>, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nA : Type u_1\ninst\u271d : Category.{?u.31721, u_1} A\nJ : GrothendieckTopology C\nX : C\nS : J.Cover X\nf\u2081 f\u2082 : S.Arrow\nW : C\np\u2081 : W \u27f6 f\u2081.Y\np\u2082 : W \u27f6 f\u2082.Y\nw : p\u2081 \u226b f\u2081.f = p\u2082 \u226b f\u2082.f\nY : C\nf : Y \u27f6 W\n\u22a2 (S.preOneHypercover.sieve\u2081 p\u2081 p\u2082).arrows f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.nextCoeff_ne_zero", "start": [588, 1], "end": [589, 19], "traced_tactics": [{"tactic": "simp [nextCoeff]", "annotated_tactic": ["simp [<a>nextCoeff</a>]", [{"full_name": "Polynomial.nextCoeff", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [580, 5], "def_end_pos": [580, 14]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np : R[X]\n\u22a2 p.nextCoeff \u2260 0 \u2194 p.natDegree \u2260 0 \u2227 p.coeff (p.natDegree - 1) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.pi_univ", "start": [715, 1], "end": [716, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.normSq_nonneg", "start": [705, 1], "end": [706, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.Ioi_mem_atTop", "start": [61, 1], "end": [63, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "full_name": "Mon_.mul_rightUnitor", "start": [440, 1], "end": [447, 47], "traced_tactics": [{"tactic": "rw [\u2190 Category.id_comp M.mul, \u2190 Category.comp_id (\u03bb_ (\ud835\udfd9_ C)).hom, tensor_comp]", "annotated_tactic": ["rw [\u2190 <a>Category.id_comp</a> M.mul, \u2190 <a>Category.comp_id</a> (\u03bb_ (\ud835\udfd9_ C)).<a>hom</a>, <a>tensor_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [168, 3], "def_end_pos": [168, 14]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "simp only [tensorHom_id, id_tensorHom]", "annotated_tactic": ["simp only [<a>tensorHom_id</a>, <a>id_tensorHom</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.tensorHom_id", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [230, 9], "def_end_pos": [230, 21]}, {"full_name": "CategoryTheory.MonoidalCategory.id_tensorHom", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [225, 9], "def_end_pos": [225, 21]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.X \u2297 M.X) \u25c1 (\u03bb_ (\ud835\udfd9_ C)).hom \u226b M.mul \u25b7 \ud835\udfd9_ C) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 3 4 => rw [rightUnitor_naturality]", "annotated_tactic": ["slice_lhs 3 4 => rw [<a>rightUnitor_naturality</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.rightUnitor_naturality", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [190, 3], "def_end_pos": [190, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.X \u2297 M.X) \u25c1 (\u03bb_ (\ud835\udfd9_ C)).hom \u226b M.mul \u25b7 \ud835\udfd9_ C) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.X \u2297 M.X) \u25c1 (\u03bb_ (\ud835\udfd9_ C)).hom \u226b (\u03c1_ (M.X \u2297 M.X)).hom \u226b M.mul =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 1 3 => rw [\u2190 rightUnitor_monoidal]", "annotated_tactic": ["slice_lhs 1 3 => rw [\u2190 <a>rightUnitor_monoidal</a>]", [{"full_name": "CategoryTheory.rightUnitor_monoidal", "def_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "def_pos": [666, 9], "def_end_pos": [666, 29]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.X \u2297 M.X) \u25c1 (\u03bb_ (\ud835\udfd9_ C)).hom \u226b (\u03c1_ (M.X \u2297 M.X)).hom \u226b M.mul =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul = ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "simp only [Category.assoc, Category.id_comp]", "annotated_tactic": ["simp only [<a>Category.assoc</a>, <a>Category.id_comp</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul = ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Basic.lean", "full_name": "ENNReal.iSup_ennreal", "start": [506, 1], "end": [508, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subterminal.lean", "full_name": "CategoryTheory.isSubterminal_of_isTerminal", "start": [86, 1], "end": [87, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "full_name": "CategoryTheory.Limits.Multicoequalizer.condition", "start": [866, 1], "end": [868, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "full_name": "slope_vadd_const", "start": [67, 1], "end": [69, 62], "traced_tactics": [{"tactic": "ext a b", "annotated_tactic": ["ext a b", []], "state_before": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\nc : PE\n\u22a2 (slope fun x => f x +\u1d65 c) = slope f", "state_after": "case h.h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\nc : PE\na b : k\n\u22a2 slope (fun x => f x +\u1d65 c) a b = slope f a b"}, {"tactic": "simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub]", "annotated_tactic": ["simp only [<a>slope</a>, <a>vadd_vsub_vadd_cancel_right</a>, <a>vsub_eq_sub</a>]", [{"full_name": "slope", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "def_pos": [30, 5], "def_end_pos": [30, 10]}, {"full_name": "vadd_vsub_vadd_cancel_right", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [201, 9], "def_end_pos": [201, 36]}, {"full_name": "vsub_eq_sub", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [74, 9], "def_end_pos": [74, 20]}]], "state_before": "case h.h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\nc : PE\na b : k\n\u22a2 slope (fun x => f x +\u1d65 c) a b = slope f a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "ArithmeticFunction.sigma_apply", "start": [884, 1], "end": [885, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Height.lean", "full_name": "Set.le_chainHeight_add_nat_iff", "start": [157, 1], "end": [159, 89], "traced_tactics": [{"tactic": "simp_rw [\u2190 tsub_le_iff_right, \u2190 ENat.coe_sub, (le_chainHeight_TFAE s (n - m)).out 0 2]", "annotated_tactic": ["simp_rw [\u2190 <a>tsub_le_iff_right</a>, \u2190 <a>ENat.coe_sub</a>, (<a>le_chainHeight_TFAE</a> s (n - m)).<a>out</a> 0 2]", [{"full_name": "tsub_le_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [62, 9], "def_end_pos": [62, 26]}, {"full_name": "ENat.coe_sub", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 16]}, {"full_name": "Set.le_chainHeight_TFAE", "def_path": "Mathlib/Order/Height.lean", "def_pos": [109, 9], "def_end_pos": [109, 28]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn m : \u2115\n\u22a2 \u2191n \u2264 s.chainHeight + \u2191m \u2194 \u2203 l \u2208 s.subchain, n \u2264 l.length + m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "antisymm_iff", "start": [40, 1], "end": [43, 28], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\na b : \u03b1\n\u22a2 a = b \u2192 r a b \u2227 r b a", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\na : \u03b1\n\u22a2 r a a \u2227 r a a"}, {"tactic": "exact \u27e8refl _, refl _\u27e9", "annotated_tactic": ["exact \u27e8<a>refl</a> _, <a>refl</a> _\u27e9", [{"full_name": "refl", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [295, 9], "def_end_pos": [295, 13]}, {"full_name": "refl", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [295, 9], "def_end_pos": [295, 13]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\na : \u03b1\n\u22a2 r a a \u2227 r a a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.parts_top_subset", "start": [269, 1], "end": [274, 13], "traced_tactics": [{"tactic": "intro b hb", "annotated_tactic": ["intro b hb", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\n\u22a2 \u22a4.parts \u2286 {a}", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb : b \u2208 \u22a4.parts\n\u22a2 b \u2208 {a}"}, {"tactic": "have hb : b \u2208 Finpartition.parts (dite _ _ _) := hb", "annotated_tactic": ["have hb : b \u2208 <a>Finpartition.parts</a> (<a>dite</a> _ _ _) := hb", [{"full_name": "Finpartition.parts", "def_path": "Mathlib/Order/Partition/Finpartition.lean", "def_pos": [67, 3], "def_end_pos": [67, 8]}, {"full_name": "dite", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [945, 21], "def_end_pos": [945, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb : b \u2208 \u22a4.parts\n\u22a2 b \u2208 {a}", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nhb : b \u2208 (if ha : a = \u22a5 then (Finpartition.empty \u03b1).copy \u22ef else indiscrete ha).parts\n\u22a2 b \u2208 {a}"}, {"tactic": "split_ifs at hb", "annotated_tactic": ["split_ifs at hb", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nhb : b \u2208 (if ha : a = \u22a5 then (Finpartition.empty \u03b1).copy \u22ef else indiscrete ha).parts\n\u22a2 b \u2208 {a}", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nh\u271d : a = \u22a5\nhb : b \u2208 ((Finpartition.empty \u03b1).copy \u22ef).parts\n\u22a2 b \u2208 {a}\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nh\u271d : \u00aca = \u22a5\nhb : b \u2208 (indiscrete h\u271d).parts\n\u22a2 b \u2208 {a}"}, {"tactic": "simp only [copy_parts, empty_parts, not_mem_empty] at hb", "annotated_tactic": ["simp only [<a>copy_parts</a>, <a>empty_parts</a>, <a>not_mem_empty</a>] at hb", [{"full_name": "Finpartition.copy_parts", "def_path": "Mathlib/Order/Partition/Finpartition.lean", "def_pos": [111, 3], "def_end_pos": [111, 8]}, {"full_name": "Finpartition.empty_parts", "def_path": "Mathlib/Order/Partition/Finpartition.lean", "def_pos": [145, 3], "def_end_pos": [145, 8]}, {"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nh\u271d : a = \u22a5\nhb : b \u2208 ((Finpartition.empty \u03b1).copy \u22ef).parts\n\u22a2 b \u2208 {a}", "state_after": "no goals"}, {"tactic": "exact hb", "annotated_tactic": ["exact hb", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\na\u271d : \u03b1\nP : Finpartition a\u271d\na : \u03b1\ninst\u271d : Decidable (a = \u22a5)\nb : \u03b1\nhb\u271d : b \u2208 \u22a4.parts\nh\u271d : \u00aca = \u22a5\nhb : b \u2208 (indiscrete h\u271d).parts\n\u22a2 b \u2208 {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_quot_le", "start": [1912, 1], "end": [1913, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/QuasiIso.lean", "full_name": "CategoryTheory.Functor.quasiIso'_of_map_quasiIso'", "start": [208, 1], "end": [214, 31], "traced_tactics": [{"tactic": "rw [\u2190 Functor.comp_map, \u2190 NatIso.naturality_2 (F.homology'FunctorIso i) f, Functor.comp_map]", "annotated_tactic": ["rw [\u2190 <a>Functor.comp_map</a>, \u2190 <a>NatIso.naturality_2</a> (F.homology'FunctorIso i) f, <a>Functor.comp_map</a>]", [{"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.NatIso.naturality_2", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [172, 9], "def_end_pos": [172, 21]}, {"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}]], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} V\ninst\u271d\u00b9\u00b3 : HasZeroMorphisms V\ninst\u271d\u00b9\u00b2 : HasZeroObject V\ninst\u271d\u00b9\u00b9 : HasEqualizers V\ninst\u271d\u00b9\u2070 : HasImages V\ninst\u271d\u2079 : HasImageMaps V\ninst\u271d\u2078 : HasCokernels V\nc : ComplexShape \u03b9\nC\u271d D\u271d E : HomologicalComplex V c\nA : Type u_2\ninst\u271d\u2077 : Category.{u_4, u_2} A\ninst\u271d\u2076 : Abelian A\nB : Type u_3\ninst\u271d\u2075 : Category.{u_5, u_3} B\ninst\u271d\u2074 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b3 : F.Additive\ninst\u271d\u00b2 : PreservesFiniteLimits F\ninst\u271d\u00b9 : PreservesFiniteColimits F\ninst\u271d : F.Faithful\nC D : HomologicalComplex A c\nf : C \u27f6 D\nhf : QuasiIso' ((F.mapHomologicalComplex c).map f)\ni : \u03b9\n\u22a2 IsIso (F.map ((homology'Functor A c i).map f))", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} V\ninst\u271d\u00b9\u00b3 : HasZeroMorphisms V\ninst\u271d\u00b9\u00b2 : HasZeroObject V\ninst\u271d\u00b9\u00b9 : HasEqualizers V\ninst\u271d\u00b9\u2070 : HasImages V\ninst\u271d\u2079 : HasImageMaps V\ninst\u271d\u2078 : HasCokernels V\nc : ComplexShape \u03b9\nC\u271d D\u271d E : HomologicalComplex V c\nA : Type u_2\ninst\u271d\u2077 : Category.{u_4, u_2} A\ninst\u271d\u2076 : Abelian A\nB : Type u_3\ninst\u271d\u2075 : Category.{u_5, u_3} B\ninst\u271d\u2074 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b3 : F.Additive\ninst\u271d\u00b2 : PreservesFiniteLimits F\ninst\u271d\u00b9 : PreservesFiniteColimits F\ninst\u271d : F.Faithful\nC D : HomologicalComplex A c\nf : C \u27f6 D\nhf : QuasiIso' ((F.mapHomologicalComplex c).map f)\ni : \u03b9\n\u22a2 IsIso\n    ((F.homology'FunctorIso i).hom.app C \u226b\n      (homology'Functor B c i).map ((F.mapHomologicalComplex c).map f) \u226b (F.homology'FunctorIso i).inv.app D)"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} V\ninst\u271d\u00b9\u00b3 : HasZeroMorphisms V\ninst\u271d\u00b9\u00b2 : HasZeroObject V\ninst\u271d\u00b9\u00b9 : HasEqualizers V\ninst\u271d\u00b9\u2070 : HasImages V\ninst\u271d\u2079 : HasImageMaps V\ninst\u271d\u2078 : HasCokernels V\nc : ComplexShape \u03b9\nC\u271d D\u271d E : HomologicalComplex V c\nA : Type u_2\ninst\u271d\u2077 : Category.{u_4, u_2} A\ninst\u271d\u2076 : Abelian A\nB : Type u_3\ninst\u271d\u2075 : Category.{u_5, u_3} B\ninst\u271d\u2074 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b3 : F.Additive\ninst\u271d\u00b2 : PreservesFiniteLimits F\ninst\u271d\u00b9 : PreservesFiniteColimits F\ninst\u271d : F.Faithful\nC D : HomologicalComplex A c\nf : C \u27f6 D\nhf : QuasiIso' ((F.mapHomologicalComplex c).map f)\ni : \u03b9\n\u22a2 IsIso\n    ((F.homology'FunctorIso i).hom.app C \u226b\n      (homology'Functor B c i).map ((F.mapHomologicalComplex c).map f) \u226b (F.homology'FunctorIso i).inv.app D)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Restrict.lean", "full_name": "Matroid.Basis.base_restrict", "start": [176, 9], "end": [177, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.essSup_measure_restrict", "start": [554, 1], "end": [565, 37], "traced_tactics": [{"tactic": "refine le_antisymm (essSup_mono_measure' Measure.restrict_le_self) ?_", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>essSup_mono_measure'</a> <a>Measure.restrict_le_self</a>) ?_", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "essSup_mono_measure'", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [194, 9], "def_end_pos": [194, 29]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [116, 9], "def_end_pos": [116, 25]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f (\u03bc.restrict s) = essSup f \u03bc", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f \u03bc \u2264 essSup f (\u03bc.restrict s)"}, {"tactic": "rw [essSup_eq_sInf (\u03bc.restrict s) f, essSup_eq_sInf \u03bc f]", "annotated_tactic": ["rw [<a>essSup_eq_sInf</a> (\u03bc.restrict s) f, <a>essSup_eq_sInf</a> \u03bc f]", [{"full_name": "essSup_eq_sInf", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}, {"full_name": "essSup_eq_sInf", "def_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f \u03bc \u2264 essSup f (\u03bc.restrict s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 sInf {a | \u03bc {x | a < f x} = 0} \u2264 sInf {a | (\u03bc.restrict s) {x | a < f x} = 0}"}, {"tactic": "refine sInf_le_sInf ?_", "annotated_tactic": ["refine <a>sInf_le_sInf</a> ?_", [{"full_name": "sInf_le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [167, 9], "def_end_pos": [167, 21]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 sInf {a | \u03bc {x | a < f x} = 0} \u2264 sInf {a | (\u03bc.restrict s) {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 {a | (\u03bc.restrict s) {x | a < f x} = 0} \u2286 {a | \u03bc {x | a < f x} = 0}"}, {"tactic": "rintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0)", "annotated_tactic": ["rintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0)", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 {a | (\u03bc.restrict s) {x | a < f x} = 0} \u2286 {a | \u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : (\u03bc.restrict s) {x | a < f x} = 0\n\u22a2 a \u2208 {a | \u03bc {x | a < f x} = 0}"}, {"tactic": "rw [Measure.restrict_apply\u2080' hs.nullMeasurableSet] at ha", "annotated_tactic": ["rw [<a>Measure.restrict_apply\u2080'</a> hs.nullMeasurableSet] at ha", [{"full_name": "MeasureTheory.Measure.restrict_apply\u2080'", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [110, 9], "def_end_pos": [110, 25]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : (\u03bc.restrict s) {x | a < f x} = 0\n\u22a2 a \u2208 {a | \u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 a \u2208 {a | \u03bc {x | a < f x} = 0}"}, {"tactic": "refine measure_zero_of_invariant hs _ ?_ ha", "annotated_tactic": ["refine <a>measure_zero_of_invariant</a> hs _ ?_ ha", [{"full_name": "MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant", "def_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "def_pos": [332, 9], "def_end_pos": [332, 34]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 a \u2208 {a | \u03bc {x | a < f x} = 0}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x}"}, {"tactic": "intro \u03b3", "annotated_tactic": ["intro \u03b3", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x}", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\n\u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\n\u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x}", "state_after": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x}"}, {"tactic": "rw [mem_smul_set_iff_inv_smul_mem]", "annotated_tactic": ["rw [<a>mem_smul_set_iff_inv_smul_mem</a>]", [{"full_name": "Set.mem_smul_set_iff_inv_smul_mem", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [900, 9], "def_end_pos": [900, 38]}]], "state_before": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x}", "state_after": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x}"}, {"tactic": "simp only [mem_setOf_eq, hf \u03b3\u207b\u00b9 x]", "annotated_tactic": ["simp only [<a>mem_setOf_eq</a>, hf \u03b3\u207b\u00b9 x]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "case h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.sup_mul_eq_of_coprime_right", "start": [681, 1], "end": [683, 37], "traced_tactics": [{"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 K = \u22a4\n\u22a2 I \u2294 J * K = I \u2294 J", "state_after": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 K = \u22a4\n\u22a2 I \u2294 K * J = I \u2294 J"}, {"tactic": "exact sup_mul_eq_of_coprime_left h", "annotated_tactic": ["exact <a>sup_mul_eq_of_coprime_left</a> h", [{"full_name": "Ideal.sup_mul_eq_of_coprime_left", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [673, 9], "def_end_pos": [673, 35]}]], "state_before": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 K = \u22a4\n\u22a2 I \u2294 K * J = I \u2294 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/FiniteDimension.lean", "full_name": "LinearMap.continuous_of_nonzero_on_open", "start": [185, 1], "end": [192, 34], "traced_tactics": [{"tactic": "refine l.continuous_of_isClosed_ker (l.isClosed_or_dense_ker.resolve_right fun hl => ?_)", "annotated_tactic": ["refine l.continuous_of_isClosed_ker (l.isClosed_or_dense_ker.resolve_right fun hl => ?_)", []], "state_before": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2082 : s.Nonempty\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\n\u22a2 Continuous \u21d1l", "state_after": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2082 : s.Nonempty\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\n\u22a2 False"}, {"tactic": "rcases hs\u2082 with \u27e8x, hx\u27e9", "annotated_tactic": ["rcases hs\u2082 with \u27e8x, hx\u27e9", []], "state_before": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2082 : s.Nonempty\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\n\u22a2 False", "state_after": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\n\u22a2 False"}, {"tactic": "have : x \u2208 interior (LinearMap.ker l : Set E)\u1d9c := by\n  rw [mem_interior_iff_mem_nhds]\n  exact mem_of_superset (hs\u2081.mem_nhds hx) hs\u2083", "annotated_tactic": ["have : x \u2208 <a>interior</a> (<a>LinearMap.ker</a> l : <a>Set</a> E)\u1d9c := by\n    rw [<a>mem_interior_iff_mem_nhds</a>]\n    exact <a>mem_of_superset</a> (hs\u2081.mem_nhds hx) hs\u2083", [{"full_name": "interior", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [111, 5], "def_end_pos": [111, 13]}, {"full_name": "LinearMap.ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [60, 5], "def_end_pos": [60, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "mem_interior_iff_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [852, 9], "def_end_pos": [852, 34]}, {"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}]], "state_before": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\n\u22a2 False", "state_after": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\nthis : x \u2208 interior (\u2191(ker l))\u1d9c\n\u22a2 False"}, {"tactic": "rwa [hl.interior_compl] at this", "annotated_tactic": ["rwa [hl.interior_compl] at this", []], "state_before": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\nthis : x \u2208 interior (\u2191(ker l))\u1d9c\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [mem_interior_iff_mem_nhds]", "annotated_tactic": ["rw [<a>mem_interior_iff_mem_nhds</a>]", [{"full_name": "mem_interior_iff_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [852, 9], "def_end_pos": [852, 34]}]], "state_before": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\n\u22a2 x \u2208 interior (\u2191(ker l))\u1d9c", "state_after": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\n\u22a2 (\u2191(ker l))\u1d9c \u2208 nhds x"}, {"tactic": "exact mem_of_superset (hs\u2081.mem_nhds hx) hs\u2083", "annotated_tactic": ["exact <a>mem_of_superset</a> (hs\u2081.mem_nhds hx) hs\u2083", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}]], "state_before": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup E\ninst\u271d\u00b9\u2070 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalAddGroup F\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2074 : AddCommGroup F'\ninst\u271d\u00b3 : Module \ud835\udd5c F'\ninst\u271d\u00b2 : TopologicalSpace F'\ninst\u271d\u00b9 : TopologicalAddGroup F'\ninst\u271d : ContinuousSMul \ud835\udd5c F'\nl : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nhs\u2081 : IsOpen s\nhs\u2083 : \u2200 x \u2208 s, l x \u2260 0\nhl : Dense \u2191(ker l)\nx : E\nhx : x \u2208 s\n\u22a2 (\u2191(ker l))\u1d9c \u2208 nhds x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "UpperSemicontinuousAt.upperSemicontinuousWithinAt", "start": [770, 1], "end": [772, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean", "full_name": "tensorQuotEquivQuotSMul_symm_mk", "start": [484, 1], "end": [487, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.predAbove_of_le_castSucc", "start": [1600, 1], "end": [1602, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convolution.lean", "full_name": "MeasureTheory.zero_convolution", "start": [483, 1], "end": [485, 71], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : AddGroup G\n\u22a2 0 \u22c6[L, \u03bc] g = 0", "state_after": "case h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 (0 \u22c6[L, \u03bc] g) x\u271d = 0 x\u271d"}, {"tactic": "simp_rw [convolution_def, Pi.zero_apply, L.map_zero\u2082, integral_zero]", "annotated_tactic": ["simp_rw [<a>convolution_def</a>, <a>Pi.zero_apply</a>, L.map_zero\u2082, <a>integral_zero</a>]", [{"full_name": "MeasureTheory.convolution_def", "def_path": "Mathlib/Analysis/Convolution.lean", "def_pos": [453, 9], "def_end_pos": [453, 24]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 22]}]], "state_before": "case h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 (0 \u22c6[L, \u03bc] g) x\u271d = 0 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.Ioo_ae_eq_Icc", "start": [445, 1], "end": [446, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Prod.lean", "full_name": "QuadraticForm.Isometry.proj_comp_single_of_same", "start": [294, 1], "end": [297, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Bounded.lean", "full_name": "BotHom.coe_id", "start": [447, 1], "end": [448, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "full_name": "Real.deriv_negMulLog", "start": [109, 1], "end": [111, 7], "traced_tactics": [{"tactic": "rw [negMulLog_eq_neg, deriv.neg, deriv_mul_log hx]", "annotated_tactic": ["rw [<a>negMulLog_eq_neg</a>, <a>deriv.neg</a>, <a>deriv_mul_log</a> hx]", [{"full_name": "Real.negMulLog_eq_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "def_pos": [85, 7], "def_end_pos": [85, 23]}, {"full_name": "deriv.neg", "def_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "def_pos": [211, 9], "def_end_pos": [211, 18]}, {"full_name": "Real.deriv_mul_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "def_pos": [45, 7], "def_end_pos": [45, 20]}]], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 deriv negMulLog x = -log x - 1", "state_after": "x : \u211d\nhx : x \u2260 0\n\u22a2 -(log x + 1) = -log x - 1"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 -(log x + 1) = -log x - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Smooth/Basic.lean", "full_name": "Algebra.Smooth.comp", "start": [393, 1], "end": [395, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "Real.le_norm_self", "start": [1449, 1], "end": [1450, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "full_name": "CategoryTheory.TwoSquare.GuitartExact.vComp_iff_of_equivalences", "start": [123, 1], "end": [154, 25], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\n\u22a2 (w.vComp w'.hom).GuitartExact \u2194 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\n\u22a2 (w.vComp w'.hom).GuitartExact \u2192 w.GuitartExact\n\ncase mpr\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\n\u22a2 w.GuitartExact \u2192 (w.vComp w'.hom).GuitartExact"}, {"tactic": "intro hww'", "annotated_tactic": ["intro hww'", []], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\n\u22a2 (w.vComp w'.hom).GuitartExact \u2192 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\n\u22a2 w.GuitartExact"}, {"tactic": "letI : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := \u27e8w'\u27e9", "annotated_tactic": ["letI : <a>CatCommSq</a> H\u2082 eL.functor eR.functor H\u2083 := \u27e8w'\u27e9", [{"full_name": "CategoryTheory.CatCommSq", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\n\u22a2 w.GuitartExact"}, {"tactic": "have hw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w' := rfl", "annotated_tactic": ["have hw' : <a>CatCommSq.iso</a> H\u2082 eL.functor eR.functor H\u2083 = w' := <a>rfl</a>", [{"full_name": "CategoryTheory.CatCommSq.iso", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\n\u22a2 w.GuitartExact"}, {"tactic": "letI : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := CatCommSq.vInvEquiv _ _ _ _ inferInstance", "annotated_tactic": ["letI : <a>CatCommSq</a> H\u2083 eL.inverse eR.inverse H\u2082 := <a>CatCommSq.vInvEquiv</a> _ _ _ _ <a>inferInstance</a>", [{"full_name": "CategoryTheory.CatCommSq", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [34, 7], "def_end_pos": [34, 16]}, {"full_name": "CategoryTheory.CatCommSq.vInvEquiv", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [130, 5], "def_end_pos": [130, 14]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\n\u22a2 w.GuitartExact"}, {"tactic": "let w'' := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082", "annotated_tactic": ["let w'' := <a>CatCommSq.iso</a> H\u2083 eL.inverse eR.inverse H\u2082", [{"full_name": "CategoryTheory.CatCommSq.iso", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u22a2 w.GuitartExact"}, {"tactic": "let \u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  Functor.associator _ _ _ \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor", "annotated_tactic": ["let \u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n      <a>Functor.associator</a> _ _ _ \u226a\u226b <a>isoWhiskerLeft</a> L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor", [{"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [172, 5], "def_end_pos": [172, 19]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u22a2 w.GuitartExact"}, {"tactic": "let \u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  Functor.associator _ _ _ \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor", "annotated_tactic": ["let \u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n      <a>Functor.associator</a> _ _ _ \u226a\u226b <a>isoWhiskerLeft</a> R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor", [{"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [172, 5], "def_end_pos": [172, 19]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\n\u22a2 w.GuitartExact"}, {"tactic": "have : w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2 := by\n  ext X\u2081\n  dsimp\n  simp? [w'', \u03b2, \u03b1] says\n    simp only [vComp'_app, Functor.comp_obj, Iso.trans_inv, isoWhiskerLeft_inv, Iso.symm_inv,\n      assoc, NatTrans.comp_app, Functor.id_obj, Functor.rightUnitor_inv_app, whiskerLeft_app,\n      Functor.associator_inv_app, comp_id, id_comp, vComp_app, Functor.map_comp,\n      Equivalence.inv_fun_map, Iso.trans_hom, isoWhiskerLeft_hom, Iso.symm_hom,\n      Functor.associator_hom_app, Functor.rightUnitor_hom_app, Iso.hom_inv_id_app_assoc,\n      w'', \u03b1, \u03b2]\n  erw [CatCommSq.vInv_iso'_hom_app]\n  simp only [hw', assoc, \u2190 eR.inverse.map_comp_assoc]\n  rw [Equivalence.counitInv_app_functor]\n  erw [\u2190 NatTrans.naturality_assoc]\n  simp [\u2190 H\u2082.map_comp]", "annotated_tactic": ["have : w = (w.vComp w'.hom).<a>vComp'</a> w''.hom \u03b1 \u03b2 := by\n      ext X\u2081\n      dsimp\n      simp? [w'', \u03b2, \u03b1] says\n        simp only [<a>vComp'_app</a>, <a>Functor.comp_obj</a>, <a>Iso.trans_inv</a>, <a>isoWhiskerLeft_inv</a>, <a>Iso.symm_inv</a>,\n          <a>assoc</a>, <a>NatTrans.comp_app</a>, <a>Functor.id_obj</a>, <a>Functor.rightUnitor_inv_app</a>, <a>whiskerLeft_app</a>,\n          <a>Functor.associator_inv_app</a>, <a>comp_id</a>, <a>id_comp</a>, <a>vComp_app</a>, <a>Functor.map_comp</a>,\n          <a>Equivalence.inv_fun_map</a>, <a>Iso.trans_hom</a>, <a>isoWhiskerLeft_hom</a>, <a>Iso.symm_hom</a>,\n          <a>Functor.associator_hom_app</a>, <a>Functor.rightUnitor_hom_app</a>, <a>Iso.hom_inv_id_app_assoc</a>,\n          w'', \u03b1, \u03b2]\n      erw [<a>CatCommSq.vInv_iso'_hom_app</a>]\n      simp only [hw', <a>assoc</a>, \u2190 eR.inverse.map_comp_assoc]\n      rw [<a>Equivalence.counitInv_app_functor</a>]\n      erw [\u2190 <a>NatTrans.naturality_assoc</a>]\n      simp [\u2190 H\u2082.map_comp]", [{"full_name": "CategoryTheory.TwoSquare.vComp'", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [104, 5], "def_end_pos": [104, 11]}, {"full_name": "CategoryTheory.TwoSquare.vComp'_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [103, 3], "def_end_pos": [103, 9]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Iso.trans_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_inv", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [183, 9], "def_end_pos": [183, 27]}, {"full_name": "CategoryTheory.Iso.symm_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [105, 9], "def_end_pos": [105, 17]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "CategoryTheory.Functor.id_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"full_name": "CategoryTheory.Functor.rightUnitor_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.whiskerLeft_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [46, 3], "def_end_pos": [46, 8]}, {"full_name": "CategoryTheory.Functor.associator_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.TwoSquare.vComp_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [88, 3], "def_end_pos": [88, 9]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Equivalence.inv_fun_map", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [234, 9], "def_end_pos": [234, 20]}, {"full_name": "CategoryTheory.Iso.trans_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_hom", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}, {"full_name": "CategoryTheory.Iso.symm_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [100, 9], "def_end_pos": [100, 17]}, {"full_name": "CategoryTheory.Functor.associator_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Functor.rightUnitor_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.Iso.hom_inv_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [63, 3], "def_end_pos": [63, 25]}, {"full_name": "CategoryTheory.CatCommSq.vInv_iso'_hom_app", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [103, 10], "def_end_pos": [103, 22]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Equivalence.counitInv_app_functor", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [166, 9], "def_end_pos": [166, 30]}, {"full_name": "CategoryTheory.NatTrans.naturality_assoc", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [60, 12], "def_end_pos": [60, 34]}]], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d\u00b9 : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis\u271d : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nthis : w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2\n\u22a2 w.GuitartExact"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d\u00b9 : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis\u271d : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nthis : w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2\n\u22a2 w.GuitartExact", "state_after": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d\u00b9 : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis\u271d : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nthis : w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2\n\u22a2 ((w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2).GuitartExact"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case mp\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d\u00b9 : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis\u271d : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nthis : w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2\n\u22a2 ((w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2).GuitartExact", "state_after": "no goals"}, {"tactic": "ext X\u2081", "annotated_tactic": ["ext X\u2081", []], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\n\u22a2 w = (w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 = ((w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2).app X\u2081"}, {"tactic": "simp? [w'', \u03b2, \u03b1] says\n  simp only [vComp'_app, Functor.comp_obj, Iso.trans_inv, isoWhiskerLeft_inv, Iso.symm_inv,\n    assoc, NatTrans.comp_app, Functor.id_obj, Functor.rightUnitor_inv_app, whiskerLeft_app,\n    Functor.associator_inv_app, comp_id, id_comp, vComp_app, Functor.map_comp,\n    Equivalence.inv_fun_map, Iso.trans_hom, isoWhiskerLeft_hom, Iso.symm_hom,\n    Functor.associator_hom_app, Functor.rightUnitor_hom_app, Iso.hom_inv_id_app_assoc,\n    w'', \u03b1, \u03b2]", "annotated_tactic": ["simp? [w'', \u03b2, \u03b1] says\n        simp only [<a>vComp'_app</a>, <a>Functor.comp_obj</a>, <a>Iso.trans_inv</a>, <a>isoWhiskerLeft_inv</a>, <a>Iso.symm_inv</a>,\n          <a>assoc</a>, <a>NatTrans.comp_app</a>, <a>Functor.id_obj</a>, <a>Functor.rightUnitor_inv_app</a>, <a>whiskerLeft_app</a>,\n          <a>Functor.associator_inv_app</a>, <a>comp_id</a>, <a>id_comp</a>, <a>vComp_app</a>, <a>Functor.map_comp</a>,\n          <a>Equivalence.inv_fun_map</a>, <a>Iso.trans_hom</a>, <a>isoWhiskerLeft_hom</a>, <a>Iso.symm_hom</a>,\n          <a>Functor.associator_hom_app</a>, <a>Functor.rightUnitor_hom_app</a>, <a>Iso.hom_inv_id_app_assoc</a>,\n          w'', \u03b1, \u03b2]", [{"full_name": "CategoryTheory.TwoSquare.vComp'_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [103, 3], "def_end_pos": [103, 9]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Iso.trans_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_inv", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [183, 9], "def_end_pos": [183, 27]}, {"full_name": "CategoryTheory.Iso.symm_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [105, 9], "def_end_pos": [105, 17]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "CategoryTheory.Functor.id_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"full_name": "CategoryTheory.Functor.rightUnitor_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.whiskerLeft_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [46, 3], "def_end_pos": [46, 8]}, {"full_name": "CategoryTheory.Functor.associator_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.TwoSquare.vComp_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [88, 3], "def_end_pos": [88, 9]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Equivalence.inv_fun_map", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [234, 9], "def_end_pos": [234, 20]}, {"full_name": "CategoryTheory.Iso.trans_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_hom", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}, {"full_name": "CategoryTheory.Iso.symm_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [100, 9], "def_end_pos": [100, 17]}, {"full_name": "CategoryTheory.Functor.associator_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Functor.rightUnitor_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.Iso.hom_inv_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [63, 3], "def_end_pos": [63, 25]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 = ((w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2).app X\u2081", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map (w'.hom.app (L\u2081.obj X\u2081)) \u226b\n          (CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082).hom.app (eL.functor.obj (L\u2081.obj X\u2081)) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "erw [CatCommSq.vInv_iso'_hom_app]", "annotated_tactic": ["erw [<a>CatCommSq.vInv_iso'_hom_app</a>]", [{"full_name": "CategoryTheory.CatCommSq.vInv_iso'_hom_app", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [103, 10], "def_end_pos": [103, 22]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map (w'.hom.app (L\u2081.obj X\u2081)) \u226b\n          (CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082).hom.app (eL.functor.obj (L\u2081.obj X\u2081)) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map (w'.hom.app (L\u2081.obj X\u2081)) \u226b\n          (eR.inverse.map (H\u2083.map (eL.counitIso.inv.app (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n              eR.inverse.map\n                  ((CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083).inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n                eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081))))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "simp only [hw', assoc, \u2190 eR.inverse.map_comp_assoc]", "annotated_tactic": ["simp only [hw', <a>assoc</a>, \u2190 eR.inverse.map_comp_assoc]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map (w'.hom.app (L\u2081.obj X\u2081)) \u226b\n          (eR.inverse.map (H\u2083.map (eL.counitIso.inv.app (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n              eR.inverse.map\n                  ((CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083).inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n                eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081))))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            (w'.hom.app (L\u2081.obj X\u2081) \u226b\n              H\u2083.map (eL.counitIso.inv.app (eL.functor.obj (L\u2081.obj X\u2081))) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "rw [Equivalence.counitInv_app_functor]", "annotated_tactic": ["rw [<a>Equivalence.counitInv_app_functor</a>]", [{"full_name": "CategoryTheory.Equivalence.counitInv_app_functor", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [166, 9], "def_end_pos": [166, 30]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            (w'.hom.app (L\u2081.obj X\u2081) \u226b\n              H\u2083.map (eL.counitIso.inv.app (eL.functor.obj (L\u2081.obj X\u2081))) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            (w'.hom.app (L\u2081.obj X\u2081) \u226b\n              H\u2083.map (eL.functor.map (eL.unit.app (L\u2081.obj X\u2081))) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "erw [\u2190 NatTrans.naturality_assoc]", "annotated_tactic": ["erw [\u2190 <a>NatTrans.naturality_assoc</a>]", [{"full_name": "CategoryTheory.NatTrans.naturality_assoc", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [60, 12], "def_end_pos": [60, 34]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            (w'.hom.app (L\u2081.obj X\u2081) \u226b\n              H\u2083.map (eL.functor.map (eL.unit.app (L\u2081.obj X\u2081))) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            ((H\u2082 \u22d9 eR.functor).map (eL.unit.app (L\u2081.obj X\u2081)) \u226b\n              w'.hom.app ((eL.functor \u22d9 eL.inverse).obj (L\u2081.obj X\u2081)) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "simp [\u2190 H\u2082.map_comp]", "annotated_tactic": ["simp [\u2190 H\u2082.map_comp]", []], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map\n            ((H\u2082 \u22d9 eR.functor).map (eL.unit.app (L\u2081.obj X\u2081)) \u226b\n              w'.hom.app ((eL.functor \u22d9 eL.inverse).obj (L\u2081.obj X\u2081)) \u226b\n                w'.inv.app (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n          eR.unitIso.inv.app (H\u2082.obj (eL.inverse.obj (eL.functor.obj (L\u2081.obj X\u2081)))) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))", "state_after": "no goals"}, {"tactic": "simp only [vComp'_app, Functor.comp_obj, Iso.trans_inv, isoWhiskerLeft_inv, Iso.symm_inv,\n  assoc, NatTrans.comp_app, Functor.id_obj, Functor.rightUnitor_inv_app, whiskerLeft_app,\n  Functor.associator_inv_app, comp_id, id_comp, vComp_app, Functor.map_comp,\n  Equivalence.inv_fun_map, Iso.trans_hom, isoWhiskerLeft_hom, Iso.symm_hom,\n  Functor.associator_hom_app, Functor.rightUnitor_hom_app, Iso.hom_inv_id_app_assoc,\n  w'', \u03b1, \u03b2]", "annotated_tactic": ["simp only [<a>vComp'_app</a>, <a>Functor.comp_obj</a>, <a>Iso.trans_inv</a>, <a>isoWhiskerLeft_inv</a>, <a>Iso.symm_inv</a>,\n          <a>assoc</a>, <a>NatTrans.comp_app</a>, <a>Functor.id_obj</a>, <a>Functor.rightUnitor_inv_app</a>, <a>whiskerLeft_app</a>,\n          <a>Functor.associator_inv_app</a>, <a>comp_id</a>, <a>id_comp</a>, <a>vComp_app</a>, <a>Functor.map_comp</a>,\n          <a>Equivalence.inv_fun_map</a>, <a>Iso.trans_hom</a>, <a>isoWhiskerLeft_hom</a>, <a>Iso.symm_hom</a>,\n          <a>Functor.associator_hom_app</a>, <a>Functor.rightUnitor_hom_app</a>, <a>Iso.hom_inv_id_app_assoc</a>,\n          w'', \u03b1, \u03b2]", [{"full_name": "CategoryTheory.TwoSquare.vComp'_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [103, 3], "def_end_pos": [103, 9]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Iso.trans_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_inv", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [183, 9], "def_end_pos": [183, 27]}, {"full_name": "CategoryTheory.Iso.symm_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [105, 9], "def_end_pos": [105, 17]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "CategoryTheory.Functor.id_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"full_name": "CategoryTheory.Functor.rightUnitor_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.whiskerLeft_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [46, 3], "def_end_pos": [46, 8]}, {"full_name": "CategoryTheory.Functor.associator_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.TwoSquare.vComp_app", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [88, 3], "def_end_pos": [88, 9]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Equivalence.inv_fun_map", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [234, 9], "def_end_pos": [234, 20]}, {"full_name": "CategoryTheory.Iso.trans_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [149, 10], "def_end_pos": [149, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft_hom", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}, {"full_name": "CategoryTheory.Iso.symm_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [100, 9], "def_end_pos": [100, 17]}, {"full_name": "CategoryTheory.Functor.associator_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [279, 3], "def_end_pos": [279, 8]}, {"full_name": "CategoryTheory.Functor.rightUnitor_hom_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.Iso.hom_inv_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [63, 3], "def_end_pos": [63, 25]}]], "state_before": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 = ((w.vComp w'.hom).vComp' w''.hom \u03b1 \u03b2).app X\u2081", "state_after": "case h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\nhww' : (w.vComp w'.hom).GuitartExact\nthis\u271d : CatCommSq H\u2082 eL.functor eR.functor H\u2083 := { iso' := w' }\nhw' : CatCommSq.iso H\u2082 eL.functor eR.functor H\u2083 = w'\nthis : CatCommSq H\u2083 eL.inverse eR.inverse H\u2082 := (CatCommSq.vInvEquiv H\u2082 eL eR H\u2083) inferInstance\nw'' : H\u2083 \u22d9 eR.inverse \u2245 eL.inverse \u22d9 H\u2082 := CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082\n\u03b1 : (L\u2081 \u22d9 eL.functor) \u22d9 eL.inverse \u2245 L\u2081 :=\n  L\u2081.associator eL.functor eL.inverse \u226a\u226b isoWhiskerLeft L\u2081 eL.unitIso.symm \u226a\u226b L\u2081.rightUnitor\n\u03b2 : (R\u2081 \u22d9 eR.functor) \u22d9 eR.inverse \u2245 R\u2081 :=\n  R\u2081.associator eR.functor eR.inverse \u226a\u226b isoWhiskerLeft R\u2081 eR.unitIso.symm \u226a\u226b R\u2081.rightUnitor\nX\u2081 : C\u2081\n\u22a2 w.app X\u2081 =\n    w.app X\u2081 \u226b\n      eR.unit.app (H\u2082.obj (L\u2081.obj X\u2081)) \u226b\n        eR.inverse.map (w'.hom.app (L\u2081.obj X\u2081)) \u226b\n          (CatCommSq.iso H\u2083 eL.inverse eR.inverse H\u2082).hom.app (eL.functor.obj (L\u2081.obj X\u2081)) \u226b\n            H\u2082.map (eL.unitIso.inv.app (L\u2081.obj X\u2081))"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case mpr\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\n\u22a2 w.GuitartExact \u2192 (w.vComp w'.hom).GuitartExact", "state_after": "case mpr\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\na\u271d : w.GuitartExact\n\u22a2 (w.vComp w'.hom).GuitartExact"}, {"tactic": "exact vComp w w'.hom", "annotated_tactic": ["exact <a>vComp</a> w w'.hom", [{"full_name": "CategoryTheory.TwoSquare.GuitartExact.vComp", "def_path": "Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean", "def_pos": [110, 10], "def_end_pos": [110, 15]}]], "state_before": "case mpr\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nD\u2081 : Type u_4\nD\u2082 : Type u_5\nD\u2083 : Type u_6\ninst\u271d\u2075 : Category.{u_11, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_7, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_8, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_12, u_4} D\u2081\ninst\u271d\u00b9 : Category.{u_9, u_5} D\u2082\ninst\u271d : Category.{u_10, u_6} D\u2083\nH\u2081 : C\u2081 \u2964 D\u2081\nL\u2081 : C\u2081 \u2964 C\u2082\nR\u2081 : D\u2081 \u2964 D\u2082\nH\u2082 : C\u2082 \u2964 D\u2082\nw : TwoSquare H\u2081 L\u2081 R\u2081 H\u2082\nL\u2082 : C\u2082 \u2964 C\u2083\nR\u2082 : D\u2082 \u2964 D\u2083\nH\u2083 : C\u2083 \u2964 D\u2083\nw'\u271d : TwoSquare H\u2082 L\u2082 R\u2082 H\u2083\neL : C\u2082 \u224c C\u2083\neR : D\u2082 \u224c D\u2083\nw' : H\u2082 \u22d9 eR.functor \u2245 eL.functor \u22d9 H\u2083\na\u271d : w.GuitartExact\n\u22a2 (w.vComp w'.hom).GuitartExact", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "full_name": "mul_lt_of_le_one_of_lt_of_nonneg", "start": [884, 1], "end": [886, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Reachable.map", "start": [2056, 11], "end": [2058, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Derivation.lean", "full_name": "Polynomial.mkDerivation_X", "start": [67, 1], "end": [67, 88], "traced_tactics": [{"tactic": "simp [mkDerivation_apply]", "annotated_tactic": ["simp [<a>mkDerivation_apply</a>]", [{"full_name": "Polynomial.mkDerivation_apply", "def_path": "Mathlib/Algebra/Polynomial/Derivation.lean", "def_pos": [62, 7], "def_end_pos": [62, 25]}]], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : Module R[X] A\ninst\u271d : IsScalarTower R R[X] A\na : A\n\u22a2 ((mkDerivation R) a) X = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.principal_singleton", "start": [2134, 1], "end": [2135, 83], "traced_tactics": [{"tactic": "simp only [mem_pure, mem_principal, singleton_subset_iff]", "annotated_tactic": ["simp only [<a>mem_pure</a>, <a>mem_principal</a>, <a>singleton_subset_iff</a>]", [{"full_name": "Filter.mem_pure", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2124, 9], "def_end_pos": [2124, 17]}, {"full_name": "Filter.mem_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [307, 17], "def_end_pos": [307, 30]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 29]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl : Filter \u03b1\na : \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 \ud835\udcdf {a} \u2194 s \u2208 pure a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.absolutelyContinuous_zero_iff", "start": [1688, 1], "end": [1690, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DualNumber.lean", "full_name": "DualNumber.eps_mul_eps", "start": [82, 1], "end": [83, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "ArithmeticFunction.IsMultiplicative.mul", "start": [673, 1], "end": [731, 9], "traced_tactics": [{"tactic": "refine \u27e8by simp [hf.1, hg.1], ?_\u27e9", "annotated_tactic": ["refine \u27e8by simp [hf.1, hg.1], ?_\u27e9", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 (f * g).IsMultiplicative", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 \u2200 {m n : \u2115}, m.Coprime n \u2192 (f * g) (m * n) = (f * g) m * (f * g) n"}, {"tactic": "simp only [mul_apply]", "annotated_tactic": ["simp only [<a>mul_apply</a>]", [{"full_name": "ArithmeticFunction.mul_apply", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [291, 9], "def_end_pos": [291, 18]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 \u2200 {m n : \u2115}, m.Coprime n \u2192 (f * g) (m * n) = (f * g) m * (f * g) n", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 \u2200 {m n : \u2115},\n    m.Coprime n \u2192\n      \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n        (\u2211 x \u2208 m.divisorsAntidiagonal, f x.1 * g x.2) * \u2211 x \u2208 n.divisorsAntidiagonal, f x.1 * g x.2"}, {"tactic": "intro m n cop", "annotated_tactic": ["intro m n cop", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 \u2200 {m n : \u2115},\n    m.Coprime n \u2192\n      \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n        (\u2211 x \u2208 m.divisorsAntidiagonal, f x.1 * g x.2) * \u2211 x \u2208 n.divisorsAntidiagonal, f x.1 * g x.2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n    (\u2211 x \u2208 m.divisorsAntidiagonal, f x.1 * g x.2) * \u2211 x \u2208 n.divisorsAntidiagonal, f x.1 * g x.2"}, {"tactic": "rw [sum_mul_sum, \u2190 sum_product']", "annotated_tactic": ["rw [<a>sum_mul_sum</a>, \u2190 <a>sum_product'</a>]", [{"full_name": "Finset.sum_mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Finset.sum_product'", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [895, 3], "def_end_pos": [895, 14]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n    (\u2211 x \u2208 m.divisorsAntidiagonal, f x.1 * g x.2) * \u2211 x \u2208 n.divisorsAntidiagonal, f x.1 * g x.2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n    \u2211 x \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2 =\n    \u2211 x \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2)", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2) =\n    \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2"}, {"tactic": "apply sum_nbij fun ((i, j), k, l) \u21a6 (i * k, j * l)", "annotated_tactic": ["apply <a>sum_nbij</a> fun ((i, j), k, l) \u21a6 (i * k, j * l)", [{"full_name": "Finset.sum_nbij", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [696, 3], "def_end_pos": [696, 14]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2211 x \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal, f x.1.1 * g x.1.2 * (f x.2.1 * g x.2.2) =\n    \u2211 x \u2208 (m * n).divisorsAntidiagonal, f x.1 * g x.2", "state_after": "case hi\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 a \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal,\n    (match a with\n      | ((i, j), k, l) => (i * k, j * l)) \u2208\n      (m * n).divisorsAntidiagonal\n\ncase i_inj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 Set.InjOn\n    (fun x =>\n      match x with\n      | ((i, j), k, l) => (i * k, j * l))\n    \u2191(m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal)\n\ncase i_surj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 Set.SurjOn\n    (fun x =>\n      match x with\n      | ((i, j), k, l) => (i * k, j * l))\n    \u2191(m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal) \u2191(m * n).divisorsAntidiagonal\n\ncase h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 a \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal,\n    f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =\n      f\n          (match a with\n            | ((i, j), k, l) => (i * k, j * l)).1 *\n        g\n          (match a with\n            | ((i, j), k, l) => (i * k, j * l)).2"}, {"tactic": "simp [hf.1, hg.1]", "annotated_tactic": ["simp [hf.1, hg.1]", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n\u22a2 (f * g) 1 = 1", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 h", "annotated_tactic": ["rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 h", []], "state_before": "case hi\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 a \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal,\n    (match a with\n      | ((i, j), k, l) => (i * k, j * l)) \u2208\n      (m * n).divisorsAntidiagonal", "state_after": "case hi.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\na1 a2 b1 b2 : \u2115\nh : ((a1, a2), b1, b2) \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (m * n).divisorsAntidiagonal"}, {"tactic": "simp only [mem_divisorsAntidiagonal, Ne, mem_product] at h", "annotated_tactic": ["simp only [<a>mem_divisorsAntidiagonal</a>, <a>Ne</a>, <a>mem_product</a>] at h", [{"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case hi.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\na1 a2 b1 b2 : \u2115\nh : ((a1, a2), b1, b2) \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (m * n).divisorsAntidiagonal", "state_after": "case hi.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\na1 a2 b1 b2 : \u2115\nh : (a1 * a2 = m \u2227 \u00acm = 0) \u2227 b1 * b2 = n \u2227 \u00acn = 0\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (m * n).divisorsAntidiagonal"}, {"tactic": "rcases h with \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9", "annotated_tactic": ["rcases h with \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9", []], "state_before": "case hi.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\na1 a2 b1 b2 : \u2115\nh : (a1 * a2 = m \u2227 \u00acm = 0) \u2227 b1 * b2 = n \u2227 \u00acn = 0\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (m * n).divisorsAntidiagonal", "state_after": "case hi.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (a1 * a2 * (b1 * b2)).divisorsAntidiagonal"}, {"tactic": "simp only [mem_divisorsAntidiagonal, Nat.mul_eq_zero, Ne]", "annotated_tactic": ["simp only [<a>mem_divisorsAntidiagonal</a>, <a>Nat.mul_eq_zero</a>, <a>Ne</a>]", [{"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Nat.mul_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [430, 9], "def_end_pos": [430, 20]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}]], "state_before": "case hi.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 (match ((a1, a2), b1, b2) with\n    | ((i, j), k, l) => (i * k, j * l)) \u2208\n    (a1 * a2 * (b1 * b2)).divisorsAntidiagonal", "state_after": "case hi.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 a1 * b1 * (a2 * b2) = a1 * a2 * (b1 * b2) \u2227 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case hi.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 a1 * b1 * (a2 * b2) = a1 * a2 * (b1 * b2) \u2227 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)", "state_after": "case hi.mk.mk.mk.intro.intro.intro.left\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 a1 * b1 * (a2 * b2) = a1 * a2 * (b1 * b2)\n\ncase hi.mk.mk.mk.intro.intro.intro.right\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)"}, {"tactic": "rw [Nat.mul_eq_zero] at *", "annotated_tactic": ["rw [<a>Nat.mul_eq_zero</a>] at *", [{"full_name": "Nat.mul_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [430, 9], "def_end_pos": [430, 20]}]], "state_before": "case hi.mk.mk.mk.intro.intro.intro.right\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)", "state_after": "case hi.mk.mk.mk.intro.intro.intro.right\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac(a1 = 0 \u2228 a2 = 0)\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00ac(b1 = 0 \u2228 b2 = 0)\n\u22a2 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)"}, {"tactic": "apply not_or_of_not ha hb", "annotated_tactic": ["apply <a>not_or_of_not</a> ha hb", [{"full_name": "not_or_of_not", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [160, 9], "def_end_pos": [160, 22]}]], "state_before": "case hi.mk.mk.mk.intro.intro.intro.right\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac(a1 = 0 \u2228 a2 = 0)\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00ac(b1 = 0 \u2228 b2 = 0)\n\u22a2 \u00ac((a1 = 0 \u2228 a2 = 0) \u2228 b1 = 0 \u2228 b2 = 0)", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case hi.mk.mk.mk.intro.intro.intro.left\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00aca1 * a2 = 0\ncop : (a1 * a2).Coprime (b1 * b2)\nhb : \u00acb1 * b2 = 0\n\u22a2 a1 * b1 * (a2 * b2) = a1 * a2 * (b1 * b2)", "state_after": "no goals"}, {"tactic": "simp only [Set.InjOn, mem_coe, mem_divisorsAntidiagonal, Ne, mem_product, Prod.mk.inj_iff]", "annotated_tactic": ["simp only [<a>Set.InjOn</a>, <a>mem_coe</a>, <a>mem_divisorsAntidiagonal</a>, <a>Ne</a>, <a>mem_product</a>, <a>Prod.mk.inj_iff</a>]", [{"full_name": "Set.InjOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [290, 5], "def_end_pos": [290, 10]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}]], "state_before": "case i_inj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 Set.InjOn\n    (fun x =>\n      match x with\n      | ((i, j), k, l) => (i * k, j * l))\n    \u2191(m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal)", "state_after": "case i_inj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 \u2983x\u2081 : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115\u2984,\n    (x\u2081.1.1 * x\u2081.1.2 = m \u2227 \u00acm = 0) \u2227 x\u2081.2.1 * x\u2081.2.2 = n \u2227 \u00acn = 0 \u2192\n      \u2200 \u2983x\u2082 : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115\u2984,\n        (x\u2082.1.1 * x\u2082.1.2 = m \u2227 \u00acm = 0) \u2227 x\u2082.2.1 * x\u2082.2.2 = n \u2227 \u00acn = 0 \u2192\n          x\u2081.1.1 * x\u2081.2.1 = x\u2082.1.1 * x\u2082.2.1 \u2227 x\u2081.1.2 * x\u2081.2.2 = x\u2082.1.2 * x\u2082.2.2 \u2192 x\u2081 = x\u2082"}, {"tactic": "rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9 \u27e8\u27e8c1, c2\u27e9, \u27e8d1, d2\u27e9\u27e9 hcd h", "annotated_tactic": ["rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9 \u27e8\u27e8c1, c2\u27e9, \u27e8d1, d2\u27e9\u27e9 hcd h", []], "state_before": "case i_inj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 \u2983x\u2081 : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115\u2984,\n    (x\u2081.1.1 * x\u2081.1.2 = m \u2227 \u00acm = 0) \u2227 x\u2081.2.1 * x\u2081.2.2 = n \u2227 \u00acn = 0 \u2192\n      \u2200 \u2983x\u2082 : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115\u2984,\n        (x\u2082.1.1 * x\u2082.1.2 = m \u2227 \u00acm = 0) \u2227 x\u2082.2.1 * x\u2082.2.2 = n \u2227 \u00acn = 0 \u2192\n          x\u2081.1.1 * x\u2081.2.1 = x\u2082.1.1 * x\u2082.2.1 \u2227 x\u2081.1.2 * x\u2081.2.2 = x\u2082.1.2 * x\u2082.2.2 \u2192 x\u2081 = x\u2082", "state_after": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh :\n  ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).2.1 = ((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).2.1 \u2227\n    ((a1, a2), b1, b2).1.2 * ((a1, a2), b1, b2).2.2 = ((c1, c2), d1, d2).1.2 * ((c1, c2), d1, d2).2.2\n\u22a2 ((a1, a2), b1, b2) = ((c1, c2), d1, d2)"}, {"tactic": "simp only [Prod.mk.inj_iff] at h", "annotated_tactic": ["simp only [<a>Prod.mk.inj_iff</a>] at h", [{"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}]], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh :\n  ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).2.1 = ((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).2.1 \u2227\n    ((a1, a2), b1, b2).1.2 * ((a1, a2), b1, b2).2.2 = ((c1, c2), d1, d2).1.2 * ((c1, c2), d1, d2).2.2\n\u22a2 ((a1, a2), b1, b2) = ((c1, c2), d1, d2)", "state_after": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 ((a1, a2), b1, b2) = ((c1, c2), d1, d2)"}, {"tactic": "ext <;> dsimp only", "annotated_tactic": ["ext <;> dsimp only", []], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 ((a1, a2), b1, b2) = ((c1, c2), d1, d2)", "state_after": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a1 = c1\n\ncase i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a2 = c2\n\ncase i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b1 = d1\n\ncase i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b2 = d2"}, {"tactic": "trans Nat.gcd (a1 * a2) (a1 * b1)", "annotated_tactic": ["trans <a>Nat.gcd</a> (a1 * a2) (a1 * b1)", [{"full_name": "Nat.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a1 = c1", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a1 = (a1 * a2).gcd (a1 * b1)\n\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a1 * b1) = c1"}, {"tactic": "rw [Nat.gcd_mul_left, cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one]", "annotated_tactic": ["rw [<a>Nat.gcd_mul_left</a>, cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, <a>mul_one</a>]", [{"full_name": "Nat.gcd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [129, 9], "def_end_pos": [129, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a1 = (a1 * a2).gcd (a1 * b1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", "annotated_tactic": ["rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a1 * b1) = c1", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a1 * b1) = c1"}, {"tactic": "rw [\u2190 hcd.1.1, h.1, Nat.gcd_mul_left,\n  cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one]", "annotated_tactic": ["rw [\u2190 hcd.1.1, h.1, <a>Nat.gcd_mul_left</a>,\n          cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, <a>mul_one</a>]", [{"full_name": "Nat.gcd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [129, 9], "def_end_pos": [129, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a1 * b1) = c1", "state_after": "no goals"}, {"tactic": "trans Nat.gcd (a1 * a2) (a2 * b2)", "annotated_tactic": ["trans <a>Nat.gcd</a> (a1 * a2) (a2 * b2)", [{"full_name": "Nat.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a2 = c2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a2 = (a1 * a2).gcd (a2 * b2)\n\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a2 * b2) = c2"}, {"tactic": "rw [mul_comm, Nat.gcd_mul_left, cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one,\n  mul_one]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>Nat.gcd_mul_left</a>, cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one,\n          <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.gcd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [129, 9], "def_end_pos": [129, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 a2 = (a1 * a2).gcd (a2 * b2)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", "annotated_tactic": ["rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a2 * b2) = c2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a2 * b2) = c2"}, {"tactic": "rw [\u2190 hcd.1.1, h.2, mul_comm, Nat.gcd_mul_left,\n  cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one, mul_one]", "annotated_tactic": ["rw [\u2190 hcd.1.1, h.2, <a>mul_comm</a>, <a>Nat.gcd_mul_left</a>,\n          cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one, <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.gcd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [129, 9], "def_end_pos": [129, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (a1 * a2).gcd (a2 * b2) = c2", "state_after": "no goals"}, {"tactic": "trans Nat.gcd (b1 * b2) (a1 * b1)", "annotated_tactic": ["trans <a>Nat.gcd</a> (b1 * b2) (a1 * b1)", [{"full_name": "Nat.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b1 = d1", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b1 = (b1 * b2).gcd (a1 * b1)\n\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a1 * b1) = d1"}, {"tactic": "rw [mul_comm, Nat.gcd_mul_right,\n  cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, one_mul]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>Nat.gcd_mul_right</a>,\n          cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, <a>one_mul</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.gcd_mul_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [134, 9], "def_end_pos": [134, 22]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b1 = (b1 * b2).gcd (a1 * b1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", "annotated_tactic": ["rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a1 * b1) = d1", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a1 * b1) = d1"}, {"tactic": "rw [\u2190 hcd.2.1, h.1, mul_comm c1 d1, Nat.gcd_mul_left,\n  cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, mul_one]", "annotated_tactic": ["rw [\u2190 hcd.2.1, h.1, <a>mul_comm</a> c1 d1, <a>Nat.gcd_mul_left</a>,\n          cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, <a>mul_one</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.gcd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [129, 9], "def_end_pos": [129, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a1 * b1) = d1", "state_after": "no goals"}, {"tactic": "trans Nat.gcd (b1 * b2) (a2 * b2)", "annotated_tactic": ["trans <a>Nat.gcd</a> (b1 * b2) (a2 * b2)", [{"full_name": "Nat.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case i_inj.mk.mk.mk.intro.intro.intro.mk.mk.mk.a.a\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b2 = d2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b2 = (b1 * b2).gcd (a2 * b2)\n\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a2 * b2) = d2"}, {"tactic": "rw [Nat.gcd_mul_right, cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one,\n  one_mul]", "annotated_tactic": ["rw [<a>Nat.gcd_mul_right</a>, cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one,\n          <a>one_mul</a>]", [{"full_name": "Nat.gcd_mul_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [134, 9], "def_end_pos": [134, 22]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 b2 = (b1 * b2).gcd (a2 * b2)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", "annotated_tactic": ["rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a2 * b2) = d2", "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a2 * b2) = d2"}, {"tactic": "rw [\u2190 hcd.2.1, h.2, Nat.gcd_mul_right,\n  cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, one_mul]", "annotated_tactic": ["rw [\u2190 hcd.2.1, h.2, <a>Nat.gcd_mul_right</a>,\n          cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, <a>one_mul</a>]", [{"full_name": "Nat.gcd_mul_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [134, 9], "def_end_pos": [134, 22]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nc1 c2 d1 d2 : \u2115\ncop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Coprime (((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2)\nhcd :\n  (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2 = ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 \u2227\n      \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0) \u2227\n    ((c1, c2), d1, d2).2.1 * ((c1, c2), d1, d2).2.2 = ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 \u2227\n      \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\nh : a1 * b1 = c1 * d1 \u2227 a2 * b2 = c2 * d2\n\u22a2 (b1 * b2).gcd (a2 * b2) = d2", "state_after": "no goals"}, {"tactic": "simp only [Set.SurjOn, Set.subset_def, mem_coe, mem_divisorsAntidiagonal, Ne, mem_product,\n  Set.mem_image, exists_prop, Prod.mk.inj_iff]", "annotated_tactic": ["simp only [<a>Set.SurjOn</a>, <a>Set.subset_def</a>, <a>mem_coe</a>, <a>mem_divisorsAntidiagonal</a>, <a>Ne</a>, <a>mem_product</a>,\n      <a>Set.mem_image</a>, <a>exists_prop</a>, <a>Prod.mk.inj_iff</a>]", [{"full_name": "Set.SurjOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [298, 5], "def_end_pos": [298, 11]}, {"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [322, 9], "def_end_pos": [322, 19]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}]], "state_before": "case i_surj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 Set.SurjOn\n    (fun x =>\n      match x with\n      | ((i, j), k, l) => (i * k, j * l))\n    \u2191(m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal) \u2191(m * n).divisorsAntidiagonal", "state_after": "case i_surj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n    x.1 * x.2 = m * n \u2227 \u00acm * n = 0 \u2192\n      \u2203 x_1,\n        ((x_1.1.1 * x_1.1.2 = m \u2227 \u00acm = 0) \u2227 x_1.2.1 * x_1.2.2 = n \u2227 \u00acn = 0) \u2227 (x_1.1.1 * x_1.2.1, x_1.1.2 * x_1.2.2) = x"}, {"tactic": "rintro \u27e8b1, b2\u27e9 h", "annotated_tactic": ["rintro \u27e8b1, b2\u27e9 h", []], "state_before": "case i_surj\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n    x.1 * x.2 = m * n \u2227 \u00acm * n = 0 \u2192\n      \u2203 x_1,\n        ((x_1.1.1 * x_1.1.2 = m \u2227 \u00acm = 0) \u2227 x_1.2.1 * x_1.2.2 = n \u2227 \u00acn = 0) \u2227 (x_1.1.1 * x_1.2.1, x_1.1.2 * x_1.2.2) = x", "state_after": "case i_surj.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : (b1, b2).1 * (b1, b2).2 = m * n \u2227 \u00acm * n = 0\n\u22a2 \u2203 x, ((x.1.1 * x.1.2 = m \u2227 \u00acm = 0) \u2227 x.2.1 * x.2.2 = n \u2227 \u00acn = 0) \u2227 (x.1.1 * x.2.1, x.1.2 * x.2.2) = (b1, b2)"}, {"tactic": "dsimp at h", "annotated_tactic": ["dsimp at h", []], "state_before": "case i_surj.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : (b1, b2).1 * (b1, b2).2 = m * n \u2227 \u00acm * n = 0\n\u22a2 \u2203 x, ((x.1.1 * x.1.2 = m \u2227 \u00acm = 0) \u2227 x.2.1 * x.2.2 = n \u2227 \u00acn = 0) \u2227 (x.1.1 * x.2.1, x.1.2 * x.2.2) = (b1, b2)", "state_after": "case i_surj.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 \u2203 x, ((x.1.1 * x.1.2 = m \u2227 \u00acm = 0) \u2227 x.2.1 * x.2.2 = n \u2227 \u00acn = 0) \u2227 (x.1.1 * x.2.1, x.1.2 * x.2.2) = (b1, b2)"}, {"tactic": "use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n))", "annotated_tactic": ["use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n))", []], "state_before": "case i_surj.mk\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 \u2203 x, ((x.1.1 * x.1.2 = m \u2227 \u00acm = 0) \u2227 x.2.1 * x.2.2 = n \u2227 \u00acn = 0) \u2227 (x.1.1 * x.2.1, x.1.2 * x.2.2) = (b1, b2)", "state_after": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 = m \u2227 \u00acm = 0) \u2227\n      ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.2 = n \u2227 \u00acn = 0) \u2227\n    (((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.1,\n        ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.2) =\n      (b1, b2)"}, {"tactic": "rw [\u2190 cop.gcd_mul _, \u2190 cop.gcd_mul _, \u2190 h.1, Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop h.1,\n  Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop.symm _]", "annotated_tactic": ["rw [\u2190 cop.gcd_mul _, \u2190 cop.gcd_mul _, \u2190 h.1, <a>Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul</a> cop h.1,\n      <a>Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul</a> cop.symm _]", [{"full_name": "Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [178, 9], "def_end_pos": [178, 45]}, {"full_name": "Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [178, 9], "def_end_pos": [178, 45]}]], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 ((((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 = m \u2227 \u00acm = 0) \u2227\n      ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.2 = n \u2227 \u00acn = 0) \u2227\n    (((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.1 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.1,\n        ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).1.2 * ((b1.gcd m, b2.gcd m), b1.gcd n, b2.gcd n).2.2) =\n      (b1, b2)", "state_after": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 ((m = m \u2227 \u00acm = 0) \u2227 n = n \u2227 \u00acn = 0) \u2227 (b1.gcd (b1 * b2), b2.gcd (b1 * b2)) = (b1, b2)\n\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 b1 * b2 = n * m"}, {"tactic": "rw [mul_comm n m, h.1]", "annotated_tactic": ["rw [<a>mul_comm</a> n m, h.1]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 b1 * b2 = n * m", "state_after": "no goals"}, {"tactic": "rw [Nat.mul_eq_zero, not_or] at h", "annotated_tactic": ["rw [<a>Nat.mul_eq_zero</a>, <a>not_or</a>] at h", [{"full_name": "Nat.mul_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [430, 9], "def_end_pos": [430, 20]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm * n = 0\n\u22a2 ((m = m \u2227 \u00acm = 0) \u2227 n = n \u2227 \u00acn = 0) \u2227 (b1.gcd (b1 * b2), b2.gcd (b1 * b2)) = (b1, b2)", "state_after": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm = 0 \u2227 \u00acn = 0\n\u22a2 ((m = m \u2227 \u00acm = 0) \u2227 n = n \u2227 \u00acn = 0) \u2227 (b1.gcd (b1 * b2), b2.gcd (b1 * b2)) = (b1, b2)"}, {"tactic": "simp [h.2.1, h.2.2]", "annotated_tactic": ["simp [h.2.1, h.2.2]", []], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\nb1 b2 : \u2115\nh : b1 * b2 = m * n \u2227 \u00acm = 0 \u2227 \u00acn = 0\n\u22a2 ((m = m \u2227 \u00acm = 0) \u2227 n = n \u2227 \u00acn = 0) \u2227 (b1.gcd (b1 * b2), b2.gcd (b1 * b2)) = (b1, b2)", "state_after": "no goals"}, {"tactic": "simp only [mem_divisorsAntidiagonal, Ne, mem_product]", "annotated_tactic": ["simp only [<a>mem_divisorsAntidiagonal</a>, <a>Ne</a>, <a>mem_product</a>]", [{"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 a \u2208 m.divisorsAntidiagonal \u00d7\u02e2 n.divisorsAntidiagonal,\n    f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =\n      f\n          (match a with\n            | ((i, j), k, l) => (i * k, j * l)).1 *\n        g\n          (match a with\n            | ((i, j), k, l) => (i * k, j * l)).2", "state_after": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 (a : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115),\n    (a.1.1 * a.1.2 = m \u2227 \u00acm = 0) \u2227 a.2.1 * a.2.2 = n \u2227 \u00acn = 0 \u2192\n      f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) = f (a.1.1 * a.2.1) * g (a.1.2 * a.2.2)"}, {"tactic": "rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9", []], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : \u2115\ncop : m.Coprime n\n\u22a2 \u2200 (a : (\u2115 \u00d7 \u2115) \u00d7 \u2115 \u00d7 \u2115),\n    (a.1.1 * a.1.2 = m \u2227 \u00acm = 0) \u2227 a.2.1 * a.2.2 = n \u2227 \u00acn = 0 \u2192\n      f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) = f (a.1.1 * a.2.1) * g (a.1.2 * a.2.2)", "state_after": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f ((a1, a2), b1, b2).1.1 * g ((a1, a2), b1, b2).1.2 * (f ((a1, a2), b1, b2).2.1 * g ((a1, a2), b1, b2).2.2) =\n    f (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).2.1) * g (((a1, a2), b1, b2).1.2 * ((a1, a2), b1, b2).2.2)"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f ((a1, a2), b1, b2).1.1 * g ((a1, a2), b1, b2).1.2 * (f ((a1, a2), b1, b2).2.1 * g ((a1, a2), b1, b2).2.2) =\n    f (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).2.1) * g (((a1, a2), b1, b2).1.2 * ((a1, a2), b1, b2).2.2)", "state_after": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f a1 * g a2 * (f b1 * g b2) = f (a1 * b1) * g (a2 * b2)"}, {"tactic": "rw [hf.map_mul_of_coprime cop.coprime_mul_right.coprime_mul_right_right,\n  hg.map_mul_of_coprime cop.coprime_mul_left.coprime_mul_left_right]", "annotated_tactic": ["rw [hf.map_mul_of_coprime cop.coprime_mul_right.coprime_mul_right_right,\n      hg.map_mul_of_coprime cop.coprime_mul_left.coprime_mul_left_right]", []], "state_before": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f a1 * g a2 * (f b1 * g b2) = f (a1 * b1) * g (a2 * b2)", "state_after": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f a1 * g a2 * (f b1 * g b2) =\n    f ((a1, a2), b1, b2).1.1 * f ((a1, a2), b1, b2).2.1 * (g ((a1, a2), b1, b2).1.2 * g ((a1, a2), b1, b2).2.2)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.mk.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : \u2115\nha : \u00ac((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2)\nhb : \u00ac((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0\n\u22a2 f a1 * g a2 * (f b1 * g b2) =\n    f ((a1, a2), b1, b2).1.1 * f ((a1, a2), b1, b2).2.1 * (g ((a1, a2), b1, b2).1.2 * g ((a1, a2), b1, b2).2.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.range_inl", "start": [147, 1], "end": [154, 39], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 range (inl R M M\u2082) = ker (snd R M M\u2082)", "state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 x \u2208 range (inl R M M\u2082) \u2194 x \u2208 ker (snd R M M\u2082)"}, {"tactic": "simp only [mem_ker, mem_range]", "annotated_tactic": ["simp only [<a>mem_ker</a>, <a>mem_range</a>]", [{"full_name": "LinearMap.mem_ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [65, 9], "def_end_pos": [65, 16]}, {"full_name": "LinearMap.mem_range", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 x \u2208 range (inl R M M\u2082) \u2194 x \u2208 ker (snd R M M\u2082)", "state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, (inl R M M\u2082) y = x) \u2194 (snd R M M\u2082) x = 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, (inl R M M\u2082) y = x) \u2194 (snd R M M\u2082) x = 0", "state_after": "case h.mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, (inl R M M\u2082) y = x) \u2192 (snd R M M\u2082) x = 0\n\ncase h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (snd R M M\u2082) x = 0 \u2192 \u2203 y, (inl R M M\u2082) y = x"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, rfl\u27e9", []], "state_before": "case h.mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, (inl R M M\u2082) y = x) \u2192 (snd R M M\u2082) x = 0", "state_after": "case h.mp.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ny : M\n\u22a2 (snd R M M\u2082) ((inl R M M\u2082) y) = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.mp.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ny : M\n\u22a2 (snd R M M\u2082) ((inl R M M\u2082) y) = 0", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (snd R M M\u2082) x = 0 \u2192 \u2203 y, (inl R M M\u2082) y = x", "state_after": "case h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\nh : (snd R M M\u2082) x = 0\n\u22a2 \u2203 y, (inl R M M\u2082) y = x"}, {"tactic": "exact \u27e8x.fst, Prod.ext rfl h.symm\u27e9", "annotated_tactic": ["exact \u27e8x.fst, <a>Prod.ext</a> <a>rfl</a> h.symm\u27e9", [{"full_name": "Prod.ext", "def_path": ".lake/packages/lean4/src/lean/Init/Ext.lean", "def_pos": [101, 16], "def_end_pos": [101, 24]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\nh : (snd R M M\u2082) x = 0\n\u22a2 \u2203 y, (inl R M M\u2082) y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.coe_iSup\u2082", "start": [785, 1], "end": [786, 78], "traced_tactics": [{"tactic": "simp_rw [coe_iSup]", "annotated_tactic": ["simp_rw [<a>coe_iSup</a>]", [{"full_name": "LowerSet.coe_iSup", "def_path": "Mathlib/Order/UpperLower/Basic.lean", "def_pos": [775, 9], "def_end_pos": [775, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 \u2191(\u2a06 i, \u2a06 j, f i j) = \u22c3 i, \u22c3 j, \u2191(f i j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.linearIndependent_extend", "start": [1476, 1], "end": [1479, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.kreplace_nodupKeys", "start": [375, 1], "end": [376, 83], "traced_tactics": [{"tactic": "simp [NodupKeys, keys_kreplace]", "annotated_tactic": ["simp [<a>NodupKeys</a>, <a>keys_kreplace</a>]", [{"full_name": "List.NodupKeys", "def_path": "Mathlib/Data/List/Sigma.lean", "def_pos": [83, 5], "def_end_pos": [83, 14]}, {"full_name": "List.keys_kreplace", "def_path": "Mathlib/Data/List/Sigma.lean", "def_pos": [368, 9], "def_end_pos": [368, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\n\u22a2 (kreplace a b l).NodupKeys \u2194 l.NodupKeys", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Group/Abs.lean", "full_name": "sub_le_of_abs_sub_le_right", "start": [441, 1], "end": [442, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Kleene.lean", "full_name": "kstar_mono", "start": [242, 1], "end": [244, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "bihimp_comm", "start": [240, 1], "end": [240, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "full_name": "UniformContinuous.div", "start": [82, 1], "end": [84, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "full_name": "AkraBazziRecurrence.eventually_atTop_sumTransform_ge", "start": [680, 1], "end": [768, 44], "traced_tactics": [{"tactic": "obtain \u27e8c\u2081, hc\u2081_mem, hc\u2081\u27e9 := R.exists_eventually_const_mul_le_r", "annotated_tactic": ["obtain \u27e8c\u2081, hc\u2081_mem, hc\u2081\u27e9 := R.exists_eventually_const_mul_le_r", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "obtain \u27e8c\u2082, hc\u2082_mem, hc\u2082\u27e9 := R.g_grows_poly.eventually_atTop_ge_nat hc\u2081_mem", "annotated_tactic": ["obtain \u27e8c\u2082, hc\u2082_mem, hc\u2082\u27e9 := R.g_grows_poly.eventually_atTop_ge_nat hc\u2081_mem", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "obtain \u27e8c\u2083, hc\u2083_mem, hc\u2083\u27e9 := R.exists_eventually_r_le_const_mul", "annotated_tactic": ["obtain \u27e8c\u2083, hc\u2083_mem, hc\u2083\u27e9 := R.exists_eventually_r_le_const_mul", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "have hc\u2081_pos : 0 < c\u2081 := hc\u2081_mem.1", "annotated_tactic": ["have hc\u2081_pos : 0 < c\u2081 := hc\u2081_mem.1", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "have hc\u2083' : 0 < (1 - c\u2083) := by have := hc\u2083_mem.2; linarith", "annotated_tactic": ["have hc\u2083' : 0 < (1 - c\u2083) := by have := hc\u2083_mem.2; linarith", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "refine \u27e8min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081^((p a b) + 1)), by positivity, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>min</a> (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081^((<a>p</a> a b) + 1)), by positivity, ?_\u27e9", [{"full_name": "Min.min", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1142, 3], "def_end_pos": [1142, 6]}, {"full_name": "AkraBazziRecurrence.p", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [546, 31], "def_end_pos": [546, 32]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\n\u22a2 \u2203 c > 0, \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (i : \u03b1), min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "filter_upwards [hc\u2081, hc\u2082, hc\u2083, R.eventually_r_pos, R.eventually_r_lt_n, eventually_gt_atTop 0]\n  with n hn\u2081 hn\u2082 hn\u2083 hrpos hr_lt_n hn_pos", "annotated_tactic": ["filter_upwards [hc\u2081, hc\u2082, hc\u2083, R.eventually_r_pos, R.eventually_r_lt_n, <a>eventually_gt_atTop</a> 0]\n    with n hn\u2081 hn\u2082 hn\u2083 hrpos hr_lt_n hn_pos", [{"full_name": "Filter.eventually_gt_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [201, 9], "def_end_pos": [201, 28]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (i : \u03b1), min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\n\u22a2 \u2200 (i : \u03b1), min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\n\u22a2 \u2200 (i : \u03b1), min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "have hrpos_i := hrpos i", "annotated_tactic": ["have hrpos_i := hrpos i", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "have g_nonneg : 0 \u2264 g n := R.g_nonneg n (by positivity)", "annotated_tactic": ["have g_nonneg : 0 \u2264 g n := R.g_nonneg n (by positivity)", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n \u2264 sumTransform (p a b) g (r i n) n"}, {"tactic": "have := hc\u2083_mem.2", "annotated_tactic": ["have := hc\u2083_mem.2", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\n\u22a2 0 < 1 - c\u2083", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nthis : c\u2083 < 1\n\u22a2 0 < 1 - c\u2083"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nthis : c\u2083 < 1\n\u22a2 0 < 1 - c\u2083", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) > 0", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\n\u22a2 \u2191n \u2265 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 sumTransform (p a b) g (r i n) n = \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, g \u2191u / \u2191u ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "gcongr with u hu", "annotated_tactic": ["gcongr with u hu", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, g \u2191u / \u2191u ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191u ^ (p a b + 1)", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "have hu' : u \u2208 Set.Icc (r i n) n := \u27e8hu.1, by omega\u27e9", "annotated_tactic": ["have hu' : u \u2208 <a>Set.Icc</a> (r i n) n := \u27e8hu.1, by omega\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "refine hn\u2082 u ?_", "annotated_tactic": ["refine hn\u2082 u ?_", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n"}, {"tactic": "rw [Set.mem_Icc]", "annotated_tactic": ["rw [<a>Set.mem_Icc</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u \u2227 \u2191u \u2264 \u2191n"}, {"tactic": "refine \u27e8?_, by norm_cast; omega\u27e9", "annotated_tactic": ["refine \u27e8?_, by norm_cast; omega\u27e9", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u \u2227 \u2191u \u2264 \u2191n", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u"}, {"tactic": "calc c\u2081 * n \u2264 r i n      := by exact hn\u2081 i\n          _ \u2264 u            := by exact_mod_cast hu'.1", "annotated_tactic": ["calc c\u2081 * n \u2264 r i n      := by exact hn\u2081 i\n                          _ \u2264 u            := by exact_mod_cast hu'.1", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 u \u2264 n", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2264 \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 u \u2264 n"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 u \u2264 n", "state_after": "no goals"}, {"tactic": "exact hn\u2081 i", "annotated_tactic": ["exact hn\u2081 i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191(r i n)", "state_after": "no goals"}, {"tactic": "exact_mod_cast hu'.1", "annotated_tactic": ["exact_mod_cast hu'.1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191(r i n) \u2264 \u2191u", "state_after": "no goals"}, {"tactic": "gcongr with u hu", "annotated_tactic": ["gcongr with u hu", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191u ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 0 < \u2191u ^ (p a b + 1)\n\ncase h.h.h.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 u \u2264 n"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 0 < \u2191u ^ (p a b + 1)"}, {"tactic": "have := calc 0 < r i n := hrpos_i\n            _ \u2264 u := hu.1", "annotated_tactic": ["have := calc 0 < r i n := hrpos_i\n                              _ \u2264 u := hu.1", []], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nthis : 0 < u\n\u22a2 0 < \u2191u ^ (p a b + 1)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nthis : 0 < u\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.h.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 u \u2264 n", "state_after": "case h.h.h.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 u \u2264 n"}, {"tactic": "exact le_of_lt hu.2", "annotated_tactic": ["exact <a>le_of_lt</a> hu.2", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h.h.h.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 u \u2264 n", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191n ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) \u2264 \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)"}, {"tactic": "exact Finset.card_nsmul_le_sum _ _ _ (fun x _ => by rfl)", "annotated_tactic": ["exact <a>Finset.card_nsmul_le_sum</a> _ _ _ (fun x _ => by rfl)", [{"full_name": "Finset.card_nsmul_le_sum", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [225, 15], "def_end_pos": [225, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) \u2264 \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\nx : \u2115\nx\u271d : x \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n / \u2191n ^ (p a b + 1) \u2264 c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rw [nsmul_eq_mul, mul_assoc]", "annotated_tactic": ["rw [<a>nsmul_eq_mul</a>, <a>mul_assoc</a>]", [{"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [81, 15], "def_end_pos": [81, 34]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1))", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    \u2191n ^ p a b * (\u2191n - \u2191(r i n)) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1))", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191(Ico (r i n) n).card = \u2191n - \u2191(r i n)"}, {"tactic": "rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]", "annotated_tactic": ["rw [<a>Nat.card_Ico</a>, <a>Nat.cast_sub</a> (<a>le_of_lt</a> <| hr_lt_n i)]", [{"full_name": "Nat.card_Ico", "def_path": "Mathlib/Order/Interval/Finset/Nat.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "Nat.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191(Ico (r i n) n).card = \u2191n - \u2191(r i n)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * (\u2191n - \u2191(r i n)) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) \u2265\n    \u2191n ^ p a b * (\u2191n - c\u2083 * \u2191n) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1))", "state_after": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191(r i n) \u2264 c\u2083 * \u2191n"}, {"tactic": "exact hn\u2083 i", "annotated_tactic": ["exact hn\u2083 i", []], "state_before": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191(r i n) \u2264 c\u2083 * \u2191n", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * (\u2191n - c\u2083 * \u2191n) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1))", "state_after": "no goals"}, {"tactic": "rw [\u2190 Real.rpow_add_one (by positivity) (p a b)]", "annotated_tactic": ["rw [\u2190 <a>Real.rpow_add_one</a> (by positivity) (<a>p</a> a b)]", [{"full_name": "Real.rpow_add_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [451, 9], "def_end_pos": [451, 21]}, {"full_name": "AkraBazziRecurrence.p", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [546, 31], "def_end_pos": [546, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    c\u2082 * (1 - c\u2083) * g \u2191n * (\u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) * (1 - c\u2083) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    c\u2082 * (1 - c\u2083) * g \u2191n * (\u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) * (1 - c\u2083) * (c\u2082 * g \u2191n / \u2191n ^ (p a b + 1)) =\n    c\u2082 * (1 - c\u2083) * g \u2191n * (\u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1))", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "rw [div_self (by positivity), mul_one]", "annotated_tactic": ["rw [<a>div_self</a> (by positivity), <a>mul_one</a>]", [{"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [295, 15], "def_end_pos": [295, 23]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 c\u2082 * (1 - c\u2083) * g \u2191n * (\u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1)) = c\u2082 * (1 - c\u2083) * g \u2191n", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) \u2260 0", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 c\u2082 * (1 - c\u2083) * g \u2191n \u2265 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) \u2264 c\u2082 * (1 - c\u2083)"}, {"tactic": "exact min_le_left _ _", "annotated_tactic": ["exact <a>min_le_left</a> _ _", [{"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 \u2264 p a b + 1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) \u2264 c\u2082 * (1 - c\u2083)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 sumTransform (p a b) g (r i n) n = \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, g \u2191u / \u2191u ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "gcongr with u hu", "annotated_tactic": ["gcongr with u hu", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, g \u2191u / \u2191u ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191u ^ (p a b + 1)", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "have hu' : u \u2208 Set.Icc (r i n) n := \u27e8hu.1, by omega\u27e9", "annotated_tactic": ["have hu' : u \u2208 <a>Set.Icc</a> (r i n) n := \u27e8hu.1, by omega\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u"}, {"tactic": "refine hn\u2082 u ?_", "annotated_tactic": ["refine hn\u2082 u ?_", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2082 * g \u2191n \u2264 g \u2191u", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n"}, {"tactic": "rw [Set.mem_Icc]", "annotated_tactic": ["rw [<a>Set.mem_Icc</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u \u2227 \u2191u \u2264 \u2191n"}, {"tactic": "refine \u27e8?_, by norm_cast; omega\u27e9", "annotated_tactic": ["refine \u27e8?_, by norm_cast; omega\u27e9", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u \u2227 \u2191u \u2264 \u2191n", "state_after": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u"}, {"tactic": "calc c\u2081 * n \u2264 r i n := by exact hn\u2081 i\n          _ \u2264 u := by exact_mod_cast hu'.1", "annotated_tactic": ["calc c\u2081 * n \u2264 r i n := by exact hn\u2081 i\n                       _ \u2264 u := by exact_mod_cast hu'.1", []], "state_before": "case h.h.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191u", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 u \u2264 n", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191u \u2264 \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 u \u2264 n"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 u \u2264 n", "state_after": "no goals"}, {"tactic": "exact hn\u2081 i", "annotated_tactic": ["exact hn\u2081 i", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 c\u2081 * \u2191n \u2264 \u2191(r i n)", "state_after": "no goals"}, {"tactic": "exact_mod_cast hu'.1", "annotated_tactic": ["exact_mod_cast hu'.1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nhu' : u \u2208 Set.Icc (r i n) n\n\u22a2 \u2191(r i n) \u2264 \u2191u", "state_after": "no goals"}, {"tactic": "gcongr n^(p a b) * (Finset.Ico (r i n) n).sum (fun _ => c\u2082 * g n / ?_) with u hu", "annotated_tactic": ["gcongr n^(<a>p</a> a b) * (<a>Finset.Ico</a> (r i n) n).<a>sum</a> (fun _ => c\u2082 * g n / ?_) with u hu", [{"full_name": "AkraBazziRecurrence.p", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [546, 31], "def_end_pos": [546, 32]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Finset.sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191u ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 0 < \u2191u ^ (p a b + 1)\n\ncase h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 \u2191u ^ (p a b + 1) \u2264 \u2191(r i n) ^ (p a b + 1)"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 0 < \u2191u ^ (p a b + 1)"}, {"tactic": "have := calc 0 < r i n := hrpos_i\n            _ \u2264 u := hu.1", "annotated_tactic": ["have := calc 0 < r i n := hrpos_i\n                           _ \u2264 u := hu.1", []], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nthis : 0 < u\n\u22a2 0 < \u2191u ^ (p a b + 1)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case h.h.hc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\nthis : 0 < u\n\u22a2 0 < \u2191u ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_Ico] at hu", "annotated_tactic": ["rw [<a>Finset.mem_Ico</a>] at hu", [{"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : u \u2208 Ico (r i n) n\n\u22a2 \u2191u ^ (p a b + 1) \u2264 \u2191(r i n) ^ (p a b + 1)", "state_after": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 \u2191u ^ (p a b + 1) \u2264 \u2191(r i n) ^ (p a b + 1)"}, {"tactic": "exact rpow_le_rpow_of_exponent_nonpos (by positivity)\n  (by exact_mod_cast hu.1) (le_of_lt hp)", "annotated_tactic": ["exact <a>rpow_le_rpow_of_exponent_nonpos</a> (by positivity)\n                 (by exact_mod_cast hu.1) (<a>le_of_lt</a> hp)", [{"full_name": "Real.rpow_le_rpow_of_exponent_nonpos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [647, 9], "def_end_pos": [647, 40]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 \u2191u ^ (p a b + 1) \u2264 \u2191(r i n) ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 0 < \u2191(r i n)", "state_after": "no goals"}, {"tactic": "exact_mod_cast hu.1", "annotated_tactic": ["exact_mod_cast hu.1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nu : \u2115\nhu : r i n \u2264 u \u2227 u < n\n\u22a2 \u2191(r i n) \u2264 \u2191u", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1) \u2265\n    \u2191n ^ p a b * (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1))", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)) \u2264 \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)"}, {"tactic": "exact Finset.card_nsmul_le_sum _ _ _ (fun x _ => by rfl)", "annotated_tactic": ["exact <a>Finset.card_nsmul_le_sum</a> _ _ _ (fun x _ => by rfl)", [{"full_name": "Finset.card_nsmul_le_sum", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [225, 15], "def_end_pos": [225, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)) \u2264 \u2211 _u \u2208 Ico (r i n) n, c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\nx : \u2115\nx\u271d : x \u2208 Ico (r i n) n\n\u22a2 c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1) \u2264 c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "rw [nsmul_eq_mul, mul_assoc]", "annotated_tactic": ["rw [<a>nsmul_eq_mul</a>, <a>mul_assoc</a>]", [{"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [81, 15], "def_end_pos": [81, 34]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * (Ico (r i n) n).card \u2022 (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)) =\n    \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1))", "state_after": "no goals"}, {"tactic": "gcongr n^(p a b) * (Finset.Ico (r i n) n).card * (c\u2082 * g n / ?_)", "annotated_tactic": ["gcongr n^(<a>p</a> a b) * (<a>Finset.Ico</a> (r i n) n).<a>card</a> * (c\u2082 * g n / ?_)", [{"full_name": "AkraBazziRecurrence.p", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [546, 31], "def_end_pos": [546, 32]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [42, 5], "def_end_pos": [42, 9]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / \u2191(r i n) ^ (p a b + 1)) \u2265\n    \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1))", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(r i n) ^ (p a b + 1) \u2264 (c\u2081 * \u2191n) ^ (p a b + 1)"}, {"tactic": "exact rpow_le_rpow_of_exponent_nonpos (by positivity) (hn\u2081 i) (le_of_lt hp)", "annotated_tactic": ["exact <a>rpow_le_rpow_of_exponent_nonpos</a> (by positivity) (hn\u2081 i) (<a>le_of_lt</a> hp)", [{"full_name": "Real.rpow_le_rpow_of_exponent_nonpos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [647, 9], "def_end_pos": [647, 40]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(r i n) ^ (p a b + 1) \u2264 (c\u2081 * \u2191n) ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 0 < c\u2081 * \u2191n", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191(Ico (r i n) n).card * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1)) =\n    \u2191n ^ p a b * (\u2191n - \u2191(r i n)) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1))", "state_after": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(Ico (r i n) n).card = \u2191n - \u2191(r i n)"}, {"tactic": "rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]", "annotated_tactic": ["rw [<a>Nat.card_Ico</a>, <a>Nat.cast_sub</a> (<a>le_of_lt</a> <| hr_lt_n i)]", [{"full_name": "Nat.card_Ico", "def_path": "Mathlib/Order/Interval/Finset/Nat.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "Nat.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(Ico (r i n) n).card = \u2191n - \u2191(r i n)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * (\u2191n - \u2191(r i n)) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1)) \u2265\n    \u2191n ^ p a b * (\u2191n - c\u2083 * \u2191n) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1))", "state_after": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(r i n) \u2264 c\u2083 * \u2191n"}, {"tactic": "exact hn\u2083 i", "annotated_tactic": ["exact hn\u2083 i", []], "state_before": "case h.h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191(r i n) \u2264 c\u2083 * \u2191n", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * (\u2191n - c\u2083 * \u2191n) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1)) =\n    \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1))", "state_after": "no goals"}, {"tactic": "rw [Real.mul_rpow (by positivity) (by positivity)]", "annotated_tactic": ["rw [<a>Real.mul_rpow</a> (by positivity) (by positivity)]", [{"full_name": "Real.mul_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [482, 9], "def_end_pos": [482, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 * \u2191n) ^ (p a b + 1)) =\n    \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 ^ (p a b + 1) * \u2191n ^ (p a b + 1)))", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 0 \u2264 c\u2081", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 0 \u2264 \u2191n", "state_after": "no goals"}, {"tactic": "rw [\u2190 Real.rpow_add_one (by positivity) (p a b)]", "annotated_tactic": ["rw [\u2190 <a>Real.rpow_add_one</a> (by positivity) (<a>p</a> a b)]", [{"full_name": "Real.rpow_add_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [451, 9], "def_end_pos": [451, 21]}, {"full_name": "AkraBazziRecurrence.p", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [546, 31], "def_end_pos": [546, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ p a b * \u2191n * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 ^ (p a b + 1) * \u2191n ^ (p a b + 1))) =\n    \u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1) * (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 ^ (p a b + 1) * \u2191n ^ (p a b + 1))) =\n    \u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1) * (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) * (1 - c\u2083) * (c\u2082 * g \u2191n / (c\u2081 ^ (p a b + 1) * \u2191n ^ (p a b + 1))) =\n    \u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1) * (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1)", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "rw [div_self (by positivity), one_mul]", "annotated_tactic": ["rw [<a>div_self</a> (by positivity), <a>one_mul</a>]", [{"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [295, 15], "def_end_pos": [295, 23]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) / \u2191n ^ (p a b + 1) * (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1) =\n    (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1) * g \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1) = (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1) * g \u2191n"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 (1 - c\u2083) * c\u2082 * g \u2191n / c\u2081 ^ (p a b + 1) = (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1) * g \u2191n", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 \u2191n ^ (p a b + 1) \u2260 0", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1) * g \u2191n \u2265 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) * g \u2191n", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) \u2264 (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)"}, {"tactic": "exact min_le_right _ _", "annotated_tactic": ["exact <a>min_le_right</a> _ _", [{"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\nc\u2081 : \u211d\nhc\u2081_mem : c\u2081 \u2208 Set.Ioo 0 1\nhc\u2081 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nc\u2082 : \u211d\nhc\u2082_mem : c\u2082 > 0\nhc\u2082 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nc\u2083 : \u211d\nhc\u2083_mem : c\u2083 \u2208 Set.Ioo 0 1\nhc\u2083 : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhc\u2081_pos : 0 < c\u2081\nhc\u2083' : 0 < 1 - c\u2083\nn : \u2115\nhn\u2081 : \u2200 (i : \u03b1), c\u2081 * \u2191n \u2264 \u2191(r i n)\nhn\u2082 : \u2200 u \u2208 Set.Icc (c\u2081 * \u2191n) \u2191n, c\u2082 * g \u2191n \u2264 g u\nhn\u2083 : \u2200 (i : \u03b1), \u2191(r i n) \u2264 c\u2083 * \u2191n\nhrpos : \u2200 (i : \u03b1), 0 < r i n\nhr_lt_n : \u2200 (i : \u03b1), r i n < n\nhn_pos : 0 < n\ni : \u03b1\nhrpos_i : 0 < r i n\ng_nonneg : 0 \u2264 g \u2191n\nhp : 0 > p a b + 1\n\u22a2 min (c\u2082 * (1 - c\u2083)) ((1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)) \u2264 (1 - c\u2083) * c\u2082 / c\u2081 ^ (p a b + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/Basic.lean", "full_name": "ModularForm.neg_apply", "start": [222, 1], "end": [223, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "full_name": "iSup\u2082_disjoint_iff", "start": [230, 1], "end": [232, 30], "traced_tactics": [{"tactic": "simp_rw [iSup_disjoint_iff]", "annotated_tactic": ["simp_rw [<a>iSup_disjoint_iff</a>]", [{"full_name": "iSup_disjoint_iff", "def_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "def_pos": [222, 9], "def_end_pos": [222, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03ba : \u03b9 \u2192 Sort w'\ninst\u271d : Frame \u03b1\ns t : Set \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 Disjoint (\u2a06 i, \u2a06 j, f i j) a \u2194 \u2200 (i : \u03b9) (j : \u03ba i), Disjoint (f i j) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.map_toSubsemiring", "start": [460, 1], "end": [462, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.zero_apply", "start": [293, 1], "end": [293, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Submodule.lean", "full_name": "IsLocalization.coeSubmodule_strictMono", "start": [113, 1], "end": [115, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Lemmas.lean", "full_name": "Polynomial.degree_map_eq_iff", "start": [292, 1], "end": [301, 91], "traced_tactics": [{"tactic": "rcases eq_or_ne p 0 with h|h", "annotated_tactic": ["rcases <a>eq_or_ne</a> p 0 with h|h", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0 \u2228 p = 0", "state_after": "case inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p = 0\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0 \u2228 p = 0\n\ncase inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0 \u2228 p = 0"}, {"tactic": "simp only [h, or_false]", "annotated_tactic": ["simp only [h, <a>or_false</a>]", [{"full_name": "or_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [121, 17], "def_end_pos": [121, 25]}]], "state_before": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0 \u2228 p = 0", "state_after": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0"}, {"tactic": "refine \u27e8fun h2 \u21a6 ?_, degree_map_eq_of_leadingCoeff_ne_zero f\u27e9", "annotated_tactic": ["refine \u27e8fun h2 \u21a6 ?_, <a>degree_map_eq_of_leadingCoeff_ne_zero</a> f\u27e9", [{"full_name": "Polynomial.degree_map_eq_of_leadingCoeff_ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [896, 9], "def_end_pos": [896, 46]}]], "state_before": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\n\u22a2 (map f p).degree = p.degree \u2194 f p.leadingCoeff \u2260 0", "state_after": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\nh2 : (map f p).degree = p.degree\n\u22a2 f p.leadingCoeff \u2260 0"}, {"tactic": "have h3 : natDegree (map f p) = natDegree p := by simp_rw [natDegree, h2]", "annotated_tactic": ["have h3 : <a>natDegree</a> (<a>map</a> f p) = <a>natDegree</a> p := by simp_rw [<a>natDegree</a>, h2]", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}, {"full_name": "Polynomial.map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [713, 5], "def_end_pos": [713, 8]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}]], "state_before": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\nh2 : (map f p).degree = p.degree\n\u22a2 f p.leadingCoeff \u2260 0", "state_after": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\nh2 : (map f p).degree = p.degree\nh3 : (map f p).natDegree = p.natDegree\n\u22a2 f p.leadingCoeff \u2260 0"}, {"tactic": "have h4 : map f p \u2260 0 := by\n  rwa [ne_eq, \u2190 degree_eq_bot, h2, degree_eq_bot]", "annotated_tactic": ["have h4 : <a>map</a> f p \u2260 0 := by\n    rwa [<a>ne_eq</a>, \u2190 <a>degree_eq_bot</a>, h2, <a>degree_eq_bot</a>]", [{"full_name": "Polynomial.map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [713, 5], "def_end_pos": [713, 8]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "state_before": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\nh2 : (map f p).degree = p.degree\nh3 : (map f p).natDegree = p.natDegree\n\u22a2 f p.leadingCoeff \u2260 0", "state_after": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nh : p \u2260 0\nh2 : (map f p).degree = p.degree\nh3 : (map f p).natDegree = p.natDegree\nh4 : map f p \u2260 0\n\u22a2 f p.leadingCoeff \u2260 0"}, {"tactic": "rwa [\u2190 coeff_natDegree, \u2190 coeff_map, \u2190 h3, coeff_natDegree, ne_eq, leadingCoeff_eq_zero]", "annotated_tactic": ["rwa [\u2190 <a>coeff_natDegree</a>, \u2190 <a>coeff_map</a>, \u2190 h3, <a>coeff_natDegree</a>, <a>ne_eq</a>, <a>leadingCoeff_eq_zero</a>]", [{"full_name": "Polynomial.coeff_natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [113, 9], "def_end_pos": [113, 24]}, {"full_name": "Polynomial.coeff_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [803, 9], "def_end_pos": [803, 18]}, {"full_name": "Polynomial.coeff_natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [113, 9], "def_end_pos": [113, 24]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Polynomial.leadingCoeff_eq_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [688, 9], "def_end_pos": [688, 29]}]], "state_before": "case inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : 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[\u2190 inv_top, preimage, inv_inv, image_top]", "annotated_tactic": ["rwa [\u2190 <a>inv_top</a>, <a>preimage</a>, <a>inv_inv</a>, <a>image_top</a>]", [{"full_name": "Rel.inv_top", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [161, 9], "def_end_pos": [161, 16]}, {"full_name": "Rel.preimage", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [226, 5], "def_end_pos": [226, 13]}, {"full_name": "Rel.inv_inv", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [70, 9], "def_end_pos": [70, 16]}, {"full_name": "Rel.image_top", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [221, 9], "def_end_pos": [221, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : s.Nonempty\n\u22a2 \u22a4.preimage s = Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_union_iff", "start": [155, 1], "end": [158, 89], "traced_tactics": [{"tactic": "simp only [union_eq_iUnion, aemeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm]", "annotated_tactic": ["simp only [<a>union_eq_iUnion</a>, <a>aemeasurable_iUnion_iff</a>, <a>Bool.forall_bool</a>, <a>cond</a>, <a>and_comm</a>]", [{"full_name": "Set.union_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 24]}, {"full_name": "aemeasurable_iUnion_iff", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [149, 9], "def_end_pos": [149, 39]}, {"full_name": "Bool.forall_bool", "def_path": "Mathlib/Data/Bool/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 20]}, {"full_name": "cond", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1006, 21], "def_end_pos": [1006, 25]}, {"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nR : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u22a2 AEMeasurable f (\u03bc.restrict (s \u222a t)) \u2194 AEMeasurable f (\u03bc.restrict s) \u2227 AEMeasurable f (\u03bc.restrict t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineSubspace.subtype\u2090\u1d62_toAffineMap", "start": [300, 1], "end": [302, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/NoetherianSpace.lean", "full_name": "TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent", "start": [223, 1], "end": [264, 90], "traced_tactics": [{"tactic": "let \u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}", "annotated_tactic": ["let \u03b9 : <a>Set</a> (<a>Set</a> \u03b1) := <a>irreducibleComponents</a> \u03b1 \\ {Z}", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [107, 5], "def_end_pos": [107, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z"}, {"tactic": "have h\u03b9 : \u03b9.Finite := NoetherianSpace.finite_irreducibleComponents.subset Set.diff_subset", "annotated_tactic": ["have h\u03b9 : \u03b9.Finite := NoetherianSpace.finite_irreducibleComponents.subset <a>Set.diff_subset</a>", [{"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1782, 9], "def_end_pos": [1782, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z"}, {"tactic": "have h\u03b9' : Finite \u03b9 := by rwa [Set.finite_coe_iff]", "annotated_tactic": ["have h\u03b9' : <a>Finite</a> \u03b9 := by rwa [<a>Set.finite_coe_iff</a>]", [{"full_name": "Finite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [81, 17], "def_end_pos": [81, 23]}, {"full_name": "Set.finite_coe_iff", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [73, 9], "def_end_pos": [73, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z"}, {"tactic": "let U := Z \\ \u22c3 (x : \u03b9), x", "annotated_tactic": ["let U := Z \\ \u22c3 (x : \u03b9), x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z"}, {"tactic": "have hU0 : U \u2260 \u2205 := fun r \u21a6 by\n  obtain \u27e8Z', hZ'\u27e9 := isIrreducible_iff_sUnion_closed.mp H.1 h\u03b9.toFinset\n    (fun z hz \u21a6 by\n      simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hz\n      exact isClosed_of_mem_irreducibleComponents _ hz.1)\n    (by\n      rw [Set.Finite.coe_toFinset, Set.sUnion_eq_iUnion]\n      rw [Set.diff_eq_empty] at r\n      exact r)\n  simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hZ'\n  exact hZ'.1.2 <| le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2", "annotated_tactic": ["have hU0 : U \u2260 \u2205 := fun r \u21a6 by\n    obtain \u27e8Z', hZ'\u27e9 := isIrreducible_iff_sUnion_closed.mp H.1 h\u03b9.toFinset\n      (fun z hz \u21a6 by\n        simp only [<a>Set.Finite.mem_toFinset</a>, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>] at hz\n        exact <a>isClosed_of_mem_irreducibleComponents</a> _ hz.1)\n      (by\n        rw [<a>Set.Finite.coe_toFinset</a>, <a>Set.sUnion_eq_iUnion</a>]\n        rw [<a>Set.diff_eq_empty</a>] at r\n        exact r)\n    simp only [<a>Set.Finite.mem_toFinset</a>, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>] at hZ'\n    exact hZ'.1.2 <| <a>le_antisymm</a> (H.2 hZ'.1.1.1 hZ'.2) hZ'.2", [{"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "isClosed_of_mem_irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [111, 9], "def_end_pos": [111, 46]}, {"full_name": "Set.Finite.coe_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [171, 19], "def_end_pos": [171, 31]}, {"full_name": "Set.sUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1312, 9], "def_end_pos": [1312, 25]}, {"full_name": "Set.diff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1867, 9], "def_end_pos": [1867, 22]}, {"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z"}, {"tactic": "refine \u27e8U, hU1 \u25b8 isOpen_compl_iff.mpr ?_, hU0, sdiff_le\u27e9", "annotated_tactic": ["refine \u27e8U, hU1 \u25b8 isOpen_compl_iff.mpr ?_, hU0, <a>sdiff_le</a>\u27e9", [{"full_name": "sdiff_le", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\nhU1 : U = (\u22c3 x, \u2191x)\u1d9c\n\u22a2 \u2203 o, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\nhU1 : U = (\u22c3 x, \u2191x)\u1d9c\n\u22a2 IsClosed (\u22c3 x, \u2191x)"}, {"tactic": "exact isClosed_iUnion_of_finite fun i \u21a6 isClosed_of_mem_irreducibleComponents i.1 i.2.1", "annotated_tactic": ["exact <a>isClosed_iUnion_of_finite</a> fun i \u21a6 <a>isClosed_of_mem_irreducibleComponents</a> i.1 i.2.1", [{"full_name": "isClosed_iUnion_of_finite", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [229, 9], "def_end_pos": [229, 34]}, {"full_name": "isClosed_of_mem_irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [111, 9], "def_end_pos": [111, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\nhU1 : U = (\u22c3 x, \u2191x)\u1d9c\n\u22a2 IsClosed (\u22c3 x, \u2191x)", "state_after": "no goals"}, {"tactic": "rwa [Set.finite_coe_iff]", "annotated_tactic": ["rwa [<a>Set.finite_coe_iff</a>]", [{"full_name": "Set.finite_coe_iff", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [73, 9], "def_end_pos": [73, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\n\u22a2 Finite \u2191\u03b9", "state_after": "no goals"}, {"tactic": "obtain \u27e8Z', hZ'\u27e9 := isIrreducible_iff_sUnion_closed.mp H.1 h\u03b9.toFinset\n  (fun z hz \u21a6 by\n    simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hz\n    exact isClosed_of_mem_irreducibleComponents _ hz.1)\n  (by\n    rw [Set.Finite.coe_toFinset, Set.sUnion_eq_iUnion]\n    rw [Set.diff_eq_empty] at r\n    exact r)", "annotated_tactic": ["obtain \u27e8Z', hZ'\u27e9 := isIrreducible_iff_sUnion_closed.mp H.1 h\u03b9.toFinset\n      (fun z hz \u21a6 by\n        simp only [<a>Set.Finite.mem_toFinset</a>, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>] at hz\n        exact <a>isClosed_of_mem_irreducibleComponents</a> _ hz.1)\n      (by\n        rw [<a>Set.Finite.coe_toFinset</a>, <a>Set.sUnion_eq_iUnion</a>]\n        rw [<a>Set.diff_eq_empty</a>] at r\n        exact r)", [{"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "isClosed_of_mem_irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [111, 9], "def_end_pos": [111, 46]}, {"full_name": "Set.Finite.coe_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [171, 19], "def_end_pos": [171, 31]}, {"full_name": "Set.sUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1312, 9], "def_end_pos": [1312, 25]}, {"full_name": "Set.diff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1867, 9], "def_end_pos": [1867, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nZ' : Set \u03b1\nhZ' : Z' \u2208 h\u03b9.toFinset \u2227 Z \u2286 Z'\n\u22a2 False"}, {"tactic": "simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hZ'", "annotated_tactic": ["simp only [<a>Set.Finite.mem_toFinset</a>, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>] at hZ'", [{"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nZ' : Set \u03b1\nhZ' : Z' \u2208 h\u03b9.toFinset \u2227 Z \u2286 Z'\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nZ' : Set \u03b1\nhZ' : Z' \u2208 \u03b9 \u2227 Z \u2286 Z'\n\u22a2 False"}, {"tactic": "exact hZ'.1.2 <| le_antisymm (H.2 hZ'.1.1.1 hZ'.2) hZ'.2", "annotated_tactic": ["exact hZ'.1.2 <| <a>le_antisymm</a> (H.2 hZ'.1.1.1 hZ'.2) hZ'.2", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nZ' : Set \u03b1\nhZ' : Z' \u2208 \u03b9 \u2227 Z \u2286 Z'\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp only [Set.Finite.mem_toFinset, Set.mem_diff, Set.mem_singleton_iff] at hz", "annotated_tactic": ["simp only [<a>Set.Finite.mem_toFinset</a>, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>] at hz", [{"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nz : Set \u03b1\nhz : z \u2208 h\u03b9.toFinset\n\u22a2 IsClosed z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nz : Set \u03b1\nhz : z \u2208 \u03b9\n\u22a2 IsClosed z"}, {"tactic": "exact isClosed_of_mem_irreducibleComponents _ hz.1", "annotated_tactic": ["exact <a>isClosed_of_mem_irreducibleComponents</a> _ hz.1", [{"full_name": "isClosed_of_mem_irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [111, 9], "def_end_pos": [111, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\nz : Set \u03b1\nhz : z \u2208 \u03b9\n\u22a2 IsClosed z", "state_after": "no goals"}, {"tactic": "rw [Set.Finite.coe_toFinset, Set.sUnion_eq_iUnion]", "annotated_tactic": ["rw [<a>Set.Finite.coe_toFinset</a>, <a>Set.sUnion_eq_iUnion</a>]", [{"full_name": "Set.Finite.coe_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [171, 19], "def_end_pos": [171, 31]}, {"full_name": "Set.sUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1312, 9], "def_end_pos": [1312, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\n\u22a2 Z \u2286 \u22c3\u2080 \u2191h\u03b9.toFinset", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\n\u22a2 Z \u2286 \u22c3 i, \u2191i"}, {"tactic": "rw [Set.diff_eq_empty] at r", "annotated_tactic": ["rw [<a>Set.diff_eq_empty</a>] at r", [{"full_name": "Set.diff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1867, 9], "def_end_pos": [1867, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : U = \u2205\n\u22a2 Z \u2286 \u22c3 i, \u2191i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : Z \u2286 \u22c3 x, \u2191x\n\u22a2 Z \u2286 \u22c3 i, \u2191i"}, {"tactic": "exact r", "annotated_tactic": ["exact r", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nr : Z \u2286 \u22c3 x, \u2191x\n\u22a2 Z \u2286 \u22c3 i, \u2191i", "state_after": "no goals"}, {"tactic": "rw [Set.compl_eq_univ_diff]", "annotated_tactic": ["rw [<a>Set.compl_eq_univ_diff</a>]", [{"full_name": "Set.compl_eq_univ_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1858, 9], "def_end_pos": [1858, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 U = (\u22c3 x, \u2191x)\u1d9c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 U = Set.univ \\ \u22c3 x, \u2191x"}, {"tactic": "refine le_antisymm (Set.diff_subset_diff le_top <| subset_refl _) ?_", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>Set.diff_subset_diff</a> <a>le_top</a> <| <a>subset_refl</a> _) ?_", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Set.diff_subset_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1844, 9], "def_end_pos": [1844, 25]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}, {"full_name": "subset_refl", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [615, 7], "def_end_pos": [615, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 U = Set.univ \\ \u22c3 x, \u2191x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \\ \u22c3 x, \u2191x \u2264 U"}, {"tactic": "rw [\u2190 Set.compl_eq_univ_diff]", "annotated_tactic": ["rw [\u2190 <a>Set.compl_eq_univ_diff</a>]", [{"full_name": "Set.compl_eq_univ_diff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1858, 9], "def_end_pos": [1858, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \\ \u22c3 x, \u2191x \u2264 U", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 (\u22c3 x, \u2191x)\u1d9c \u2264 U"}, {"tactic": "refine Set.compl_subset_iff_union.mpr (le_antisymm le_top ?_)", "annotated_tactic": ["refine Set.compl_subset_iff_union.mpr (<a>le_antisymm</a> <a>le_top</a> ?_)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 (\u22c3 x, \u2191x)\u1d9c \u2264 U", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \u2264 (\u22c3 x, \u2191x) \u222a U"}, {"tactic": "rw [Set.union_comm, \u2190 Set.sUnion_eq_iUnion, \u2190 Set.sUnion_insert]", "annotated_tactic": ["rw [<a>Set.union_comm</a>, \u2190 <a>Set.sUnion_eq_iUnion</a>, \u2190 <a>Set.sUnion_insert</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [753, 9], "def_end_pos": [753, 19]}, {"full_name": "Set.sUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1312, 9], "def_end_pos": [1312, 25]}, {"full_name": "Set.sUnion_insert", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1131, 9], "def_end_pos": [1131, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \u2264 (\u22c3 x, \u2191x) \u222a U", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \u2264 \u22c3\u2080 insert U \u03b9"}, {"tactic": "rintro a -", "annotated_tactic": ["rintro a -", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\n\u22a2 Set.univ \u2264 \u22c3\u2080 insert U \u03b9", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9"}, {"tactic": "by_cases h : a \u2208 U", "annotated_tactic": ["by_cases h : a \u2208 U", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2208 U\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 U\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9"}, {"tactic": "exact \u27e8U, Set.mem_insert _ _, h\u27e9", "annotated_tactic": ["exact \u27e8U, <a>Set.mem_insert</a> _ _, h\u27e9", [{"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2208 U\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "no goals"}, {"tactic": "rw [Set.mem_diff, Decidable.not_and_iff_or_not_not, not_not, Set.mem_iUnion] at h", "annotated_tactic": ["rw [<a>Set.mem_diff</a>, <a>Decidable.not_and_iff_or_not_not</a>, <a>not_not</a>, <a>Set.mem_iUnion</a>] at h", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Decidable.not_and_iff_or_not_not", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [453, 9], "def_end_pos": [453, 41]}, {"full_name": "Classical.not_not", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [135, 17], "def_end_pos": [135, 24]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 U\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z \u2228 \u2203 i, a \u2208 \u2191i\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9"}, {"tactic": "rcases h with (h|\u27e8i, hi\u27e9)", "annotated_tactic": ["rcases h with (h|\u27e8i, hi\u27e9)", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z \u2228 \u2203 i, a \u2208 \u2191i\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9\n\ncase neg.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\ni : \u2191\u03b9\nhi : a \u2208 \u2191i\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9"}, {"tactic": "refine \u27e8irreducibleComponent a, Or.inr ?_, mem_irreducibleComponent\u27e9", "annotated_tactic": ["refine \u27e8<a>irreducibleComponent</a> a, <a>Or.inr</a> ?_, <a>mem_irreducibleComponent</a>\u27e9", [{"full_name": "irreducibleComponent", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [130, 5], "def_end_pos": [130, 25]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "mem_irreducibleComponent", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [141, 9], "def_end_pos": [141, 33]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 irreducibleComponent a \u2208 \u03b9"}, {"tactic": "simp only [\u03b9, Set.mem_diff, Set.mem_singleton_iff]", "annotated_tactic": ["simp only [\u03b9, <a>Set.mem_diff</a>, <a>Set.mem_singleton_iff</a>]", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 irreducibleComponent a \u2208 \u03b9", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 irreducibleComponent a \u2208 irreducibleComponents \u03b1 \u2227 \u00acirreducibleComponent a = Z"}, {"tactic": "refine \u27e8irreducibleComponent_mem_irreducibleComponents _, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>irreducibleComponent_mem_irreducibleComponents</a> _, ?_\u27e9", [{"full_name": "irreducibleComponent_mem_irreducibleComponents", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [154, 9], "def_end_pos": [154, 55]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 irreducibleComponent a \u2208 irreducibleComponents \u03b1 \u2227 \u00acirreducibleComponent a = Z", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 \u00acirreducibleComponent a = Z"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\nh : a \u2209 Z\n\u22a2 \u00acirreducibleComponent a = Z", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\na : \u03b1\nH : irreducibleComponent a \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {irreducibleComponent a}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := irreducibleComponent a \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\nh : a \u2209 irreducibleComponent a\n\u22a2 False"}, {"tactic": "exact h mem_irreducibleComponent", "annotated_tactic": ["exact h <a>mem_irreducibleComponent</a>", [{"full_name": "mem_irreducibleComponent", "def_path": "Mathlib/Topology/Irreducible.lean", "def_pos": [141, 9], "def_end_pos": [141, 33]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\na : \u03b1\nH : irreducibleComponent a \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {irreducibleComponent a}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := irreducibleComponent a \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\nh : a \u2209 irreducibleComponent a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact \u27e8i, Or.inr i.2, hi\u27e9", "annotated_tactic": ["exact \u27e8i, <a>Or.inr</a> i.2, hi\u27e9", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case neg.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nZ : Set \u03b1\nH : Z \u2208 irreducibleComponents \u03b1\n\u03b9 : Set (Set \u03b1) := irreducibleComponents \u03b1 \\ {Z}\nh\u03b9 : \u03b9.Finite\nh\u03b9' : Finite \u2191\u03b9\nU : Set \u03b1 := Z \\ \u22c3 x, \u2191x\nhU0 : U \u2260 \u2205\na : \u03b1\ni : \u2191\u03b9\nhi : a \u2208 \u2191i\n\u22a2 a \u2208 \u22c3\u2080 insert U \u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Intervals.lean", "full_name": "List.Ico.filter_le_of_top_le", "start": [192, 1], "end": [195, 57], "traced_tactics": [{"tactic": "rw [decide_eq_true_eq]", "annotated_tactic": ["rw [<a>decide_eq_true_eq</a>]", [{"full_name": "decide_eq_true_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [244, 17], "def_end_pos": [244, 34]}]], "state_before": "n m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acdecide (l \u2264 k) = true", "state_after": "n m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acl \u2264 k"}, {"tactic": "exact not_le_of_gt (lt_of_lt_of_le (mem.1 hk).2 hml)", "annotated_tactic": ["exact <a>not_le_of_gt</a> (<a>lt_of_lt_of_le</a> (<a>mem</a>.1 hk).2 hml)", [{"full_name": "not_le_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [148, 9], "def_end_pos": [148, 21]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "List.Ico.mem", "def_path": "Mathlib/Data/List/Intervals.lean", "def_pos": [62, 9], "def_end_pos": [62, 12]}]], "state_before": "n m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acl \u2264 k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean", "full_name": "TensorAlgebra.tprod_apply", "start": [333, 1], "end": [334, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.map_unop_mul", "start": [298, 1], "end": [306, 89], "traced_tactics": [{"tactic": "rw [\u2190 map_comp, map_op_mul, \u2190 map_comp, \u2190 map_comp, LinearEquiv.comp_coe,\n  LinearEquiv.symm_trans_self, LinearEquiv.refl_toLinearMap, map_id, map_id, map_id]", "annotated_tactic": ["rw [\u2190 <a>map_comp</a>, <a>map_op_mul</a>, \u2190 <a>map_comp</a>, \u2190 <a>map_comp</a>, <a>LinearEquiv.comp_coe</a>,\n      <a>LinearEquiv.symm_trans_self</a>, <a>LinearEquiv.refl_toLinearMap</a>, <a>map_id</a>, <a>map_id</a>, <a>map_id</a>]", [{"full_name": "Submodule.map_comp", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [106, 9], "def_end_pos": [106, 17]}, {"full_name": "Submodule.map_op_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [276, 9], "def_end_pos": [276, 19]}, {"full_name": "Submodule.map_comp", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [106, 9], "def_end_pos": [106, 17]}, {"full_name": "Submodule.map_comp", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [106, 9], "def_end_pos": [106, 17]}, {"full_name": "LinearEquiv.comp_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [482, 9], "def_end_pos": [482, 17]}, {"full_name": "LinearEquiv.symm_trans_self", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [471, 9], "def_end_pos": [471, 24]}, {"full_name": "LinearEquiv.refl_toLinearMap", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [477, 9], "def_end_pos": [477, 25]}, {"full_name": "Submodule.map_id", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [102, 9], "def_end_pos": [102, 15]}, {"full_name": "Submodule.map_id", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [102, 9], "def_end_pos": [102, 15]}, {"full_name": "Submodule.map_id", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [102, 9], "def_end_pos": [102, 15]}]], "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM\u271d N\u271d P Q : Submodule R A\nm n : A\nM N : Submodule R A\u1d50\u1d52\u1d56\nthis : Function.Injective \u21d1\u2191(opLinearEquiv R)\n\u22a2 map (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R).symm) (M * N)) =\n    map (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R).symm) N * map (\u2191(opLinearEquiv R).symm) M)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.toReal_le_of_le_ofReal", "start": [338, 1], "end": [341, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Const.lean", "full_name": "MvQPF.Const.get_map", "start": [69, 1], "end": [69, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measure_union_add_inter\u2080'", "start": [310, 1], "end": [312, 71], "traced_tactics": [{"tactic": "rw [union_comm, inter_comm, measure_union_add_inter\u2080 t hs, add_comm]", "annotated_tactic": ["rw [<a>union_comm</a>, <a>inter_comm</a>, <a>measure_union_add_inter\u2080</a> t hs, <a>add_comm</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [753, 9], "def_end_pos": [753, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}, {"full_name": "MeasureTheory.measure_union_add_inter\u2080", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [304, 9], "def_end_pos": [304, 33]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nhs : NullMeasurableSet s \u03bc\nt : Set \u03b1\n\u22a2 \u03bc (s \u222a t) + \u03bc (s \u2229 t) = \u03bc s + \u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/RiemannZeta.lean", "full_name": "riemannZeta_neg_two_mul_nat_add_one", "start": [158, 1], "end": [159, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Complex.abs_cpow_inv_nat", "start": [338, 1], "end": [339, 46], "traced_tactics": [{"tactic": "rw [\u2190 abs_cpow_real]", "annotated_tactic": ["rw [\u2190 <a>abs_cpow_real</a>]", [{"full_name": "Complex.abs_cpow_real", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [333, 9], "def_end_pos": [333, 22]}]], "state_before": "x : \u2102\nn : \u2115\n\u22a2 abs (x ^ (\u2191n)\u207b\u00b9) = abs x ^ (\u2191n)\u207b\u00b9", "state_after": "x : \u2102\nn : \u2115\n\u22a2 abs (x ^ (\u2191n)\u207b\u00b9) = abs (x ^ \u2191(\u2191n)\u207b\u00b9)"}, {"tactic": "simp [-abs_cpow_real]", "annotated_tactic": ["simp [-<a>abs_cpow_real</a>]", [{"full_name": "Complex.abs_cpow_real", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [333, 9], "def_end_pos": [333, 22]}]], "state_before": "x : \u2102\nn : \u2115\n\u22a2 abs (x ^ (\u2191n)\u207b\u00b9) = abs (x ^ \u2191(\u2191n)\u207b\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "contDiff_prod_mk_left", "start": [1735, 1], "end": [1736, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.add_pos_iff_pos_or_pos", "start": [310, 1], "end": [310, 69], "traced_tactics": [{"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\n\u22a2 0 < m + n \u2194 0 < m \u2228 0 < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Basic.lean", "full_name": "Finset.nonempty_Ico", "start": [62, 1], "end": [63, 49], "traced_tactics": [{"tactic": "rw [\u2190 coe_nonempty, coe_Ico, Set.nonempty_Ico]", "annotated_tactic": ["rw [\u2190 <a>coe_nonempty</a>, <a>coe_Ico</a>, <a>Set.nonempty_Ico</a>]", [{"full_name": "Finset.coe_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 21]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [338, 9], "def_end_pos": [338, 16]}, {"full_name": "Set.nonempty_Ico", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [271, 9], "def_end_pos": [271, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 (Ico a b).Nonempty \u2194 a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean", "full_name": "Topology.WithLowerSet.toLowerSet_inj", "start": [133, 1], "end": [133, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjointed.lean", "full_name": "disjointed_eq_inter_compl", "start": [173, 1], "end": [175, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "not_summable_of_ratio_norm_eventually_ge", "start": [604, 1], "end": [623, 11], "traced_tactics": [{"tactic": "rw [eventually_atTop] at h", "annotated_tactic": ["rw [<a>eventually_atTop</a>] at h", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2203\u1da0 (n : \u2115) in atTop, \u2016f n\u2016 \u2260 0\nh : \u2200\u1da0 (n : \u2115) in atTop, r * \u2016f n\u2016 \u2264 \u2016f (n + 1)\u2016\n\u22a2 \u00acSummable f", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2203\u1da0 (n : \u2115) in atTop, \u2016f n\u2016 \u2260 0\nh : \u2203 a, \u2200 b \u2265 a, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f"}, {"tactic": "rcases h with \u27e8N\u2080, hN\u2080\u27e9", "annotated_tactic": ["rcases h with \u27e8N\u2080, hN\u2080\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2203\u1da0 (n : \u2115) in atTop, \u2016f n\u2016 \u2260 0\nh : \u2203 a, \u2200 b \u2265 a, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f", "state_after": "case intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2203\u1da0 (n : \u2115) in atTop, \u2016f n\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f"}, {"tactic": "rw [frequently_atTop] at hf", "annotated_tactic": ["rw [<a>frequently_atTop</a>] at hf", [{"full_name": "Filter.frequently_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [365, 9], "def_end_pos": [365, 25]}]], "state_before": "case intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2203\u1da0 (n : \u2115) in atTop, \u2016f n\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f", "state_after": "case intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f"}, {"tactic": "rcases hf N\u2080 with \u27e8N, hNN\u2080 : N\u2080 \u2264 N, hN\u27e9", "annotated_tactic": ["rcases hf N\u2080 with \u27e8N, hNN\u2080 : N\u2080 \u2264 N, hN\u27e9", []], "state_before": "case intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\n\u22a2 \u00acSummable f", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\n\u22a2 \u00acSummable f"}, {"tactic": "rw [\u2190 @summable_nat_add_iff \u03b1 _ _ _ _ N]", "annotated_tactic": ["rw [\u2190 @<a>summable_nat_add_iff</a> \u03b1 _ _ _ _ N]", [{"full_name": "summable_nat_add_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\n\u22a2 \u00acSummable f", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\n\u22a2 \u00acSummable fun n => f (n + N)"}, {"tactic": "refine mt Summable.tendsto_atTop_zero\n  fun h' \u21a6 not_tendsto_atTop_of_tendsto_nhds (tendsto_norm_zero.comp h') ?_", "annotated_tactic": ["refine <a>mt</a> <a>Summable.tendsto_atTop_zero</a>\n    fun h' \u21a6 <a>not_tendsto_atTop_of_tendsto_nhds</a> (tendsto_norm_zero.comp h') ?_", [{"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "Summable.tendsto_atTop_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [307, 3], "def_end_pos": [307, 14]}, {"full_name": "not_tendsto_atTop_of_tendsto_nhds", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [438, 9], "def_end_pos": [438, 42]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\n\u22a2 \u00acSummable fun n => f (n + N)", "state_after": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) atTop atTop"}, {"tactic": "convert tendsto_atTop_of_geom_le _ hr _", "annotated_tactic": ["convert <a>tendsto_atTop_of_geom_le</a> _ hr _", [{"full_name": "tendsto_atTop_of_geom_le", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [216, 9], "def_end_pos": [216, 33]}]], "state_before": "case intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) atTop atTop", "state_after": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 0 < ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0\n\ncase intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), r * ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) n \u2264 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) (n + 1)"}, {"tactic": "refine lt_of_le_of_ne (norm_nonneg _) ?_", "annotated_tactic": ["refine <a>lt_of_le_of_ne</a> (<a>norm_nonneg</a> _) ?_", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [198, 9], "def_end_pos": [198, 23]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}]], "state_before": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 0 < ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0", "state_after": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 0 \u2260 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0"}, {"tactic": "intro h''", "annotated_tactic": ["intro h''", []], "state_before": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 0 \u2260 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0", "state_after": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nh'' : 0 = ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0\n\u22a2 False"}, {"tactic": "specialize hN\u2080 N hNN\u2080", "annotated_tactic": ["specialize hN\u2080 N hNN\u2080", []], "state_before": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nh'' : 0 = ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0\n\u22a2 False", "state_after": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 N : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nh'' : 0 = ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0\nhN\u2080 : r * \u2016f N\u2016 \u2264 \u2016f (N + 1)\u2016\n\u22a2 False"}, {"tactic": "simp only [comp_apply, zero_add] at h''", "annotated_tactic": ["simp only [<a>comp_apply</a>, <a>zero_add</a>] at h''", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 N : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nh'' : 0 = ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) 0\nhN\u2080 : r * \u2016f N\u2016 \u2264 \u2016f (N + 1)\u2016\n\u22a2 False", "state_after": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 N : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nhN\u2080 : r * \u2016f N\u2016 \u2264 \u2016f (N + 1)\u2016\nh'' : 0 = \u2016f N\u2016\n\u22a2 False"}, {"tactic": "exact hN h''.symm", "annotated_tactic": ["exact hN h''.symm", []], "state_before": "case intro.intro.intro.convert_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 N : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\nhN\u2080 : r * \u2016f N\u2016 \u2264 \u2016f (N + 1)\u2016\nh'' : 0 = \u2016f N\u2016\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), r * ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) n \u2264 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) (n + 1)", "state_after": "case intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 r * ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) i \u2264 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) (i + 1)"}, {"tactic": "dsimp only [comp_apply]", "annotated_tactic": ["dsimp only [<a>comp_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "case intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 r * ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) i \u2264 ((fun a => \u2016a\u2016) \u2218 fun n => f (n + N)) (i + 1)", "state_after": "case intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 r * \u2016f (i + N)\u2016 \u2264 \u2016f (i + 1 + N)\u2016"}, {"tactic": "convert hN\u2080 (i + N) (hNN\u2080.trans (N.le_add_left i)) using 3", "annotated_tactic": ["convert hN\u2080 (i + N) (hNN\u2080.trans (N.le_add_left i)) using 3", []], "state_before": "case intro.intro.intro.convert_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 r * \u2016f (i + N)\u2016 \u2264 \u2016f (i + 1 + N)\u2016", "state_after": "case h.e'_4.h.e'_3.h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 i + 1 + N = i + N + 1"}, {"tactic": "ac_rfl", "annotated_tactic": ["ac_rfl", []], "state_before": "case h.e'_4.h.e'_3.h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : SeminormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr : 1 < r\nhf : \u2200 (a : \u2115), \u2203 b \u2265 a, \u2016f b\u2016 \u2260 0\nN\u2080 : \u2115\nhN\u2080 : \u2200 b \u2265 N\u2080, r * \u2016f b\u2016 \u2264 \u2016f (b + 1)\u2016\nN : \u2115\nhNN\u2080 : N\u2080 \u2264 N\nhN : \u2016f N\u2016 \u2260 0\nh' : Tendsto (fun n => f (n + N)) atTop (\ud835\udcdd 0)\ni : \u2115\n\u22a2 i + 1 + N = i + N + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "full_name": "MultilinearMap.map_update_sum", "start": [655, 1], "end": [660, 39], "traced_tactics": [{"tactic": "induction' t using Finset.induction with a t has ih h", "annotated_tactic": ["induction' t using <a>Finset.induction</a> with a t has ih h", [{"full_name": "Finset.induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1267, 19], "def_end_pos": [1267, 28]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin n.succ \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1\u271d : \u03b9 \u2192 Type u_1\ng\u271d : (i : \u03b9) \u2192 \u03b1\u271d i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1\u271d i)\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b9\nt : Finset \u03b1\ni : \u03b9\ng : \u03b1 \u2192 M\u2081 i\nm : (i : \u03b9) \u2192 M\u2081 i\n\u22a2 f (update m i (\u2211 a \u2208 t, g a)) = \u2211 a \u2208 t, f (update m i (g a))", "state_after": "case empty\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin n.succ \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1\u271d : \u03b9 \u2192 Type u_1\ng\u271d : (i : \u03b9) \u2192 \u03b1\u271d i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1\u271d i)\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\ng : \u03b1 \u2192 M\u2081 i\nm : (i : \u03b9) \u2192 M\u2081 i\n\u22a2 f (update m i (\u2211 a \u2208 \u2205, g a)) = \u2211 a \u2208 \u2205, f (update m i (g a))\n\ncase insert\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin n.succ \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1\u271d : \u03b9 \u2192 Type u_1\ng\u271d : (i : \u03b9) \u2192 \u03b1\u271d i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1\u271d i)\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\ng : \u03b1 \u2192 M\u2081 i\nm : (i : \u03b9) \u2192 M\u2081 i\na : \u03b1\nt : Finset \u03b1\nhas : a \u2209 t\nih : f (update m i (\u2211 a \u2208 t, g a)) = \u2211 a \u2208 t, f (update m i (g a))\n\u22a2 f (update m i (\u2211 a \u2208 insert a t, g a)) = \u2211 a \u2208 insert a t, f (update m i (g a))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin n.succ \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1\u271d : \u03b9 \u2192 Type u_1\ng\u271d : (i : \u03b9) \u2192 \u03b1\u271d i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1\u271d i)\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\ng : \u03b1 \u2192 M\u2081 i\nm : (i : \u03b9) \u2192 M\u2081 i\n\u22a2 f (update m i (\u2211 a \u2208 \u2205, g a)) = \u2211 a \u2208 \u2205, f (update m i (g a))", "state_after": "no goals"}, {"tactic": "simp [Finset.sum_insert has, ih]", "annotated_tactic": ["simp [<a>Finset.sum_insert</a> has, ih]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [350, 3], "def_end_pos": [350, 14]}]], "state_before": "case insert\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin n.succ \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1\u271d : \u03b9 \u2192 Type u_1\ng\u271d : (i : \u03b9) \u2192 \u03b1\u271d i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1\u271d i)\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\ng : \u03b1 \u2192 M\u2081 i\nm : (i : \u03b9) \u2192 M\u2081 i\na : \u03b1\nt : Finset \u03b1\nhas : a \u2209 t\nih : f (update m i (\u2211 a \u2208 t, g a)) = \u2211 a \u2208 t, f (update m i (g a))\n\u22a2 f (update m i (\u2211 a \u2208 insert a t, g a)) = \u2211 a \u2208 insert a t, f (update m i (g a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsGreatest.upperBounds_eq", "start": [310, 1], "end": [311, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.casesOn'_some", "start": [379, 1], "end": [380, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.coe_eq_coe", "start": [132, 1], "end": [132, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Gradient/Basic.lean", "full_name": "HasDerivAt.hasGradientAt'", "start": [198, 1], "end": [199, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Subset.lean", "full_name": "Set.image_val_union_self_right_eq", "start": [106, 1], "end": [108, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "full_name": "Ordinal.log_eq_zero", "start": [347, 1], "end": [354, 92], "traced_tactics": [{"tactic": "rcases eq_or_ne o 0 with (rfl | ho)", "annotated_tactic": ["rcases <a>eq_or_ne</a> o 0 with (rfl | ho)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "b o : Ordinal.{u_1}\nhbo : o < b\n\u22a2 log b o = 0", "state_after": "case inl\nb : Ordinal.{u_1}\nhbo : 0 < b\n\u22a2 log b 0 = 0\n\ncase inr\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\n\u22a2 log b o = 0"}, {"tactic": "rcases le_or_lt b 1 with hb | hb", "annotated_tactic": ["rcases <a>le_or_lt</a> b 1 with hb | hb", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}]], "state_before": "case inr\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\n\u22a2 log b o = 0", "state_after": "case inr.inl\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\nhb : b \u2264 1\n\u22a2 log b o = 0\n\ncase inr.inr\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\nhb : 1 < b\n\u22a2 log b o = 0"}, {"tactic": "exact log_zero_right b", "annotated_tactic": ["exact <a>log_zero_right</a> b", [{"full_name": "Ordinal.log_zero_right", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [284, 9], "def_end_pos": [284, 23]}]], "state_before": "case inl\nb : Ordinal.{u_1}\nhbo : 0 < b\n\u22a2 log b 0 = 0", "state_after": "no goals"}, {"tactic": "rcases le_one_iff.1 hb with (rfl | rfl)", "annotated_tactic": ["rcases <a>le_one_iff</a>.1 hb with (rfl | rfl)", [{"full_name": "Ordinal.le_one_iff", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 19]}]], "state_before": "case inr.inl\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\nhb : b \u2264 1\n\u22a2 log b o = 0", "state_after": "case inr.inl.inl\no : Ordinal.{u_1}\nho : o \u2260 0\nhbo : o < 0\nhb : 0 \u2264 1\n\u22a2 log 0 o = 0\n\ncase inr.inl.inr\no : Ordinal.{u_1}\nho : o \u2260 0\nhbo : o < 1\nhb : 1 \u2264 1\n\u22a2 log 1 o = 0"}, {"tactic": "exact log_zero_left o", "annotated_tactic": ["exact <a>log_zero_left</a> o", [{"full_name": "Ordinal.log_zero_left", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [279, 9], "def_end_pos": [279, 22]}]], "state_before": "case inr.inl.inl\no : Ordinal.{u_1}\nho : o \u2260 0\nhbo : o < 0\nhb : 0 \u2264 1\n\u22a2 log 0 o = 0", "state_after": "no goals"}, {"tactic": "exact log_one_left o", "annotated_tactic": ["exact <a>log_one_left</a> o", [{"full_name": "Ordinal.log_one_left", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [295, 9], "def_end_pos": [295, 21]}]], "state_before": "case inr.inl.inr\no : Ordinal.{u_1}\nho : o \u2260 0\nhbo : o < 1\nhb : 1 \u2264 1\n\u22a2 log 1 o = 0", "state_after": "no goals"}, {"tactic": "rwa [\u2190 Ordinal.le_zero, \u2190 lt_succ_iff, succ_zero, \u2190 lt_opow_iff_log_lt hb ho, opow_one]", "annotated_tactic": ["rwa [\u2190 <a>Ordinal.le_zero</a>, \u2190 <a>lt_succ_iff</a>, <a>succ_zero</a>, \u2190 <a>lt_opow_iff_log_lt</a> hb ho, <a>opow_one</a>]", [{"full_name": "Ordinal.le_zero", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [400, 19], "def_end_pos": [400, 26]}, {"full_name": "Order.lt_succ_iff", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [356, 9], "def_end_pos": [356, 20]}, {"full_name": "Ordinal.succ_zero", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 18]}, {"full_name": "Ordinal.lt_opow_iff_log_lt", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [339, 9], "def_end_pos": [339, 27]}, {"full_name": "Ordinal.opow_one", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [78, 9], "def_end_pos": [78, 17]}]], "state_before": "case inr.inr\nb o : Ordinal.{u_1}\nhbo : o < b\nho : o \u2260 0\nhb : 1 < b\n\u22a2 log b o = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exponential.lean", "full_name": "hasFDerivAt_exp", "start": [187, 1], "end": [188, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.subset_insert", "start": [1223, 9], "end": [1223, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.snd_prod", "start": [1107, 1], "end": [1110, 67], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 (\u03bc.prod \u03bd).snd = \u03bd", "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 (\u03bc.prod \u03bd).snd s = \u03bd s"}, {"tactic": "rw [snd_apply hs, \u2190 univ_prod, prod_prod, measure_univ, one_mul]", "annotated_tactic": ["rw [<a>snd_apply</a> hs, \u2190 <a>univ_prod</a>, <a>prod_prod</a>, <a>measure_univ</a>, <a>one_mul</a>]", [{"full_name": "MeasureTheory.Measure.snd_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1083, 9], "def_end_pos": [1083, 18]}, {"full_name": "Set.univ_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [101, 9], "def_end_pos": [101, 18]}, {"full_name": "MeasureTheory.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 18]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [234, 3], "def_end_pos": [234, 15]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 (\u03bc.prod \u03bd).snd s = \u03bd s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "full_name": "UpperHalfPlane.dist_triangle", "start": [108, 11], "end": [114, 23], "traced_tactics": [{"tactic": "rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff,\n  div_mul_eq_mul_div]", "annotated_tactic": ["rw [<a>dist_le_iff_le_sinh</a>, <a>sinh_half_dist_add_dist</a>, <a>div_mul_eq_div_div</a> _ _ (<a>dist</a> _ _), <a>le_div_iff</a>,\n    <a>div_mul_eq_mul_div</a>]", [{"full_name": "UpperHalfPlane.dist_le_iff_le_sinh", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "def_pos": [91, 9], "def_end_pos": [91, 28]}, {"full_name": "UpperHalfPlane.sinh_half_dist_add_dist", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "def_pos": [76, 9], "def_end_pos": [76, 32]}, {"full_name": "div_mul_eq_div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [791, 9], "def_end_pos": [791, 27]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [82, 3], "def_end_pos": [82, 7]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}, {"full_name": "div_mul_eq_mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 27]}]], "state_before": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 dist a c \u2264 dist a b + dist b c", "state_after": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 dist \u2191a \u2191c * dist (\u2191b) ((starRingEnd \u2102) \u2191b) / (2 * \u221a(a.im * c.im)) \u2264\n    (dist \u2191a \u2191b * dist (\u2191c) ((starRingEnd \u2102) \u2191b) + dist \u2191b \u2191c * dist (\u2191a) ((starRingEnd \u2102) \u2191b)) / (2 * \u221a(a.im * c.im))\n\nz w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 0 < dist (\u2191b) ((starRingEnd \u2102) \u2191b)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 dist \u2191a \u2191c * dist (\u2191b) ((starRingEnd \u2102) \u2191b) / (2 * \u221a(a.im * c.im)) \u2264\n    (dist \u2191a \u2191b * dist (\u2191c) ((starRingEnd \u2102) \u2191b) + dist \u2191b \u2191c * dist (\u2191a) ((starRingEnd \u2102) \u2191b)) / (2 * \u221a(a.im * c.im))", "state_after": "case hab\nz w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 dist \u2191a \u2191c * dist (\u2191b) ((starRingEnd \u2102) \u2191b) \u2264\n    dist \u2191a \u2191b * dist (\u2191c) ((starRingEnd \u2102) \u2191b) + dist \u2191b \u2191c * dist (\u2191a) ((starRingEnd \u2102) \u2191b)"}, {"tactic": "exact EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : \u2102) b c (conj (b : \u2102))", "annotated_tactic": ["exact <a>EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist</a> (a : \u2102) b c (conj (b : \u2102))", [{"full_name": "EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist", "def_path": "Mathlib/Geometry/Euclidean/Inversion/Basic.lean", "def_pos": [186, 9], "def_end_pos": [186, 42]}]], "state_before": "case hab\nz w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 dist \u2191a \u2191c * dist (\u2191b) ((starRingEnd \u2102) \u2191b) \u2264\n    dist \u2191a \u2191b * dist (\u2191c) ((starRingEnd \u2102) \u2191b) + dist \u2191b \u2191c * dist (\u2191a) ((starRingEnd \u2102) \u2191b)", "state_after": "no goals"}, {"tactic": "rw [dist_comm, dist_pos, Ne, Complex.conj_eq_iff_im]", "annotated_tactic": ["rw [<a>dist_comm</a>, <a>dist_pos</a>, <a>Ne</a>, <a>Complex.conj_eq_iff_im</a>]", [{"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "dist_pos", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 17]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Complex.conj_eq_iff_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [612, 9], "def_end_pos": [612, 23]}]], "state_before": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 0 < dist (\u2191b) ((starRingEnd \u2102) \u2191b)", "state_after": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 \u00ac(\u2191b).im = 0"}, {"tactic": "exact b.im_ne_zero", "annotated_tactic": ["exact b.im_ne_zero", []], "state_before": "z w : \u210d\nr R : \u211d\na b c : \u210d\n\u22a2 \u00ac(\u2191b).im = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.Pivot.listTransvecCol_mul_last_row_drop", "start": [371, 1], "end": [380, 22], "traced_tactics": [{"tactic": "refine Nat.decreasingInduction' ?_ hk ?_", "annotated_tactic": ["refine <a>Nat.decreasingInduction'</a> ?_ hk ?_", [{"full_name": "Nat.decreasingInduction'", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [1034, 5], "def_end_pos": [1034, 25]}]], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 ((List.drop k (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "case refine_1\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 \u2200 k_1 < r,\n    k \u2264 k_1 \u2192\n      ((List.drop (k_1 + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i \u2192\n        ((List.drop k_1 (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n\ncase refine_2\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 ((List.drop r (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i"}, {"tactic": "intro n hn _ IH", "annotated_tactic": ["intro n hn _ IH", []], "state_before": "case refine_1\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 \u2200 k_1 < r,\n    k \u2264 k_1 \u2192\n      ((List.drop (k_1 + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i \u2192\n        ((List.drop k_1 (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n\u22a2 ((List.drop n (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i"}, {"tactic": "have hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn", "annotated_tactic": ["have hn' : n < (<a>listTransvecCol</a> M).<a>length</a> := by simpa [<a>listTransvecCol</a>] using hn", [{"full_name": "Matrix.Pivot.listTransvecCol", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [358, 5], "def_end_pos": [358, 20]}, {"full_name": "List.length", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2316, 5], "def_end_pos": [2316, 16]}, {"full_name": "Matrix.Pivot.listTransvecCol", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [358, 5], "def_end_pos": [358, 20]}]], "state_before": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n\u22a2 ((List.drop n (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\nhn' : n < (listTransvecCol M).length\n\u22a2 ((List.drop n (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i"}, {"tactic": "rw [List.drop_eq_getElem_cons hn']", "annotated_tactic": ["rw [<a>List.drop_eq_getElem_cons</a> hn']", [{"full_name": "List.drop_eq_getElem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1691, 9], "def_end_pos": [1691, 29]}]], "state_before": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\nhn' : n < (listTransvecCol M).length\n\u22a2 ((List.drop n (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\nhn' : n < (listTransvecCol M).length\n\u22a2 (((listTransvecCol M)[n] :: List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i"}, {"tactic": "simpa [listTransvecCol, Matrix.mul_assoc]", "annotated_tactic": ["simpa [<a>listTransvecCol</a>, <a>Matrix.mul_assoc</a>]", [{"full_name": "Matrix.Pivot.listTransvecCol", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [358, 5], "def_end_pos": [358, 20]}, {"full_name": "Matrix.mul_assoc", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1201, 19], "def_end_pos": [1201, 28]}]], "state_before": "case refine_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\nhn' : n < (listTransvecCol M).length\n\u22a2 (((listTransvecCol M)[n] :: List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "no goals"}, {"tactic": "simpa [listTransvecCol] using hn", "annotated_tactic": ["simpa [<a>listTransvecCol</a>] using hn", [{"full_name": "Matrix.Pivot.listTransvecCol", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [358, 5], "def_end_pos": [358, 20]}]], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i\n\u22a2 n < (listTransvecCol M).length", "state_after": "no goals"}, {"tactic": "simp only [listTransvecCol, List.length_ofFn, le_refl, List.drop_eq_nil_of_le, List.prod_nil,\n  Matrix.one_mul]", "annotated_tactic": ["simp only [<a>listTransvecCol</a>, <a>List.length_ofFn</a>, <a>le_refl</a>, <a>List.drop_eq_nil_of_le</a>, <a>List.prod_nil</a>,\n      <a>Matrix.one_mul</a>]", [{"full_name": "Matrix.Pivot.listTransvecCol", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [358, 5], "def_end_pos": [358, 20]}, {"full_name": "List.length_ofFn", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}, {"full_name": "List.drop_eq_nil_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [742, 9], "def_end_pos": [742, 26]}, {"full_name": "List.prod_nil", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [66, 9], "def_end_pos": [66, 17]}, {"full_name": "Matrix.one_mul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1150, 19], "def_end_pos": [1150, 26]}]], "state_before": "case refine_2\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 ((List.drop r (listTransvecCol M)).prod * M) (inr ()) i = M (inr ()) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.map_comp_map", "start": [1102, 1], "end": [1105, 94], "traced_tactics": [{"tactic": "apply algHom_ext", "annotated_tactic": ["apply <a>algHom_ext</a>", [{"full_name": "TrivSqZeroExt.algHom_ext", "def_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "def_pos": [910, 9], "def_end_pos": [910, 19]}]], "state_before": "S : Type u_1\nR R' : Type u\nM : Type v\ninst\u271d\u00b2\u2076 : CommSemiring S\ninst\u271d\u00b2\u2075 : Semiring R\ninst\u271d\u00b2\u2074 : CommSemiring R'\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Algebra S R\ninst\u271d\u00b2\u00b9 : Algebra S R'\ninst\u271d\u00b2\u2070 : Module S M\ninst\u271d\u00b9\u2079 : Module R M\ninst\u271d\u00b9\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2076 : IsScalarTower S R M\ninst\u271d\u00b9\u2075 : IsScalarTower S R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2074 : Module R' M\ninst\u271d\u00b9\u00b3 : Module R'\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b2 : IsCentralScalar R' M\ninst\u271d\u00b9\u00b9 : IsScalarTower S R' M\nA : Type u_2\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra S A\ninst\u271d\u2078 : Algebra R' A\nN : Type u_3\nP : Type u_4\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R' N\ninst\u271d\u2075 : Module R'\u1d50\u1d52\u1d56 N\ninst\u271d\u2074 : IsCentralScalar R' N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R' P\ninst\u271d\u00b9 : Module R'\u1d50\u1d52\u1d56 P\ninst\u271d : IsCentralScalar R' P\nf : M \u2192\u2097[R'] N\ng : N \u2192\u2097[R'] P\n\u22a2 map (g \u2218\u2097 f) = (map g).comp (map f)", "state_after": "case h\nS : Type u_1\nR R' : Type u\nM : Type v\ninst\u271d\u00b2\u2076 : CommSemiring S\ninst\u271d\u00b2\u2075 : Semiring R\ninst\u271d\u00b2\u2074 : CommSemiring R'\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Algebra S R\ninst\u271d\u00b2\u00b9 : Algebra S R'\ninst\u271d\u00b2\u2070 : Module S M\ninst\u271d\u00b9\u2079 : Module R M\ninst\u271d\u00b9\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2076 : IsScalarTower S R M\ninst\u271d\u00b9\u2075 : IsScalarTower S R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2074 : Module R' M\ninst\u271d\u00b9\u00b3 : Module R'\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b2 : IsCentralScalar R' M\ninst\u271d\u00b9\u00b9 : IsScalarTower S R' M\nA : Type u_2\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra S A\ninst\u271d\u2078 : Algebra R' A\nN : Type u_3\nP : Type u_4\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R' N\ninst\u271d\u2075 : Module R'\u1d50\u1d52\u1d56 N\ninst\u271d\u2074 : IsCentralScalar R' N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R' P\ninst\u271d\u00b9 : Module R'\u1d50\u1d52\u1d56 P\ninst\u271d : IsCentralScalar R' P\nf : M \u2192\u2097[R'] N\ng : N \u2192\u2097[R'] P\n\u22a2 \u2200 (m : M), (map (g \u2218\u2097 f)) (inr m) = ((map g).comp (map f)) (inr m)"}, {"tactic": "simp only [map_inr, LinearMap.coe_comp, Function.comp_apply, AlgHom.coe_comp, forall_const]", "annotated_tactic": ["simp only [<a>map_inr</a>, <a>LinearMap.coe_comp</a>, <a>Function.comp_apply</a>, <a>AlgHom.coe_comp</a>, <a>forall_const</a>]", [{"full_name": "TrivSqZeroExt.map_inr", "def_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "def_pos": [1066, 9], "def_end_pos": [1066, 16]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [557, 9], "def_end_pos": [557, 17]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "AlgHom.coe_comp", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [320, 9], "def_end_pos": [320, 17]}, {"full_name": "forall_const", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [265, 17], "def_end_pos": [265, 29]}]], "state_before": "case h\nS : Type u_1\nR R' : Type u\nM : Type v\ninst\u271d\u00b2\u2076 : CommSemiring S\ninst\u271d\u00b2\u2075 : Semiring R\ninst\u271d\u00b2\u2074 : CommSemiring R'\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Algebra S R\ninst\u271d\u00b2\u00b9 : Algebra S R'\ninst\u271d\u00b2\u2070 : Module S M\ninst\u271d\u00b9\u2079 : Module R M\ninst\u271d\u00b9\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2076 : IsScalarTower S R M\ninst\u271d\u00b9\u2075 : IsScalarTower S R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2074 : Module R' M\ninst\u271d\u00b9\u00b3 : Module R'\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b2 : IsCentralScalar R' M\ninst\u271d\u00b9\u00b9 : IsScalarTower S R' M\nA : Type u_2\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra S A\ninst\u271d\u2078 : Algebra R' A\nN : Type u_3\nP : Type u_4\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R' N\ninst\u271d\u2075 : Module R'\u1d50\u1d52\u1d56 N\ninst\u271d\u2074 : IsCentralScalar R' N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R' P\ninst\u271d\u00b9 : Module R'\u1d50\u1d52\u1d56 P\ninst\u271d : IsCentralScalar R' P\nf : M \u2192\u2097[R'] N\ng : N \u2192\u2097[R'] P\n\u22a2 \u2200 (m : M), (map (g \u2218\u2097 f)) (inr m) = ((map g).comp (map f)) (inr m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "Dense.biUnion_uniformity_ball", "start": [1000, 1], "end": [1004, 22], "traced_tactics": [{"tactic": "refine iUnion\u2082_eq_univ_iff.2 fun y => ?_", "annotated_tactic": ["refine <a>iUnion\u2082_eq_univ_iff</a>.2 fun y => ?_", [{"full_name": "Set.iUnion\u2082_eq_univ_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 28]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nU : Set (\u03b1 \u00d7 \u03b1)\nhs : Dense s\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u22c3 x \u2208 s, ball x U = univ", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nU : Set (\u03b1 \u00d7 \u03b1)\nhs : Dense s\nhU : U \u2208 \ud835\udce4 \u03b1\ny : \u03b1\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), y \u2208 ball i U"}, {"tactic": "rcases hs.inter_nhds_nonempty (mem_nhds_right y hU) with \u27e8x, hxs, hxy : (x, y) \u2208 U\u27e9", "annotated_tactic": ["rcases hs.inter_nhds_nonempty (<a>mem_nhds_right</a> y hU) with \u27e8x, hxs, hxy : (x, y) \u2208 U\u27e9", [{"full_name": "mem_nhds_right", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [827, 9], "def_end_pos": [827, 23]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nU : Set (\u03b1 \u00d7 \u03b1)\nhs : Dense s\nhU : U \u2208 \ud835\udce4 \u03b1\ny : \u03b1\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), y \u2208 ball i U", "state_after": "case intro.intro\n\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nU : Set (\u03b1 \u00d7 \u03b1)\nhs : Dense s\nhU : U \u2208 \ud835\udce4 \u03b1\ny x : \u03b1\nhxs : x \u2208 s\nhxy : (x, y) \u2208 U\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), y \u2208 ball i U"}, {"tactic": "exact \u27e8x, hxs, hxy\u27e9", "annotated_tactic": ["exact \u27e8x, hxs, hxy\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nU : Set (\u03b1 \u00d7 \u03b1)\nhs : Dense s\nhU : U \u2208 \ud835\udce4 \u03b1\ny x : \u03b1\nhxs : x \u2208 s\nhxy : (x, y) \u2208 U\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), y \u2208 ball i U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "units_smul_eq_self_iff", "start": [571, 1], "end": [573, 101], "traced_tactics": [{"tactic": "induction' v using Module.Ray.ind with v hv", "annotated_tactic": ["induction' v using <a>Module.Ray.ind</a> with v hv", [{"full_name": "Module.Ray.ind", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [259, 9], "def_end_pos": [259, 23]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nu : R\u02e3\nv : Module.Ray R M\n\u22a2 u \u2022 v = v \u2194 0 < \u2191u", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nu : R\u02e3\nv : M\nhv : v \u2260 0\n\u22a2 u \u2022 rayOfNeZero R v hv = rayOfNeZero R v hv \u2194 0 < \u2191u"}, {"tactic": "simp only [smul_rayOfNeZero, ray_eq_iff, Units.smul_def, sameRay_smul_left_iff_of_ne hv u.ne_zero]", "annotated_tactic": ["simp only [<a>smul_rayOfNeZero</a>, <a>ray_eq_iff</a>, <a>Units.smul_def</a>, <a>sameRay_smul_left_iff_of_ne</a> hv u.ne_zero]", [{"full_name": "smul_rayOfNeZero", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [337, 9], "def_end_pos": [337, 25]}, {"full_name": "ray_eq_iff", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [271, 9], "def_end_pos": [271, 19]}, {"full_name": "Units.smul_def", "def_path": "Mathlib/GroupTheory/GroupAction/Units.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}, {"full_name": "sameRay_smul_left_iff_of_ne", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [544, 9], "def_end_pos": [544, 36]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nu : R\u02e3\nv : M\nhv : v \u2260 0\n\u22a2 u \u2022 rayOfNeZero R v hv = rayOfNeZero R v hv \u2194 0 < \u2191u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Perm.lean", "full_name": "List.nodup_permutations'Aux_iff", "start": [838, 1], "end": [865, 38], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, nodup_permutations'Aux_of_not_mem _ _\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, <a>nodup_permutations'Aux_of_not_mem</a> _ _\u27e9", [{"full_name": "List.nodup_permutations'Aux_of_not_mem", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [825, 9], "def_end_pos": [825, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\n\u22a2 (permutations'Aux x s).Nodup \u2194 x \u2209 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\n\u22a2 x \u2209 s"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\n\u22a2 x \u2209 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\nH : x \u2208 s\n\u22a2 False"}, {"tactic": "obtain \u27e8k, hk, hk'\u27e9 := nthLe_of_mem H", "annotated_tactic": ["obtain \u27e8k, hk, hk'\u27e9 := <a>nthLe_of_mem</a> H", [{"full_name": "List.nthLe_of_mem", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\nH : x \u2208 s\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 False"}, {"tactic": "rw [nodup_iff_nthLe_inj] at h", "annotated_tactic": ["rw [<a>nodup_iff_nthLe_inj</a>] at h", [{"full_name": "List.nodup_iff_nthLe_inj", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [111, 9], "def_end_pos": [111, 28]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh : (permutations'Aux x s).Nodup\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 False"}, {"tactic": "refine k.succ_ne_self.symm $ h k (k + 1) ?_ ?_ ?_", "annotated_tactic": ["refine k.succ_ne_self.symm $ h k (k + 1) ?_ ?_ ?_", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 False", "state_after": "case intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 k < (permutations'Aux x s).length\n\ncase intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 k + 1 < (permutations'Aux x s).length\n\ncase intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 (permutations'Aux x s).nthLe k ?intro.intro.refine_1 = (permutations'Aux x s).nthLe (k + 1) ?intro.intro.refine_2"}, {"tactic": "rw [nthLe_permutations'Aux, nthLe_permutations'Aux]", "annotated_tactic": ["rw [<a>nthLe_permutations'Aux</a>, <a>nthLe_permutations'Aux</a>]", [{"full_name": "List.nthLe_permutations'Aux", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [783, 9], "def_end_pos": [783, 31]}, {"full_name": "List.nthLe_permutations'Aux", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [783, 9], "def_end_pos": [783, 31]}]], "state_before": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 (permutations'Aux x s).nthLe k \u22ef = (permutations'Aux x s).nthLe (k + 1) \u22ef", "state_after": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 insertNth k x s = insertNth (k + 1) x s"}, {"tactic": "have hl : length (insertNth k x s) = length (insertNth (k + 1) x s) := by\n  rw [length_insertNth _ _ hk.le, length_insertNth _ _ (Nat.succ_le_of_lt hk)]", "annotated_tactic": ["have hl : <a>length</a> (<a>insertNth</a> k x s) = <a>length</a> (<a>insertNth</a> (k + 1) x s) := by\n    rw [<a>length_insertNth</a> _ _ hk.le, <a>length_insertNth</a> _ _ (<a>Nat.succ_le_of_lt</a> hk)]", [{"full_name": "List.length", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2316, 5], "def_end_pos": [2316, 16]}, {"full_name": "List.insertNth", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [284, 5], "def_end_pos": [284, 14]}, {"full_name": "List.length", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2316, 5], "def_end_pos": [2316, 16]}, {"full_name": "List.insertNth", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [284, 5], "def_end_pos": [284, 14]}, {"full_name": "List.length_insertNth", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [46, 9], "def_end_pos": [46, 25]}, {"full_name": "List.length_insertNth", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [46, 9], "def_end_pos": [46, 25]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}]], "state_before": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 insertNth k x s = insertNth (k + 1) x s", "state_after": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\n\u22a2 insertNth k x s = insertNth (k + 1) x s"}, {"tactic": "refine ext_nthLe hl fun n hn hn' => ?_", "annotated_tactic": ["refine <a>ext_nthLe</a> hl fun n hn hn' => ?_", [{"full_name": "List.ext_nthLe", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1358, 9], "def_end_pos": [1358, 18]}]], "state_before": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\n\u22a2 insertNth k x s = insertNth (k + 1) x s", "state_after": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'"}, {"tactic": "rcases lt_trichotomy n k with (H | rfl | H)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> n k with (H | rfl | H)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}]], "state_before": "case intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'", "state_after": "case intro.intro.refine_3.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : n < k\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'\n\ncase intro.intro.refine_3.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nn : \u2115\nhk : n < s.length\nhk' : s.nthLe n hk = x\nhl : (insertNth n x s).length = (insertNth (n + 1) x s).length\nhn : n < (insertNth n x s).length\nhn' : n < (insertNth (n + 1) x s).length\n\u22a2 (insertNth n x s).nthLe n hn = (insertNth (n + 1) x s).nthLe n hn'\n\ncase intro.intro.refine_3.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : k < n\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'"}, {"tactic": "simpa [Nat.lt_succ_iff] using hk.le", "annotated_tactic": ["simpa [<a>Nat.lt_succ_iff</a>] using hk.le", [{"full_name": "Nat.lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [573, 19], "def_end_pos": [573, 30]}]], "state_before": "case intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 k < (permutations'Aux x s).length", "state_after": "no goals"}, {"tactic": "simpa using hk", "annotated_tactic": ["simpa using hk", []], "state_before": "case intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 k + 1 < (permutations'Aux x s).length", "state_after": "no goals"}, {"tactic": "rw [length_insertNth _ _ hk.le, length_insertNth _ _ (Nat.succ_le_of_lt hk)]", "annotated_tactic": ["rw [<a>length_insertNth</a> _ _ hk.le, <a>length_insertNth</a> _ _ (<a>Nat.succ_le_of_lt</a> hk)]", [{"full_name": "List.length_insertNth", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [46, 9], "def_end_pos": [46, 25]}, {"full_name": "List.length_insertNth", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [46, 9], "def_end_pos": [46, 25]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\n\u22a2 (insertNth k x s).length = (insertNth (k + 1) x s).length", "state_after": "no goals"}, {"tactic": "rw [nthLe_insertNth_of_lt _ _ _ _ H (H.trans hk),\n  nthLe_insertNth_of_lt _ _ _ _ (H.trans (Nat.lt_succ_self _))]", "annotated_tactic": ["rw [<a>nthLe_insertNth_of_lt</a> _ _ _ _ H (H.trans hk),\n      <a>nthLe_insertNth_of_lt</a> _ _ _ _ (H.trans (<a>Nat.lt_succ_self</a> _))]", [{"full_name": "List.nthLe_insertNth_of_lt", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [157, 9], "def_end_pos": [157, 30]}, {"full_name": "List.nthLe_insertNth_of_lt", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [157, 9], "def_end_pos": [157, 30]}, {"full_name": "Nat.lt_succ_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [358, 17], "def_end_pos": [358, 29]}]], "state_before": "case intro.intro.refine_3.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : n < k\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'", "state_after": "no goals"}, {"tactic": "rw [nthLe_insertNth_self _ _ _ hk.le, nthLe_insertNth_of_lt _ _ _ _ (Nat.lt_succ_self _) hk,\n  hk']", "annotated_tactic": ["rw [<a>nthLe_insertNth_self</a> _ _ _ hk.le, <a>nthLe_insertNth_of_lt</a> _ _ _ _ (<a>Nat.lt_succ_self</a> _) hk,\n      hk']", [{"full_name": "List.nthLe_insertNth_self", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [182, 9], "def_end_pos": [182, 29]}, {"full_name": "List.nthLe_insertNth_of_lt", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [157, 9], "def_end_pos": [157, 30]}, {"full_name": "Nat.lt_succ_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [358, 17], "def_end_pos": [358, 29]}]], "state_before": "case intro.intro.refine_3.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH : x \u2208 s\nn : \u2115\nhk : n < s.length\nhk' : s.nthLe n hk = x\nhl : (insertNth n x s).length = (insertNth (n + 1) x s).length\nhn : n < (insertNth n x s).length\nhn' : n < (insertNth (n + 1) x s).length\n\u22a2 (insertNth n x s).nthLe n hn = (insertNth (n + 1) x s).nthLe n hn'", "state_after": "no goals"}, {"tactic": "rcases (Nat.succ_le_of_lt H).eq_or_lt with (rfl | H')", "annotated_tactic": ["rcases (<a>Nat.succ_le_of_lt</a> H).<a>eq_or_lt</a> with (rfl | H')", [{"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [395, 7], "def_end_pos": [395, 21]}]], "state_before": "case intro.intro.refine_3.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : k < n\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'", "state_after": "case intro.intro.refine_3.inr.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = (insertNth (k + 1) x s).nthLe k.succ hn'\n\ncase intro.intro.refine_3.inr.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : k < n\nH' : k.succ < n\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'"}, {"tactic": "rw [nthLe_insertNth_self _ _ _ (Nat.succ_le_of_lt hk)]", "annotated_tactic": ["rw [<a>nthLe_insertNth_self</a> _ _ _ (<a>Nat.succ_le_of_lt</a> hk)]", [{"full_name": "List.nthLe_insertNth_self", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [182, 9], "def_end_pos": [182, 29]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}]], "state_before": "case intro.intro.refine_3.inr.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = (insertNth (k + 1) x s).nthLe k.succ hn'", "state_after": "case intro.intro.refine_3.inr.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = x"}, {"tactic": "convert hk' using 1", "annotated_tactic": ["convert hk' using 1", []], "state_before": "case intro.intro.refine_3.inr.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = x", "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = s.nthLe k hk"}, {"tactic": "exact nthLe_insertNth_add_succ _ _ _ 0 _", "annotated_tactic": ["exact <a>nthLe_insertNth_add_succ</a> _ _ _ 0 _", [{"full_name": "List.nthLe_insertNth_add_succ", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [205, 9], "def_end_pos": [205, 33]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nhn : k.succ < (insertNth k x s).length\nhn' : k.succ < (insertNth (k + 1) x s).length\nH : k < k.succ\n\u22a2 (insertNth k x s).nthLe k.succ hn = s.nthLe k hk", "state_after": "no goals"}, {"tactic": "obtain \u27e8m, rfl\u27e9 := Nat.exists_eq_add_of_lt H'", "annotated_tactic": ["obtain \u27e8m, rfl\u27e9 := <a>Nat.exists_eq_add_of_lt</a> H'", [{"full_name": "Nat.exists_eq_add_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}]], "state_before": "case intro.intro.refine_3.inr.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nn : \u2115\nhn : n < (insertNth k x s).length\nhn' : n < (insertNth (k + 1) x s).length\nH : k < n\nH' : k.succ < n\n\u22a2 (insertNth k x s).nthLe n hn = (insertNth (k + 1) x s).nthLe n hn'", "state_after": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn : k.succ + m + 1 < (insertNth k x s).length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn = (insertNth (k + 1) x s).nthLe (k.succ + m + 1) hn'"}, {"tactic": "erw [length_insertNth _ _ hk.le, Nat.succ_lt_succ_iff, Nat.succ_add] at hn", "annotated_tactic": ["erw [<a>length_insertNth</a> _ _ hk.le, <a>Nat.succ_lt_succ_iff</a>, <a>Nat.succ_add</a>] at hn", [{"full_name": "List.length_insertNth", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [46, 9], "def_end_pos": [46, 25]}, {"full_name": "Nat.succ_lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [637, 9], "def_end_pos": [637, 25]}, {"full_name": "Nat.succ_add", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 17]}]], "state_before": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn : k.succ + m + 1 < (insertNth k x s).length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn = (insertNth (k + 1) x s).nthLe (k.succ + m + 1) hn'", "state_after": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn\u271d = (insertNth (k + 1) x s).nthLe (k.succ + m + 1) hn'"}, {"tactic": "rw [nthLe_insertNth_add_succ]", "annotated_tactic": ["rw [<a>nthLe_insertNth_add_succ</a>]", [{"full_name": "List.nthLe_insertNth_add_succ", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [205, 9], "def_end_pos": [205, 33]}]], "state_before": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn\u271d = (insertNth (k + 1) x s).nthLe (k.succ + m + 1) hn'", "state_after": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn\u271d = s.nthLe (k + 1 + m) ?intro.intro.refine_3.inr.inr.inr.intro.hk'\n\ncase intro.intro.refine_3.inr.inr.inr.intro.hk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k + 1 + m < s.length"}, {"tactic": "convert nthLe_insertNth_add_succ s x k m.succ (by simpa using hn) using 2", "annotated_tactic": ["convert <a>nthLe_insertNth_add_succ</a> s x k m.succ (by simpa using hn) using 2", [{"full_name": "List.nthLe_insertNth_add_succ", "def_path": "Mathlib/Data/List/InsertNth.lean", "def_pos": [205, 9], "def_end_pos": [205, 33]}]], "state_before": "case intro.intro.refine_3.inr.inr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 (insertNth k x s).nthLe (k.succ + m + 1) hn\u271d = s.nthLe (k + 1 + m) ?intro.intro.refine_3.inr.inr.inr.intro.hk'", "state_after": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k.succ + m + 1 = k + m.succ + 1\n\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k + 1 + m = k + m.succ"}, {"tactic": "simpa using hn", "annotated_tactic": ["simpa using hn", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k + m.succ < s.length", "state_after": "no goals"}, {"tactic": "simp [Nat.add_assoc, Nat.add_left_comm]", "annotated_tactic": ["simp [<a>Nat.add_assoc</a>, <a>Nat.add_left_comm</a>]", [{"full_name": "Nat.add_assoc", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [168, 19], "def_end_pos": [168, 28]}, {"full_name": "Nat.add_left_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [173, 19], "def_end_pos": [173, 32]}]], "state_before": "case h.e'_2.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k.succ + m + 1 = k + m.succ + 1", "state_after": "no goals"}, {"tactic": "simp [Nat.add_left_comm, Nat.add_comm]", "annotated_tactic": ["simp [<a>Nat.add_left_comm</a>, <a>Nat.add_comm</a>]", [{"full_name": "Nat.add_left_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [173, 19], "def_end_pos": [173, 32]}, {"full_name": "Nat.add_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [160, 19], "def_end_pos": [160, 27]}]], "state_before": "case h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k + 1 + m = k + m.succ", "state_after": "no goals"}, {"tactic": "simpa [Nat.succ_add] using hn", "annotated_tactic": ["simpa [<a>Nat.succ_add</a>] using hn", [{"full_name": "Nat.succ_add", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 17]}]], "state_before": "case intro.intro.refine_3.inr.inr.inr.intro.hk'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : List \u03b1\nx : \u03b1\nh :\n  \u2200 (i j : \u2115) (h\u2081 : i < (permutations'Aux x s).length) (h\u2082 : j < (permutations'Aux x s).length),\n    (permutations'Aux x s).nthLe i h\u2081 = (permutations'Aux x s).nthLe j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < s.length\nhk' : s.nthLe k hk = x\nhl : (insertNth k x s).length = (insertNth (k + 1) x s).length\nm : \u2115\nhn\u271d : k.succ + m + 1 < (insertNth k x s).length\nhn : (k + m).succ < s.length\nhn' : k.succ + m + 1 < (insertNth (k + 1) x s).length\nH : k < k.succ + m + 1\nH' : k.succ < k.succ + m + 1\n\u22a2 k + 1 + m < s.length", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.liftRel_bind", "start": [1136, 1], "end": [1151, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "equiv_iff_sameRay", "start": [246, 1], "end": [247, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finite/Card.lean", "full_name": "Finite.card_le_of_injective", "start": [98, 1], "end": [102, 91], "traced_tactics": [{"tactic": "haveI := Fintype.ofFinite \u03b2", "annotated_tactic": ["haveI := <a>Fintype.ofFinite</a> \u03b2", [{"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [449, 19], "def_end_pos": [449, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis : Fintype \u03b2\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2"}, {"tactic": "haveI := Fintype.ofInjective f hf", "annotated_tactic": ["haveI := <a>Fintype.ofInjective</a> f hf", [{"full_name": "Fintype.ofInjective", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [566, 19], "def_end_pos": [566, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis : Fintype \u03b2\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis\u271d : Fintype \u03b2\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2"}, {"tactic": "simpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_le_of_injective f hf", "annotated_tactic": ["simpa only [<a>Nat.card_eq_fintype_card</a>, <a>ge_iff_le</a>] using <a>Fintype.card_le_of_injective</a> f hf", [{"full_name": "Nat.card_eq_fintype_card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [40, 9], "def_end_pos": [40, 29]}, {"full_name": "ge_iff_le", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1947, 17], "def_end_pos": [1947, 26]}, {"full_name": "Fintype.card_le_of_injective", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [483, 9], "def_end_pos": [483, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis\u271d : Fintype \u03b2\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bornology/Hom.lean", "full_name": "LocallyBoundedMap.id_comp", "start": [184, 1], "end": [185, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean", "full_name": "EllipticCurve.ofJ1728_j", "start": [703, 1], "end": [706, 12], "traced_tactics": [{"tactic": "field_simp [j, ofJ1728, @val_unitOfInvertible _ _ _ <| invertibleNeg _,\n  WeierstrassCurve.ofJ1728_c\u2084]", "annotated_tactic": ["field_simp [<a>j</a>, <a>ofJ1728</a>, @<a>val_unitOfInvertible</a> _ _ _ <| <a>invertibleNeg</a> _,\n    <a>WeierstrassCurve.ofJ1728_c\u2084</a>]", [{"full_name": "EllipticCurve.j", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean", "def_pos": [578, 5], "def_end_pos": [578, 6]}, {"full_name": "EllipticCurve.ofJ1728", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean", "def_pos": [698, 5], "def_end_pos": [698, 12]}, {"full_name": "val_unitOfInvertible", "def_path": "Mathlib/Algebra/Group/Invertible/Basic.lean", "def_pos": [25, 3], "def_end_pos": [25, 8]}, {"full_name": "invertibleNeg", "def_path": "Mathlib/Algebra/Ring/Invertible.lean", "def_pos": [20, 5], "def_end_pos": [20, 18]}, {"full_name": "WeierstrassCurve.ofJ1728_c\u2084", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean", "def_pos": [533, 7], "def_end_pos": [533, 17]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\n\u22a2 (ofJ1728 R).j = 1728", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\n\u22a2 -(1728 * 2 ^ 6) = (-48) ^ 3"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\n\u22a2 -(1728 * 2 ^ 6) = (-48) ^ 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "full_name": "LieModule.mem_weightSpace", "start": [180, 1], "end": [182, 40], "traced_tactics": [{"tactic": "simp [weightSpace, mem_weightSpaceOf]", "annotated_tactic": ["simp [<a>weightSpace</a>, <a>mem_weightSpaceOf</a>]", [{"full_name": "LieModule.weightSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [177, 5], "def_end_pos": [177, 16]}, {"full_name": "LieModule.mem_weightSpaceOf", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [164, 9], "def_end_pos": [164, 26]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : L \u2192 R\nm : M\n\u22a2 m \u2208 weightSpace M \u03c7 \u2194 \u2200 (x : L), \u2203 k, (((toEnd R L M) x - \u03c7 x \u2022 1) ^ k) m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.induction_on", "start": [367, 11], "end": [370, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "IsClosed.exists_closed_singleton", "start": [371, 1], "end": [375, 42], "traced_tactics": [{"tactic": "obtain \u27e8V, Vsub, Vne, Vcls, hV\u27e9 := hS.exists_minimal_nonempty_closed_subset hne", "annotated_tactic": ["obtain \u27e8V, Vsub, Vne, Vcls, hV\u27e9 := hS.exists_minimal_nonempty_closed_subset hne", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T0Space X\ninst\u271d : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\n\u22a2 \u2203 x \u2208 S, IsClosed {x}", "state_after": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T0Space X\ninst\u271d : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\nV : Set X\nVsub : V \u2286 S\nVne : V.Nonempty\nVcls : IsClosed V\nhV : \u2200 V' \u2286 V, V'.Nonempty \u2192 IsClosed V' \u2192 V' = V\n\u22a2 \u2203 x \u2208 S, IsClosed {x}"}, {"tactic": "rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with \u27e8x, rfl\u27e9", "annotated_tactic": ["rcases <a>minimal_nonempty_closed_eq_singleton</a> Vcls Vne hV with \u27e8x, rfl\u27e9", [{"full_name": "minimal_nonempty_closed_eq_singleton", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [363, 9], "def_end_pos": [363, 45]}]], "state_before": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T0Space X\ninst\u271d : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\nV : Set X\nVsub : V \u2286 S\nVne : V.Nonempty\nVcls : IsClosed V\nhV : \u2200 V' \u2286 V, V'.Nonempty \u2192 IsClosed V' \u2192 V' = V\n\u22a2 \u2203 x \u2208 S, IsClosed {x}", "state_after": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T0Space X\ninst\u271d : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\nx : X\nVsub : {x} \u2286 S\nVne : {x}.Nonempty\nVcls : IsClosed {x}\nhV : \u2200 V' \u2286 {x}, V'.Nonempty \u2192 IsClosed V' \u2192 V' = {x}\n\u22a2 \u2203 x \u2208 S, IsClosed {x}"}, {"tactic": "exact \u27e8x, Vsub (mem_singleton x), Vcls\u27e9", "annotated_tactic": ["exact \u27e8x, Vsub (<a>mem_singleton</a> x), Vcls\u27e9", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T0Space X\ninst\u271d : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\nx : X\nVsub : {x} \u2286 S\nVne : {x}.Nonempty\nVcls : IsClosed {x}\nhV : \u2200 V' \u2286 {x}, V'.Nonempty \u2192 IsClosed V' \u2192 V' = {x}\n\u22a2 \u2203 x \u2208 S, IsClosed {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_inf_distrib_right", "start": [497, 1], "end": [498, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "fderivWithin_csin", "start": [405, 1], "end": [407, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.coe_ofLe", "start": [633, 1], "end": [634, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean", "full_name": "Polynomial.cyclotomic_expand_eq_cyclotomic_mul", "start": [36, 1], "end": [72, 85], "traced_tactics": [{"tactic": "rcases Nat.eq_zero_or_pos n with (rfl | hnpos)", "annotated_tactic": ["rcases <a>Nat.eq_zero_or_pos</a> n with (rfl | hnpos)", [{"full_name": "Nat.eq_zero_or_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 23]}]], "state_before": "p n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R", "state_after": "case inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d : CommRing R\nhdiv : \u00acp \u2223 0\n\u22a2 (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R * cyclotomic 0 R\n\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R"}, {"tactic": "haveI := NeZero.of_pos hnpos", "annotated_tactic": ["haveI := <a>NeZero.of_pos</a> hnpos", [{"full_name": "NeZero.of_pos", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [101, 9], "def_end_pos": [101, 15]}]], "state_before": "case inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R", "state_after": "case inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R"}, {"tactic": "suffices expand \u2124 p (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124 by\n  rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, Polynomial.map_mul, map_cyclotomic_int,\n    map_cyclotomic]", "annotated_tactic": ["suffices <a>expand</a> \u2124 p (<a>cyclotomic</a> n \u2124) = <a>cyclotomic</a> (n * p) \u2124 * <a>cyclotomic</a> n \u2124 by\n    rw [\u2190 <a>map_cyclotomic_int</a>, \u2190 <a>map_expand</a>, this, <a>Polynomial.map_mul</a>, <a>map_cyclotomic_int</a>,\n      <a>map_cyclotomic</a>]", [{"full_name": "Polynomial.expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [36, 19], "def_end_pos": [36, 25]}, {"full_name": "Polynomial.cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [253, 5], "def_end_pos": [253, 15]}, {"full_name": "Polynomial.cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [253, 5], "def_end_pos": [253, 15]}, {"full_name": "Polynomial.cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [253, 5], "def_end_pos": [253, 15]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 23]}]], "state_before": "case inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R", "state_after": "case inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 (expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124"}, {"tactic": "refine eq_of_monic_of_dvd_of_natDegree_le ((cyclotomic.monic _ \u2124).mul (cyclotomic.monic _ \u2124))\n  ((cyclotomic.monic n \u2124).expand hp.pos) ?_ ?_", "annotated_tactic": ["refine <a>eq_of_monic_of_dvd_of_natDegree_le</a> ((<a>cyclotomic.monic</a> _ \u2124).<a>mul</a> (<a>cyclotomic.monic</a> _ \u2124))\n    ((<a>cyclotomic.monic</a> n \u2124).<a>expand</a> hp.pos) ?_ ?_", [{"full_name": "Polynomial.eq_of_monic_of_dvd_of_natDegree_le", "def_path": "Mathlib/Algebra/Polynomial/RingDivision.lean", "def_pos": [670, 9], "def_end_pos": [670, 43]}, {"full_name": "Polynomial.cyclotomic.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 25]}, {"full_name": "Polynomial.Monic.mul", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}, {"full_name": "Polynomial.cyclotomic.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 25]}, {"full_name": "Polynomial.cyclotomic.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 25]}, {"full_name": "Polynomial.Monic.expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [182, 11], "def_end_pos": [182, 23]}]], "state_before": "case inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 (expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124", "state_after": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u2124 * cyclotomic n \u2124 \u2223 (expand \u2124 p) (cyclotomic n \u2124)\n\ncase inr.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 ((expand \u2124 p) (cyclotomic n \u2124)).natDegree \u2264 (cyclotomic (n * p) \u2124 * cyclotomic n \u2124).natDegree"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d : CommRing R\nhdiv : \u00acp \u2223 0\n\u22a2 (expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R * cyclotomic 0 R", "state_after": "no goals"}, {"tactic": "rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, Polynomial.map_mul, map_cyclotomic_int,\n  map_cyclotomic]", "annotated_tactic": ["rw [\u2190 <a>map_cyclotomic_int</a>, \u2190 <a>map_expand</a>, this, <a>Polynomial.map_mul</a>, <a>map_cyclotomic_int</a>,\n      <a>map_cyclotomic</a>]", [{"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 23]}]], "state_before": "p n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis\u271d : NeZero n\nthis : (expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124\n\u22a2 (expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R", "state_after": "no goals"}, {"tactic": "refine (IsPrimitive.Int.dvd_iff_map_cast_dvd_map_cast _ _\n  (IsPrimitive.mul (cyclotomic.isPrimitive (n * p) \u2124) (cyclotomic.isPrimitive n \u2124))\n  ((cyclotomic.monic n \u2124).expand hp.pos).isPrimitive).2 ?_", "annotated_tactic": ["refine (<a>IsPrimitive.Int.dvd_iff_map_cast_dvd_map_cast</a> _ _\n      (<a>IsPrimitive.mul</a> (<a>cyclotomic.isPrimitive</a> (n * p) \u2124) (<a>cyclotomic.isPrimitive</a> n \u2124))\n      ((<a>cyclotomic.monic</a> n \u2124).<a>expand</a> hp.pos).<a>isPrimitive</a>).2 ?_", [{"full_name": "Polynomial.IsPrimitive.Int.dvd_iff_map_cast_dvd_map_cast", "def_path": "Mathlib/RingTheory/Polynomial/GaussLemma.lean", "def_pos": [333, 9], "def_end_pos": [333, 54]}, {"full_name": "Polynomial.IsPrimitive.mul", "def_path": "Mathlib/RingTheory/Polynomial/Content.lean", "def_pos": [403, 9], "def_end_pos": [403, 24]}, {"full_name": "Polynomial.cyclotomic.isPrimitive", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [326, 9], "def_end_pos": [326, 31]}, {"full_name": "Polynomial.cyclotomic.isPrimitive", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [326, 9], "def_end_pos": [326, 31]}, {"full_name": "Polynomial.cyclotomic.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 25]}, {"full_name": "Polynomial.Monic.expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [182, 11], "def_end_pos": [182, 23]}, {"full_name": "Polynomial.Monic.isPrimitive", "def_path": "Mathlib/RingTheory/Polynomial/Content.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u2124 * cyclotomic n \u2124 \u2223 (expand \u2124 p) (cyclotomic n \u2124)", "state_after": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 map (Int.castRingHom \u211a) (cyclotomic (n * p) \u2124 * cyclotomic n \u2124) \u2223\n    map (Int.castRingHom \u211a) ((expand \u2124 p) (cyclotomic n \u2124))"}, {"tactic": "rw [Polynomial.map_mul, map_cyclotomic_int, map_cyclotomic_int, map_expand, map_cyclotomic_int]", "annotated_tactic": ["rw [<a>Polynomial.map_mul</a>, <a>map_cyclotomic_int</a>, <a>map_cyclotomic_int</a>, <a>map_expand</a>, <a>map_cyclotomic_int</a>]", [{"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}]], "state_before": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 map (Int.castRingHom \u211a) (cyclotomic (n * p) \u2124 * cyclotomic n \u2124) \u2223\n    map (Int.castRingHom \u211a) ((expand \u2124 p) (cyclotomic n \u2124))", "state_after": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a * cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "refine IsCoprime.mul_dvd (cyclotomic.isCoprime_rat fun h => ?_) ?_ ?_", "annotated_tactic": ["refine <a>IsCoprime.mul_dvd</a> (<a>cyclotomic.isCoprime_rat</a> fun h => ?_) ?_ ?_", [{"full_name": "IsCoprime.mul_dvd", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 26]}, {"full_name": "Polynomial.cyclotomic.isCoprime_rat", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [213, 9], "def_end_pos": [213, 33]}]], "state_before": "case inr.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a * cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n\n\u22a2 False\n\ncase inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)\n\ncase inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "replace h : n * p = n * 1 := by simp [h]", "annotated_tactic": ["replace h : n * p = n * 1 := by simp [h]", []], "state_before": "case inr.refine_1.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n\n\u22a2 False", "state_after": "case inr.refine_1.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n * 1\n\u22a2 False"}, {"tactic": "exact Nat.Prime.ne_one hp (mul_left_cancel\u2080 hnpos.ne' h)", "annotated_tactic": ["exact <a>Nat.Prime.ne_one</a> hp (<a>mul_left_cancel\u2080</a> hnpos.ne' h)", [{"full_name": "Nat.Prime.ne_one", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [85, 9], "def_end_pos": [85, 21]}, {"full_name": "mul_left_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [52, 9], "def_end_pos": [52, 25]}]], "state_before": "case inr.refine_1.refine_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n * 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "p n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n\n\u22a2 n * p = n * 1", "state_after": "no goals"}, {"tactic": "have hpos : 0 < n * p := mul_pos hnpos hp.pos", "annotated_tactic": ["have hpos : 0 < n * p := <a>mul_pos</a> hnpos hp.pos", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [406, 7], "def_end_pos": [406, 14]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "have hprim := Complex.isPrimitiveRoot_exp _ hpos.ne'", "annotated_tactic": ["have hprim := <a>Complex.isPrimitiveRoot_exp</a> _ hpos.ne'", [{"full_name": "Complex.isPrimitiveRoot_exp", "def_path": "Mathlib/RingTheory/RootsOfUnity/Complex.lean", "def_pos": [53, 9], "def_end_pos": [53, 28]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "rw [cyclotomic_eq_minpoly_rat hprim hpos]", "annotated_tactic": ["rw [<a>cyclotomic_eq_minpoly_rat</a> hprim hpos]", [{"full_name": "Polynomial.cyclotomic_eq_minpoly_rat", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [192, 9], "def_end_pos": [192, 34]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 cyclotomic (n * p) \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "refine minpoly.dvd \u211a _ ?_", "annotated_tactic": ["refine <a>minpoly.dvd</a> \u211a _ ?_", [{"full_name": "minpoly.dvd", "def_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "def_pos": [68, 9], "def_end_pos": [68, 12]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 (aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)))) ((expand \u211a p) (cyclotomic n \u211a)) = 0"}, {"tactic": "rw [aeval_def, \u2190 eval_map, map_expand, map_cyclotomic, expand_eval, \u2190 IsRoot.def,\n  @isRoot_cyclotomic_iff]", "annotated_tactic": ["rw [<a>aeval_def</a>, \u2190 <a>eval_map</a>, <a>map_expand</a>, <a>map_cyclotomic</a>, <a>expand_eval</a>, \u2190 <a>IsRoot.def</a>,\n        @<a>isRoot_cyclotomic_iff</a>]", [{"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}, {"full_name": "Polynomial.eval_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [964, 9], "def_end_pos": [964, 17]}, {"full_name": "Polynomial.map_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "Polynomial.map_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 23]}, {"full_name": "Polynomial.expand_eval", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [195, 9], "def_end_pos": [195, 20]}, {"full_name": "Polynomial.IsRoot.def", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [497, 9], "def_end_pos": [497, 19]}, {"full_name": "Polynomial.isRoot_cyclotomic_iff", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [99, 9], "def_end_pos": [99, 30]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 (aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)))) ((expand \u211a p) (cyclotomic n \u211a)) = 0", "state_after": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)) ^ p) n"}, {"tactic": "convert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n)", "annotated_tactic": ["convert <a>IsPrimitiveRoot.pow_of_dvd</a> hprim hp.ne_zero (<a>dvd_mul_left</a> p n)", [{"full_name": "IsPrimitiveRoot.pow_of_dvd", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 19]}, {"full_name": "dvd_mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 21]}]], "state_before": "case inr.refine_1.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)) ^ p) n", "state_after": "case h.e'_4\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 n = n * p / p"}, {"tactic": "rw [Nat.mul_div_cancel _ (Nat.Prime.pos hp)]", "annotated_tactic": ["rw [<a>Nat.mul_div_cancel</a> _ (<a>Nat.Prime.pos</a> hp)]", [{"full_name": "Nat.mul_div_cancel", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [349, 19], "def_end_pos": [349, 33]}, {"full_name": "Nat.Prime.pos", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}]], "state_before": "case h.e'_4\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 n = n * p / p", "state_after": "no goals"}, {"tactic": "have hprim := Complex.isPrimitiveRoot_exp _ hnpos.ne.symm", "annotated_tactic": ["have hprim := <a>Complex.isPrimitiveRoot_exp</a> _ hnpos.ne.symm", [{"full_name": "Complex.isPrimitiveRoot_exp", "def_path": "Mathlib/RingTheory/RootsOfUnity/Complex.lean", "def_pos": [53, 9], "def_end_pos": [53, 28]}]], "state_before": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)"}, {"tactic": "rw [cyclotomic_eq_minpoly_rat hprim hnpos]", "annotated_tactic": ["rw [<a>cyclotomic_eq_minpoly_rat</a> hprim hnpos]", [{"full_name": "Polynomial.cyclotomic_eq_minpoly_rat", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [192, 9], "def_end_pos": [192, 34]}]], "state_before": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 cyclotomic n \u211a \u2223 (expand \u211a p) (cyclotomic n \u211a)", "state_after": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) \u2223\n    (expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))"}, {"tactic": "refine minpoly.dvd \u211a _ ?_", "annotated_tactic": ["refine <a>minpoly.dvd</a> \u211a _ ?_", [{"full_name": "minpoly.dvd", "def_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "def_pos": [68, 9], "def_end_pos": [68, 12]}]], "state_before": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) \u2223\n    (expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))", "state_after": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 (aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))\n      ((expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))) =\n    0"}, {"tactic": "rw [aeval_def, \u2190 eval_map, map_expand, expand_eval, \u2190 IsRoot.def, \u2190\n  cyclotomic_eq_minpoly_rat hprim hnpos, map_cyclotomic, @isRoot_cyclotomic_iff]", "annotated_tactic": ["rw [<a>aeval_def</a>, \u2190 <a>eval_map</a>, <a>map_expand</a>, <a>expand_eval</a>, \u2190 <a>IsRoot.def</a>, \u2190\n        <a>cyclotomic_eq_minpoly_rat</a> hprim hnpos, <a>map_cyclotomic</a>, @<a>isRoot_cyclotomic_iff</a>]", [{"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}, {"full_name": "Polynomial.eval_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [964, 9], "def_end_pos": [964, 17]}, {"full_name": "Polynomial.map_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [185, 9], "def_end_pos": [185, 19]}, {"full_name": "Polynomial.expand_eval", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [195, 9], "def_end_pos": [195, 20]}, {"full_name": "Polynomial.IsRoot.def", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [497, 9], "def_end_pos": [497, 19]}, {"full_name": "Polynomial.cyclotomic_eq_minpoly_rat", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [192, 9], "def_end_pos": [192, 34]}, {"full_name": "Polynomial.map_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 23]}, {"full_name": "Polynomial.isRoot_cyclotomic_iff", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean", "def_pos": [99, 9], "def_end_pos": [99, 30]}]], "state_before": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 (aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))\n      ((expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))) =\n    0", "state_after": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n) ^ p) n"}, {"tactic": "exact IsPrimitiveRoot.pow_of_prime hprim hp hdiv", "annotated_tactic": ["exact <a>IsPrimitiveRoot.pow_of_prime</a> hprim hp hdiv", [{"full_name": "IsPrimitiveRoot.pow_of_prime", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [432, 9], "def_end_pos": [432, 21]}]], "state_before": "case inr.refine_1.refine_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n) ^ p) n", "state_after": "no goals"}, {"tactic": "rw [natDegree_expand, natDegree_cyclotomic,\n  natDegree_mul (cyclotomic_ne_zero _ \u2124) (cyclotomic_ne_zero _ \u2124), natDegree_cyclotomic,\n  natDegree_cyclotomic, mul_comm n,\n  Nat.totient_mul ((Nat.Prime.coprime_iff_not_dvd hp).2 hdiv), Nat.totient_prime hp,\n  mul_comm (p - 1), \u2190 Nat.mul_succ, Nat.sub_one, Nat.succ_pred_eq_of_pos hp.pos]", "annotated_tactic": ["rw [<a>natDegree_expand</a>, <a>natDegree_cyclotomic</a>,\n      <a>natDegree_mul</a> (<a>cyclotomic_ne_zero</a> _ \u2124) (<a>cyclotomic_ne_zero</a> _ \u2124), <a>natDegree_cyclotomic</a>,\n      <a>natDegree_cyclotomic</a>, <a>mul_comm</a> n,\n      <a>Nat.totient_mul</a> ((<a>Nat.Prime.coprime_iff_not_dvd</a> hp).2 hdiv), <a>Nat.totient_prime</a> hp,\n      <a>mul_comm</a> (p - 1), \u2190 <a>Nat.mul_succ</a>, <a>Nat.sub_one</a>, <a>Nat.succ_pred_eq_of_pos</a> hp.pos]", [{"full_name": "Polynomial.natDegree_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [153, 9], "def_end_pos": [153, 25]}, {"full_name": "Polynomial.natDegree_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 29]}, {"full_name": "Polynomial.natDegree_mul", "def_path": "Mathlib/Algebra/Polynomial/RingDivision.lean", "def_pos": [124, 9], "def_end_pos": [124, 22]}, {"full_name": "Polynomial.cyclotomic_ne_zero", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 27]}, {"full_name": "Polynomial.cyclotomic_ne_zero", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 27]}, {"full_name": "Polynomial.natDegree_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 29]}, {"full_name": "Polynomial.natDegree_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [349, 9], "def_end_pos": [349, 29]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.totient_mul", "def_path": "Mathlib/Data/Nat/Totient.lean", "def_pos": [134, 9], "def_end_pos": [134, 20]}, {"full_name": "Nat.Prime.coprime_iff_not_dvd", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [565, 9], "def_end_pos": [565, 34]}, {"full_name": "Nat.totient_prime", "def_path": "Mathlib/Data/Nat/Totient.lean", "def_pos": [223, 9], "def_end_pos": [223, 22]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.mul_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 17]}, {"full_name": "Nat.sub_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [95, 19], "def_end_pos": [95, 26]}, {"full_name": "Nat.succ_pred_eq_of_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 28]}]], "state_before": "case inr.refine_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 ((expand \u2124 p) (cyclotomic n \u2124)).natDegree \u2264 (cyclotomic (n * p) \u2124 * cyclotomic n \u2124).natDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/RatFunc/Basic.lean", "full_name": "RatFunc.liftRingHom_apply_div", "start": [665, 1], "end": [667, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/Homogeneous.lean", "full_name": "MvPolynomial.homogeneousComponent_zero", "start": [502, 1], "end": [503, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.Lex.not_inr_lt_inl", "start": [372, 1], "end": [373, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "LinearIsometryEquiv.comp_hasStrictFDerivAt_iff", "start": [348, 1], "end": [350, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictAnti.prod_map", "start": [1242, 1], "end": [1245, 81], "traced_tactics": [{"tactic": "simp only [Prod.lt_iff]", "annotated_tactic": ["simp only [<a>Prod.lt_iff</a>]", [{"full_name": "Prod.lt_iff", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1325, 9], "def_end_pos": [1325, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictAnti f\nhg : StrictAnti g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a < b \u2192 Prod.map f g b < Prod.map f g a", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictAnti f\nhg : StrictAnti g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a.1 < b.1 \u2227 a.2 \u2264 b.2 \u2228 a.1 \u2264 b.1 \u2227 a.2 < b.2 \u2192\n    (Prod.map f g b).1 < (Prod.map f g a).1 \u2227 (Prod.map f g b).2 \u2264 (Prod.map f g a).2 \u2228\n      (Prod.map f g b).1 \u2264 (Prod.map f g a).1 \u2227 (Prod.map f g b).2 < (Prod.map f g a).2"}, {"tactic": "exact Or.imp (And.imp hf.imp hg.antitone.imp) (And.imp hf.antitone.imp hg.imp)", "annotated_tactic": ["exact <a>Or.imp</a> (<a>And.imp</a> hf.imp hg.antitone.imp) (<a>And.imp</a> hf.antitone.imp hg.imp)", [{"full_name": "Or.imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [172, 9], "def_end_pos": [172, 15]}, {"full_name": "And.imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [39, 9], "def_end_pos": [39, 16]}, {"full_name": "And.imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [39, 9], "def_end_pos": [39, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictAnti f\nhg : StrictAnti g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a.1 < b.1 \u2227 a.2 \u2264 b.2 \u2228 a.1 \u2264 b.1 \u2227 a.2 < b.2 \u2192\n    (Prod.map f g b).1 < (Prod.map f g a).1 \u2227 (Prod.map f g b).2 \u2264 (Prod.map f g a).2 \u2228\n      (Prod.map f g b).1 \u2264 (Prod.map f g a).1 \u2227 (Prod.map f g b).2 < (Prod.map f g a).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "full_name": "PiLp.norm_equiv_symm_const", "start": [937, 1], "end": [940, 79], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2\u271d : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\n\u03b2 : Type u_5\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\n\u22a2 \u2191(\u2191(Fintype.card \u03b9) ^ (1 / p).toReal * \u2016b\u2016\u208a) = \u2191\u2191(Fintype.card \u03b9) ^ (1 / p).toReal * \u2016b\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/NormNum/Eq.lean", "full_name": "Mathlib.Meta.NormNum.isInt_eq_false", "start": [22, 1], "end": [24, 71], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : CharZero \u03b1\nn\u271d\u00b9 n\u271d : \u2124\nh : decide (n\u271d\u00b9 = n\u271d) = false\n\u22a2 \u00ac\u2191n\u271d\u00b9 = \u2191n\u271d", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : CharZero \u03b1\nn\u271d\u00b9 n\u271d : \u2124\nh : decide (n\u271d\u00b9 = n\u271d) = false\n\u22a2 \u00acn\u271d\u00b9 = n\u271d"}, {"tactic": "exact of_decide_eq_false h", "annotated_tactic": ["exact <a>of_decide_eq_false</a> h", [{"full_name": "of_decide_eq_false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [889, 9], "def_end_pos": [889, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : CharZero \u03b1\nn\u271d\u00b9 n\u271d : \u2124\nh : decide (n\u271d\u00b9 = n\u271d) = false\n\u22a2 \u00acn\u271d\u00b9 = n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Module/Defs.lean", "full_name": "PosSMulReflectLT.of_pos", "start": [551, 1], "end": [558, 26], "traced_tactics": [{"tactic": "obtain ha | ha := ha.eq_or_lt", "annotated_tactic": ["obtain ha | ha := ha.eq_or_lt", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081\u271d b\u2082\u271d : \u03b2\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 : \u2200 (a : \u03b1), 0 < a \u2192 \u2200 (b\u2081 b\u2082 : \u03b2), a \u2022 b\u2081 < a \u2022 b\u2082 \u2192 b\u2081 < b\u2082\na : \u03b1\nha : 0 \u2264 a\nb\u2081 b\u2082 : \u03b2\nh : a \u2022 b\u2081 < a \u2022 b\u2082\n\u22a2 b\u2081 < b\u2082", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081\u271d b\u2082\u271d : \u03b2\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 : \u2200 (a : \u03b1), 0 < a \u2192 \u2200 (b\u2081 b\u2082 : \u03b2), a \u2022 b\u2081 < a \u2022 b\u2082 \u2192 b\u2081 < b\u2082\na : \u03b1\nha\u271d : 0 \u2264 a\nb\u2081 b\u2082 : \u03b2\nh : a \u2022 b\u2081 < a \u2022 b\u2082\nha : 0 = a\n\u22a2 b\u2081 < b\u2082\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081\u271d b\u2082\u271d : \u03b2\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 : \u2200 (a : \u03b1), 0 < a \u2192 \u2200 (b\u2081 b\u2082 : \u03b2), a \u2022 b\u2081 < a \u2022 b\u2082 \u2192 b\u2081 < b\u2082\na : \u03b1\nha\u271d : 0 \u2264 a\nb\u2081 b\u2082 : \u03b2\nh : a \u2022 b\u2081 < a \u2022 b\u2082\nha : 0 < a\n\u22a2 b\u2081 < b\u2082"}, {"tactic": "simp [\u2190 ha] at h", "annotated_tactic": ["simp [\u2190 ha] at h", []], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081\u271d b\u2082\u271d : \u03b2\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 : \u2200 (a : \u03b1), 0 < a \u2192 \u2200 (b\u2081 b\u2082 : \u03b2), a \u2022 b\u2081 < a \u2022 b\u2082 \u2192 b\u2081 < b\u2082\na : \u03b1\nha\u271d : 0 \u2264 a\nb\u2081 b\u2082 : \u03b2\nh : a \u2022 b\u2081 < a \u2022 b\u2082\nha : 0 = a\n\u22a2 b\u2081 < b\u2082", "state_after": "no goals"}, {"tactic": "exact h\u2080 _ ha _ _ h", "annotated_tactic": ["exact h\u2080 _ ha _ _ h", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081\u271d b\u2082\u271d : \u03b2\ninst\u271d\u2074 : Zero \u03b1\ninst\u271d\u00b3 : Zero \u03b2\ninst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 : \u2200 (a : \u03b1), 0 < a \u2192 \u2200 (b\u2081 b\u2082 : \u03b2), a \u2022 b\u2081 < a \u2022 b\u2082 \u2192 b\u2081 < b\u2082\na : \u03b1\nha\u271d : 0 \u2264 a\nb\u2081 b\u2082 : \u03b2\nh : a \u2022 b\u2081 < a \u2022 b\u2082\nha : 0 < a\n\u22a2 b\u2081 < b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Action/Basic.lean", "full_name": "RingHom.smul_def", "start": [84, 11], "end": [85, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Disjoint.inter_right'", "start": [2658, 1], "end": [2659, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Disjoint.exists_mem_filter_basis", "start": [661, 1], "end": [663, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "analyticAt_const", "start": [528, 1], "end": [529, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIcoDiv_add_left", "start": [319, 1], "end": [320, 36], "traced_tactics": [{"tactic": "rw [add_comm, toIcoDiv_add_right]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>toIcoDiv_add_right</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "toIcoDiv_add_right", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [299, 9], "def_end_pos": [299, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "Ideal.pow_le_prime_iff", "start": [1292, 1], "end": [1294, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean", "full_name": "AkraBazziRecurrence.GrowsPolynomially.of_isTheta", "start": [662, 1], "end": [708, 33], "traced_tactics": [{"tactic": "intro b hb", "annotated_tactic": ["intro b hb", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\n\u22a2 GrowsPolynomially f", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have hb_pos := hb.1", "annotated_tactic": ["have hb_pos := hb.1", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have hf_lb := isBigO_iff''.mp hf.isBigO_symm", "annotated_tactic": ["have hf_lb := isBigO_iff''.mp hf.isBigO_symm", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_lb : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, c * \u2016g x\u2016 \u2264 \u2016f x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have hf_ub := isBigO_iff'.mp hf.isBigO", "annotated_tactic": ["have hf_ub := isBigO_iff'.mp hf.isBigO", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_lb : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, c * \u2016g x\u2016 \u2264 \u2016f x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_lb : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, c * \u2016g x\u2016 \u2264 \u2016f x\u2016\nhf_ub : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "obtain \u27e8c\u2081, hc\u2081_pos : 0 < c\u2081, hf_lb\u27e9 := hf_lb", "annotated_tactic": ["obtain \u27e8c\u2081, hc\u2081_pos : 0 < c\u2081, hf_lb\u27e9 := hf_lb", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_lb : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, c * \u2016g x\u2016 \u2264 \u2016f x\u2016\nhf_ub : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_ub : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "obtain \u27e8c\u2082, hc\u2082_pos : 0 < c\u2082, hf_ub\u27e9 := hf_ub", "annotated_tactic": ["obtain \u27e8c\u2082, hc\u2082_pos : 0 < c\u2082, hf_ub\u27e9 := hf_ub", []], "state_before": "case intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nhf_ub : \u2203 c > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have hg := hg.norm b hb", "annotated_tactic": ["have hg := hg.norm b hb", []], "state_before": "case intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nhg :\n  \u2203 c\u2081 > 0,\n    \u2203 c\u2082 > 0,\n      \u2200\u1da0 (x : \u211d) in atTop,\n        \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2081 * (fun x => \u2016g x\u2016) x) (c\u2082 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "obtain \u27e8c\u2083, hc\u2083_pos : 0 < c\u2083, hg\u27e9 := hg", "annotated_tactic": ["obtain \u27e8c\u2083, hc\u2083_pos : 0 < c\u2083, hg\u27e9 := hg", []], "state_before": "case intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nhg :\n  \u2203 c\u2081 > 0,\n    \u2203 c\u2082 > 0,\n      \u2200\u1da0 (x : \u211d) in atTop,\n        \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2081 * (fun x => \u2016g x\u2016) x) (c\u2082 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nhg :\n  \u2203 c\u2082 > 0,\n    \u2200\u1da0 (x : \u211d) in atTop,\n      \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2082 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "obtain \u27e8c\u2084, hc\u2084_pos : 0 < c\u2084, hg\u27e9 := hg", "annotated_tactic": ["obtain \u27e8c\u2084, hc\u2084_pos : 0 < c\u2084, hg\u27e9 := hg", []], "state_before": "case intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nhg :\n  \u2203 c\u2082 > 0,\n    \u2200\u1da0 (x : \u211d) in atTop,\n      \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2082 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have h_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083 := by positivity", "annotated_tactic": ["have h_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083 := by positivity", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "have h_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9 := by positivity", "annotated_tactic": ["have h_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9 := by positivity", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)"}, {"tactic": "refine \u27e8c\u2081 * c\u2082\u207b\u00b9 * c\u2083, h_lb_pos, ?_\u27e9", "annotated_tactic": ["refine \u27e8c\u2081 * c\u2082\u207b\u00b9 * c\u2083, h_lb_pos, ?_\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2203 c\u2081 > 0, \u2203 c\u2082 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2203 c\u2082_1 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082_1 * f x)"}, {"tactic": "refine \u27e8c\u2082 * c\u2084 * c\u2081\u207b\u00b9, h_ub_pos, ?_\u27e9", "annotated_tactic": ["refine \u27e8c\u2082 * c\u2084 * c\u2081\u207b\u00b9, h_ub_pos, ?_\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2203 c\u2082_1 > 0, \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082_1 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "have c\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1 := inv_mul_cancel (by positivity)", "annotated_tactic": ["have c\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1 := <a>inv_mul_cancel</a> (by positivity)", [{"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "have c\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1 := inv_mul_cancel (by positivity)", "annotated_tactic": ["have c\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1 := <a>inv_mul_cancel</a> (by positivity)", [{"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',\n                (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,\n                (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,\n                (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,\n                eventually_ge_atTop 0]\n  with x hf_pos h_lb h_ub hg_bound hx_pos", "annotated_tactic": ["filter_upwards [(tendsto_id.const_mul_atTop hb_pos).<a>eventually_forall_ge_atTop</a> hf',\n                  (tendsto_id.const_mul_atTop hb_pos).<a>eventually_forall_ge_atTop</a> hf_lb,\n                  (tendsto_id.const_mul_atTop hb_pos).<a>eventually_forall_ge_atTop</a> hf_ub,\n                  (tendsto_id.const_mul_atTop hb_pos).<a>eventually_forall_ge_atTop</a> hg,\n                  <a>eventually_ge_atTop</a> 0]\n    with x hf_pos h_lb h_ub hg_bound hx_pos", [{"full_name": "Filter.Tendsto.eventually_forall_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [264, 9], "def_end_pos": [264, 43]}, {"full_name": "Filter.Tendsto.eventually_forall_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [264, 9], "def_end_pos": [264, 43]}, {"full_name": "Filter.Tendsto.eventually_forall_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [264, 9], "def_end_pos": [264, 43]}, {"full_name": "Filter.Tendsto.eventually_forall_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [264, 9], "def_end_pos": [264, 43]}, {"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\n\u22a2 \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "intro u hu", "annotated_tactic": ["intro u hu", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\n\u22a2 \u2200 u \u2208 Set.Icc (b * x) x, f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "have hbx : b * x \u2264 x :=\n  calc b * x \u2264 1 * x    := by gcongr; exact le_of_lt hb.2\n           _ = x        := by rw [one_mul]", "annotated_tactic": ["have hbx : b * x \u2264 x :=\n    calc b * x \u2264 1 * x    := by gcongr; exact <a>le_of_lt</a> hb.2\n             _ = x        := by rw [<a>one_mul</a>]", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "have hg_bound := hg_bound x hbx", "annotated_tactic": ["have hg_bound := hg_bound x hbx", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)"}, {"tactic": "refine \u27e8?lb, ?ub\u27e9", "annotated_tactic": ["refine \u27e8?lb, ?ub\u27e9", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2208 Set.Icc (c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x) (c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x)", "state_after": "case lb\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 f u\n\ncase ub\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x"}, {"tactic": "case lb => calc\n  c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016) := by\n        rw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx\n    _ = (c\u2082\u207b\u00b9 * c\u2082) * c\u2081 * (c\u2083 * \u2016g x\u2016) := by ring\n    _ = c\u2081 * (c\u2083 * \u2016g x\u2016) := by simp [c\u2082_cancel]\n    _ \u2264 c\u2081 * \u2016g u\u2016 := by gcongr; exact (hg_bound u hu).1\n    _ \u2264 f u := by\n        rw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]\n        exact h_lb u hu.1", "annotated_tactic": ["case lb => calc\n    c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016) := by\n          rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]; gcongr; exact h_ub x hbx\n      _ = (c\u2082\u207b\u00b9 * c\u2082) * c\u2081 * (c\u2083 * \u2016g x\u2016) := by ring\n      _ = c\u2081 * (c\u2083 * \u2016g x\u2016) := by simp [c\u2082_cancel]\n      _ \u2264 c\u2081 * \u2016g u\u2016 := by gcongr; exact (hg_bound u hu).1\n      _ \u2264 f u := by\n          rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]\n          exact h_lb u hu.1", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "case lb\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 f u\n\ncase ub\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x", "state_after": "case ub\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x"}, {"tactic": "case ub => calc\n  f u \u2264 c\u2082 * \u2016g u\u2016 := by rw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]; exact h_ub u hu.1\n    _ \u2264 c\u2082 * (c\u2084 * \u2016g x\u2016) := by gcongr; exact (hg_bound u hu).2\n    _ = c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016 := by simp [c\u2081_cancel]; ring\n    _ = c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016) := by ring\n    _ \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x := by\n              gcongr\n              rw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]\n              exact h_lb x hbx", "annotated_tactic": ["case ub => calc\n    f u \u2264 c\u2082 * \u2016g u\u2016 := by rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]; exact h_ub u hu.1\n      _ \u2264 c\u2082 * (c\u2084 * \u2016g x\u2016) := by gcongr; exact (hg_bound u hu).2\n      _ = c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016 := by simp [c\u2081_cancel]; ring\n      _ = c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016) := by ring\n      _ \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x := by\n                gcongr\n                rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]\n                exact h_lb x hbx", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "case ub\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\n\u22a2 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\n\u22a2 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\n\u22a2 c\u2082 \u2260 0", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\n\u22a2 c\u2081 \u2260 0", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 b * x \u2264 1 * x", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 b \u2264 1"}, {"tactic": "exact le_of_lt hb.2", "annotated_tactic": ["exact <a>le_of_lt</a> hb.2", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 b \u2264 1", "state_after": "no goals"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\n\u22a2 1 * x = x", "state_after": "no goals"}, {"tactic": "calc\nc\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016) := by\nrw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx\n_ = (c\u2082\u207b\u00b9 * c\u2082) * c\u2081 * (c\u2083 * \u2016g x\u2016) := by ring\n_ = c\u2081 * (c\u2083 * \u2016g x\u2016) := by simp [c\u2082_cancel]\n_ \u2264 c\u2081 * \u2016g u\u2016 := by gcongr; exact (hg_bound u hu).1\n_ \u2264 f u := by\nrw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]\nexact h_lb u hu.1", "annotated_tactic": ["calc\n    c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016) := by\n          rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]; gcongr; exact h_ub x hbx\n      _ = (c\u2082\u207b\u00b9 * c\u2082) * c\u2081 * (c\u2083 * \u2016g x\u2016) := by ring\n      _ = c\u2081 * (c\u2083 * \u2016g x\u2016) := by simp [c\u2082_cancel]\n      _ \u2264 c\u2081 * \u2016g u\u2016 := by gcongr; exact (hg_bound u hu).1\n      _ \u2264 f u := by\n          rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]\n          exact h_lb u hu.1", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 f u", "state_after": "no goals"}, {"tactic": "rw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * f x \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016)", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * \u2016f x\u2016 \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * \u2016f x\u2016 \u2264 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016"}, {"tactic": "exact h_ub x hbx", "annotated_tactic": ["exact h_ub x hbx", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * c\u2082\u207b\u00b9 * c\u2083 * (c\u2082 * \u2016g x\u2016) = c\u2082\u207b\u00b9 * c\u2082 * c\u2081 * (c\u2083 * \u2016g x\u2016)", "state_after": "no goals"}, {"tactic": "simp [c\u2082_cancel]", "annotated_tactic": ["simp [c\u2082_cancel]", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082\u207b\u00b9 * c\u2082 * c\u2081 * (c\u2083 * \u2016g x\u2016) = c\u2081 * (c\u2083 * \u2016g x\u2016)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * (c\u2083 * \u2016g x\u2016) \u2264 c\u2081 * \u2016g u\u2016", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2083 * \u2016g x\u2016 \u2264 \u2016g u\u2016"}, {"tactic": "exact (hg_bound u hu).1", "annotated_tactic": ["exact (hg_bound u hu).1", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2083 * \u2016g x\u2016 \u2264 \u2016g u\u2016", "state_after": "no goals"}, {"tactic": "rw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g u\u2016 \u2264 f u", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g u\u2016 \u2264 \u2016f u\u2016"}, {"tactic": "exact h_lb u hu.1", "annotated_tactic": ["exact h_lb u hu.1", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g u\u2016 \u2264 \u2016f u\u2016", "state_after": "no goals"}, {"tactic": "calc\nf u \u2264 c\u2082 * \u2016g u\u2016 := by rw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]; exact h_ub u hu.1\n_ \u2264 c\u2082 * (c\u2084 * \u2016g x\u2016) := by gcongr; exact (hg_bound u hu).2\n_ = c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016 := by simp [c\u2081_cancel]; ring\n_ = c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016) := by ring\n_ \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x := by\n   gcongr\n   rw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]\n   exact h_lb x hbx", "annotated_tactic": ["calc\n    f u \u2264 c\u2082 * \u2016g u\u2016 := by rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]; exact h_ub u hu.1\n      _ \u2264 c\u2082 * (c\u2084 * \u2016g x\u2016) := by gcongr; exact (hg_bound u hu).2\n      _ = c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016 := by simp [c\u2081_cancel]; ring\n      _ = c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016) := by ring\n      _ \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x := by\n                gcongr\n                rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]\n                exact h_lb x hbx", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x", "state_after": "no goals"}, {"tactic": "rw [\u2190 Real.norm_of_nonneg (hf_pos u hu.1)]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos u hu.1)]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 f u \u2264 c\u2082 * \u2016g u\u2016", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016f u\u2016 \u2264 c\u2082 * \u2016g u\u2016"}, {"tactic": "exact h_ub u hu.1", "annotated_tactic": ["exact h_ub u hu.1", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016f u\u2016 \u2264 c\u2082 * \u2016g u\u2016", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * \u2016g u\u2016 \u2264 c\u2082 * (c\u2084 * \u2016g x\u2016)", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016g u\u2016 \u2264 c\u2084 * \u2016g x\u2016"}, {"tactic": "exact (hg_bound u hu).2", "annotated_tactic": ["exact (hg_bound u hu).2", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 \u2016g u\u2016 \u2264 c\u2084 * \u2016g x\u2016", "state_after": "no goals"}, {"tactic": "simp [c\u2081_cancel]", "annotated_tactic": ["simp [c\u2081_cancel]", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * (c\u2084 * \u2016g x\u2016) = c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016", "state_after": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * (c\u2084 * |g x|) = c\u2082 * c\u2084 * |g x|"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * (c\u2084 * |g x|) = c\u2082 * c\u2084 * |g x|", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * c\u2084 * (c\u2081\u207b\u00b9 * c\u2081) * \u2016g x\u2016 = c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * (c\u2081 * \u2016g x\u2016) \u2264 c\u2082 * c\u2084 * c\u2081\u207b\u00b9 * f x", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g x\u2016 \u2264 f x"}, {"tactic": "rw [\u2190 Real.norm_of_nonneg (hf_pos x hbx)]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_of_nonneg</a> (hf_pos x hbx)]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g x\u2016 \u2264 f x", "state_after": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016"}, {"tactic": "exact h_lb x hbx", "annotated_tactic": ["exact h_lb x hbx", []], "state_before": "case h\nf\u271d f g : \u211d \u2192 \u211d\nhg\u271d : GrowsPolynomially g\nhf : f =\u0398[atTop] g\nhf' : \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x\nb : \u211d\nhb : b \u2208 Set.Ioo 0 1\nhb_pos : 0 < b\nc\u2081 : \u211d\nhc\u2081_pos : 0 < c\u2081\nhf_lb : \u2200\u1da0 (x : \u211d) in atTop, c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016\nc\u2082 : \u211d\nhc\u2082_pos : 0 < c\u2082\nhf_ub : \u2200\u1da0 (x : \u211d) in atTop, \u2016f x\u2016 \u2264 c\u2082 * \u2016g x\u2016\nc\u2083 : \u211d\nhc\u2083_pos : 0 < c\u2083\nc\u2084 : \u211d\nhc\u2084_pos : 0 < c\u2084\nhg :\n  \u2200\u1da0 (x : \u211d) in atTop,\n    \u2200 u \u2208 Set.Icc (b * x) x, (fun x => \u2016g x\u2016) u \u2208 Set.Icc (c\u2083 * (fun x => \u2016g x\u2016) x) (c\u2084 * (fun x => \u2016g x\u2016) x)\nh_lb_pos : 0 < c\u2081 * c\u2082\u207b\u00b9 * c\u2083\nh_ub_pos : 0 < c\u2082 * c\u2084 * c\u2081\u207b\u00b9\nc\u2082_cancel : c\u2082\u207b\u00b9 * c\u2082 = 1\nc\u2081_cancel : c\u2081\u207b\u00b9 * c\u2081 = 1\nx : \u211d\nhf_pos : \u2200 (y : \u211d), b * id x \u2264 y \u2192 0 \u2264 f y\nh_lb : \u2200 (y : \u211d), b * id x \u2264 y \u2192 c\u2081 * \u2016g y\u2016 \u2264 \u2016f y\u2016\nh_ub : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2016f y\u2016 \u2264 c\u2082 * \u2016g y\u2016\nhg_bound\u271d : \u2200 (y : \u211d), b * id x \u2264 y \u2192 \u2200 u \u2208 Set.Icc (b * y) y, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g y\u2016) (c\u2084 * \u2016g y\u2016)\nhx_pos : 0 \u2264 x\nu : \u211d\nhu : u \u2208 Set.Icc (b * x) x\nhbx : b * x \u2264 x\nhg_bound : \u2200 u \u2208 Set.Icc (b * x) x, \u2016g u\u2016 \u2208 Set.Icc (c\u2083 * \u2016g x\u2016) (c\u2084 * \u2016g x\u2016)\n\u22a2 c\u2081 * \u2016g x\u2016 \u2264 \u2016f x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.coe_coe", "start": [127, 11], "end": [129, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SemiconjSup.lean", "full_name": "IsOrderRightAdjoint.orderIso_comp", "start": [68, 1], "end": [70, 57], "traced_tactics": [{"tactic": "simpa [e.le_symm_apply] using h (e.symm y)", "annotated_tactic": ["simpa [e.le_symm_apply] using h (e.symm y)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b2 \u2243o \u03b3\ny : \u03b3\n\u22a2 IsLUB {x | (\u21d1e \u2218 f) x \u2264 y} ((g \u2218 \u21d1e.symm) y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Dedup.lean", "full_name": "List.mem_dedup", "start": [44, 1], "end": [48, 56], "traced_tactics": [{"tactic": "simpa only [dedup, forall_mem_ne, not_not] using this", "annotated_tactic": ["simpa only [<a>dedup</a>, <a>forall_mem_ne</a>, <a>not_not</a>] using this", [{"full_name": "List.dedup", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [335, 5], "def_end_pos": [335, 10]}, {"full_name": "List.forall_mem_ne", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [29, 9], "def_end_pos": [29, 22]}, {"full_name": "Classical.not_not", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [135, 17], "def_end_pos": [135, 24]}]], "state_before": "case refine_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nthis : (\u00ac\u2200 b \u2208 pwFilter (fun x x_1 => x \u2260 x_1) l, a \u2260 b) \u2194 \u00ac\u2200 b \u2208 l, a \u2260 b\n\u22a2 a \u2208 l.dedup \u2194 a \u2208 l", "state_after": "no goals"}, {"tactic": "intros x y z xz", "annotated_tactic": ["intros x y z xz", []], "state_before": "case refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 \u2200 {x y z : \u03b1}, (fun x x_1 => x \u2260 x_1) x z \u2192 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z", "state_after": "case refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nx y z : \u03b1\nxz : x \u2260 z\n\u22a2 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z"}, {"tactic": "exact not_and_or.1 <| mt (fun h \u21a6 h.1.trans h.2) xz", "annotated_tactic": ["exact <a>not_and_or</a>.1 <| <a>mt</a> (fun h \u21a6 h.1.<a>trans</a> h.2) xz", [{"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "case refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nx y z : \u03b1\nxz : x \u2260 z\n\u22a2 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Sheaf/Smooth.lean", "full_name": "smoothSheafCommRing.eval_germ", "start": [375, 1], "end": [379, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "full_name": "HasDerivAt.mul_const", "start": [248, 1], "end": [251, 23], "traced_tactics": [{"tactic": "rw [\u2190 hasDerivWithinAt_univ] at *", "annotated_tactic": ["rw [\u2190 <a>hasDerivWithinAt_univ</a>] at *", [{"full_name": "hasDerivWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [400, 9], "def_end_pos": [400, 30]}]], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivAt c c' x\nd : \ud835\udd38\n\u22a2 HasDerivAt (fun y => c y * d) (c' * d) x", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nd : \ud835\udd38\n\u22a2 HasDerivWithinAt (fun y => c y * d) (c' * d) univ x"}, {"tactic": "exact hc.mul_const d", "annotated_tactic": ["exact hc.mul_const d", []], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nd : \ud835\udd38\n\u22a2 HasDerivWithinAt (fun y => c y * d) (c' * d) univ x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Marginal.lean", "full_name": "MeasureTheory.lmarginal_update_of_mem", "start": [110, 1], "end": [116, 26], "traced_tactics": [{"tactic": "apply lmarginal_congr", "annotated_tactic": ["apply <a>lmarginal_congr</a>", [{"full_name": "MeasureTheory.lmarginal_congr", "def_path": "Mathlib/MeasureTheory/Integral/Marginal.lean", "def_pos": [105, 9], "def_end_pos": [105, 24]}]], "state_before": "\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\n\u22a2 (\u222b\u22ef\u222b\u207b_s, f \u2202\u03bc) (update x i y) = (\u222b\u22ef\u222b\u207b_s, f \u2202\u03bc) x", "state_after": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\n\u22a2 \u2200 i_1 \u2209 s, update x i y i_1 = x i_1"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\n\u22a2 \u2200 i_1 \u2209 s, update x i y i_1 = x i_1", "state_after": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\nj : \u03b4\nhj : j \u2209 s\n\u22a2 update x i y j = x j"}, {"tactic": "have : j \u2260 i := by rintro rfl; exact hj hi", "annotated_tactic": ["have : j \u2260 i := by rintro rfl; exact hj hi", []], "state_before": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\nj : \u03b4\nhj : j \u2209 s\n\u22a2 update x i y j = x j", "state_after": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\nj : \u03b4\nhj : j \u2209 s\nthis : j \u2260 i\n\u22a2 update x i y j = x j"}, {"tactic": "apply update_noteq this", "annotated_tactic": ["apply <a>update_noteq</a> this", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [563, 9], "def_end_pos": [563, 21]}]], "state_before": "case h\n\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\nj : \u03b4\nhj : j \u2209 s\nthis : j \u2260 i\n\u22a2 update x i y j = x j", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni\u271d i : \u03b4\nhi : i \u2208 s\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\ny : \u03c0 i\nj : \u03b4\nhj : j \u2209 s\n\u22a2 j \u2260 i", "state_after": "\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni : \u03b4\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\nj : \u03b4\nhj : j \u2209 s\nhi : j \u2208 s\ny : \u03c0 j\n\u22a2 False"}, {"tactic": "exact hj hi", "annotated_tactic": ["exact hj hi", []], "state_before": "\u03b4 : Type u_1\n\u03b4' : Type u_2\n\u03c0 : \u03b4 \u2192 Type u_3\ninst\u271d\u00b2 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ninst\u271d : DecidableEq \u03b4\ns t : Finset \u03b4\nf\u271d g : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx\u271d y\u271d : (i : \u03b4) \u2192 \u03c0 i\ni : \u03b4\nf : ((i : \u03b4) \u2192 \u03c0 i) \u2192 \u211d\u22650\u221e\nx : (i : \u03b4) \u2192 \u03c0 i\nj : \u03b4\nhj : j \u2209 s\nhi : j \u2208 s\ny : \u03c0 j\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Basic.lean", "full_name": "Finset.Ioc_subset_Ioc_left", "start": [196, 1], "end": [197, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.mul_bot_of_pos", "start": [1162, 1], "end": [1165, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean", "full_name": "ContinuousLinearMap.norm_compL_le", "start": [351, 1], "end": [352, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Icc_eq_singleton_iff", "start": [782, 1], "end": [789, 21], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b = {c} \u2194 a = c \u2227 b = c", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\n\u22a2 a = c \u2227 b = c\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 a = c \u2227 b = c \u2192 Icc a b = {c}"}, {"tactic": "have hab : a \u2264 b := nonempty_Icc.1 (h.symm.subst <| singleton_nonempty c)", "annotated_tactic": ["have hab : a \u2264 b := <a>nonempty_Icc</a>.1 (h.symm.subst <| <a>singleton_nonempty</a> c)", [{"full_name": "Set.nonempty_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 21]}, {"full_name": "Set.singleton_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1272, 9], "def_end_pos": [1272, 27]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\n\u22a2 a = c \u2227 b = c", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\nhab : a \u2264 b\n\u22a2 a = c \u2227 b = c"}, {"tactic": "exact\n  \u27e8eq_of_mem_singleton <| h.subst <| left_mem_Icc.2 hab,\n    eq_of_mem_singleton <| h.subst <| right_mem_Icc.2 hab\u27e9", "annotated_tactic": ["exact\n      \u27e8<a>eq_of_mem_singleton</a> <| h.subst <| <a>left_mem_Icc</a>.2 hab,\n        <a>eq_of_mem_singleton</a> <| h.subst <| <a>right_mem_Icc</a>.2 hab\u27e9", [{"full_name": "Set.eq_of_mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1250, 9], "def_end_pos": [1250, 28]}, {"full_name": "Set.left_mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [191, 9], "def_end_pos": [191, 21]}, {"full_name": "Set.eq_of_mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1250, 9], "def_end_pos": [1250, 28]}, {"full_name": "Set.right_mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\nhab : a \u2264 b\n\u22a2 a = c \u2227 b = c", "state_after": "no goals"}, {"tactic": "rintro \u27e8rfl, rfl\u27e9", "annotated_tactic": ["rintro \u27e8rfl, rfl\u27e9", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 a = c \u2227 b = c \u2192 Icc a b = {c}", "state_after": "case refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\nb : \u03b1\n\u22a2 Icc b b = {b}"}, {"tactic": "exact Icc_self _", "annotated_tactic": ["exact <a>Icc_self</a> _", [{"full_name": "Set.Icc_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [773, 9], "def_end_pos": [773, 17]}]], "state_before": "case refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\nb : \u03b1\n\u22a2 Icc b b = {b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sym/Basic.lean", "full_name": "Sym.ofVector_cons", "start": [154, 1], "end": [156, 6], "traced_tactics": [{"tactic": "cases v", "annotated_tactic": ["cases v", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nv : Vector \u03b1 n\n\u22a2 ofVector (a ::\u1d65 v) = a ::\u209b ofVector v", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : val\u271d.length = n\n\u22a2 ofVector (a ::\u1d65 \u27e8val\u271d, property\u271d\u27e9) = a ::\u209b ofVector \u27e8val\u271d, property\u271d\u27e9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : val\u271d.length = n\n\u22a2 ofVector (a ::\u1d65 \u27e8val\u271d, property\u271d\u27e9) = a ::\u209b ofVector \u27e8val\u271d, property\u271d\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "OrderIso.sumLexDualAntidistrib_inl", "start": [708, 1], "end": [710, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "closure_iUnion\u2082_le_nat", "start": [537, 1], "end": [539, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.le_comap_of_map_le", "start": [478, 1], "end": [479, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "full_name": "contMDiffWithinAt_iff_of_mem_source", "start": [412, 1], "end": [419, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Indexes.lean", "full_name": "List.le_findIdx_of_not", "start": [312, 1], "end": [315, 71], "traced_tactics": [{"tactic": "by_contra! f", "annotated_tactic": ["by_contra! f", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ni : \u2115\nh : i < xs.length\nh2 : \u2200 (j : \u2115) (hji : j < i), \u00acp (xs.get \u27e8j, \u22ef\u27e9) = true\n\u22a2 i \u2264 findIdx p xs", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ni : \u2115\nh : i < xs.length\nh2 : \u2200 (j : \u2115) (hji : j < i), \u00acp (xs.get \u27e8j, \u22ef\u27e9) = true\nf : findIdx p xs < i\n\u22a2 False"}, {"tactic": "exact absurd (@findIdx_get _ p xs (f.trans h)) (h2 (xs.findIdx p) f)", "annotated_tactic": ["exact <a>absurd</a> (@<a>findIdx_get</a> _ p xs (f.trans h)) (h2 (xs.findIdx p) f)", [{"full_name": "absurd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "List.findIdx_get", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [687, 9], "def_end_pos": [687, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ni : \u2115\nh : i < xs.length\nh2 : \u2200 (j : \u2115) (hji : j < i), \u00acp (xs.get \u27e8j, \u22ef\u27e9) = true\nf : findIdx p xs < i\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_zero", "start": [142, 1], "end": [142, 55], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u222b\u207b (x : \u03b1), 0 \u2202\u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Module.lean", "full_name": "Complex.coe_basisOneI", "start": [156, 1], "end": [164, 101], "traced_tactics": [{"tactic": "fin_cases i <;> fin_cases j <;>\n  simp [coe_basisOneI_repr, Finsupp.single_eq_of_ne, Matrix.cons_val_zero,\n    Matrix.cons_val_one, Matrix.head_cons, Fin.one_eq_zero_iff, Ne, not_false_iff, I_re,\n    Nat.succ_succ_ne_one, one_im, I_im, one_re, Finsupp.single_eq_same, Fin.zero_eq_one_iff]", "annotated_tactic": ["fin_cases i <;> fin_cases j <;>\n          -- Porting note: removed `only`, consider squeezing again\n          simp [<a>coe_basisOneI_repr</a>, <a>Finsupp.single_eq_of_ne</a>, <a>Matrix.cons_val_zero</a>,\n            <a>Matrix.cons_val_one</a>, <a>Matrix.head_cons</a>, <a>Fin.one_eq_zero_iff</a>, <a>Ne</a>, <a>not_false_iff</a>, <a>I_re</a>,\n            <a>Nat.succ_succ_ne_one</a>, <a>one_im</a>, <a>I_im</a>, <a>one_re</a>, <a>Finsupp.single_eq_same</a>, <a>Fin.zero_eq_one_iff</a>]", [{"full_name": "Complex.coe_basisOneI_repr", "def_path": "Mathlib/Data/Complex/Module.lean", "def_pos": [150, 9], "def_end_pos": [150, 27]}, {"full_name": "Finsupp.single_eq_of_ne", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 24]}, {"full_name": "Matrix.cons_val_zero", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}, {"full_name": "Matrix.cons_val_one", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [209, 9], "def_end_pos": [209, 21]}, {"full_name": "Matrix.head_cons", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [152, 9], "def_end_pos": [152, 18]}, {"full_name": "Fin.one_eq_zero_iff", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [627, 9], "def_end_pos": [627, 24]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "not_false_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 22]}, {"full_name": "Complex.I_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [284, 9], "def_end_pos": [284, 13]}, {"full_name": "Nat.succ_succ_ne_one", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [108, 7], "def_end_pos": [108, 23]}, {"full_name": "Complex.one_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 15]}, {"full_name": "Complex.I_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [290, 9], "def_end_pos": [290, 13]}, {"full_name": "Complex.one_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Finsupp.single_eq_same", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [304, 9], "def_end_pos": [304, 23]}, {"full_name": "Fin.zero_eq_one_iff", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [632, 9], "def_end_pos": [632, 24]}]], "state_before": "R : Type u_1\nS : Type u_2\ni j : Fin 2\n\u22a2 (basisOneI.repr (![1, I] i)) j = (Finsupp.single i 1) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Finset.noncommProd_commute", "start": [376, 1], "end": [382, 15], "traced_tactics": [{"tactic": "apply Multiset.noncommProd_commute", "annotated_tactic": ["apply <a>Multiset.noncommProd_commute</a>", [{"full_name": "Multiset.noncommProd_commute", "def_path": "Mathlib/Data/Finset/NoncommProd.lean", "def_pos": [223, 9], "def_end_pos": [223, 28]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 x \u2208 s, Commute y (f x)\n\u22a2 Commute y (s.noncommProd f comm)", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 x \u2208 s, Commute y (f x)\n\u22a2 \u2200 x \u2208 Multiset.map f s.val, Commute y x"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 x \u2208 s, Commute y (f x)\n\u22a2 \u2200 x \u2208 Multiset.map f s.val, Commute y x", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 x \u2208 s, Commute y\u271d (f x)\ny : \u03b2\n\u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y"}, {"tactic": "rw [Multiset.mem_map]", "annotated_tactic": ["rw [<a>Multiset.mem_map</a>]", [{"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1277, 9], "def_end_pos": [1277, 16]}]], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 x \u2208 s, Commute y\u271d (f x)\ny : \u03b2\n\u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y", "state_after": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 x \u2208 s, Commute y\u271d (f x)\ny : \u03b2\n\u22a2 (\u2203 a \u2208 s.val, f a = y) \u2192 Commute y\u271d y"}, {"tactic": "rintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9", []], "state_before": "case h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 x \u2208 s, Commute y\u271d (f x)\ny : \u03b2\n\u22a2 (\u2203 a \u2208 s.val, f a = y) \u2192 Commute y\u271d y", "state_after": "case h.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 x \u2208 s, Commute y (f x)\nx : \u03b1\nhx : x \u2208 s.val\n\u22a2 Commute y (f x)"}, {"tactic": "exact h x hx", "annotated_tactic": ["exact h x hx", []], "state_before": "case h.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : FunLike F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : (\u2191s).Pairwise fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 x \u2208 s, Commute y (f x)\nx : \u03b1\nhx : x \u2208 s.val\n\u22a2 Commute y (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": 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"Mathlib/AlgebraicGeometry/Restrict.lean", "def_pos": [70, 7], "def_end_pos": [70, 28]}, {"full_name": "AlgebraicGeometry.Scheme.basicOpen_res_eq", "def_path": "Mathlib/AlgebraicGeometry/Scheme.lean", "def_pos": [473, 9], "def_end_pos": [473, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v, u\u2081} C\nX\u271d X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191\u0393(X \u2223_\u1d64 U, \u22a4)\n\u22a2 \u03b9Opens U ''\u1d41 (X \u2223_\u1d64 U).basicOpen r = X.basicOpen r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "PMF.pure_bind", "start": [132, 1], "end": [136, 14], "traced_tactics": [{"tactic": "have : \u2200 b a', ite (a' = a) (f a' b) 0 = ite (a' = a) (f a b) 0 := fun b a' => by\n  split_ifs with h <;> simp [h]", "annotated_tactic": ["have : \u2200 b a', <a>ite</a> (a' = a) (f a' b) 0 = <a>ite</a> (a' = a) (f a b) 0 := fun b a' => by\n    split_ifs with h <;> simp [h]", [{"full_name": "ite", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [970, 21], "def_end_pos": [970, 24]}, {"full_name": "ite", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [970, 21], "def_end_pos": [970, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\n\u22a2 (pure a).bind f = f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\nthis : \u2200 (b : \u03b2) (a' : \u03b1), (if a' = a then (f a') b else 0) = if a' = a then (f a) b else 0\n\u22a2 (pure a).bind f = f a"}, {"tactic": "ext b", "annotated_tactic": ["ext b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\nthis : \u2200 (b : \u03b2) (a' : \u03b1), (if a' = a then (f a') b else 0) = if a' = a then (f a) b else 0\n\u22a2 (pure a).bind f = f a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\nthis : \u2200 (b : \u03b2) (a' : \u03b1), (if a' = a then (f a') b else 0) = if a' = a then (f a) b else 0\nb : \u03b2\n\u22a2 ((pure a).bind f) b = (f a) b"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\nthis : \u2200 (b : \u03b2) (a' : \u03b1), (if a' = a then (f a') b else 0) = if a' = a then (f a) b else 0\nb : \u03b2\n\u22a2 ((pure a).bind f) b = (f a) b", "state_after": "no goals"}, {"tactic": "split_ifs with h <;> simp [h]", "annotated_tactic": ["split_ifs with h <;> simp [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf\u271d : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\na : \u03b1\nf : \u03b1 \u2192 PMF \u03b2\nb : \u03b2\na' : \u03b1\n\u22a2 (if a' = a then (f a') b else 0) = if a' = a then (f a) b else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "full_name": "MeasureTheory.Measure.rnDeriv_withDensity_left", "start": [138, 1], "end": [150, 22], "traced_tactics": [{"tactic": "let \u03bc' := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)", "annotated_tactic": ["let \u03bc' := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "have h\u03bc'\u03bd : \u03bc' \u226a \u03bd := withDensity_absolutelyContinuous _ _", "annotated_tactic": ["have h\u03bc'\u03bd : \u03bc' \u226a \u03bd := <a>withDensity_absolutelyContinuous</a> _ _", [{"full_name": "MeasureTheory.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [150, 9], "def_end_pos": [150, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "have h := rnDeriv_withDensity_left_of_absolutelyContinuous h\u03bc'\u03bd hf\u03bd", "annotated_tactic": ["have h := <a>rnDeriv_withDensity_left_of_absolutelyContinuous</a> h\u03bc'\u03bd hf\u03bd", [{"full_name": "MeasureTheory.Measure.rnDeriv_withDensity_left_of_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [122, 7], "def_end_pos": [122, 55]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "have h1 : \u03bc'.rnDeriv \u03bd =\u1d50[\u03bd] \u03bc.rnDeriv \u03bd :=\n  Measure.rnDeriv_withDensity _ (Measure.measurable_rnDeriv _ _)", "annotated_tactic": ["have h1 : \u03bc'.rnDeriv \u03bd =\u1d50[\u03bd] \u03bc.rnDeriv \u03bd :=\n    <a>Measure.rnDeriv_withDensity</a> _ (<a>Measure.measurable_rnDeriv</a> _ _)", [{"full_name": "MeasureTheory.Measure.rnDeriv_withDensity", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [561, 9], "def_end_pos": [561, 28]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "have h2 : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1d50[\u03bd] (\u03bc.withDensity f).rnDeriv \u03bd := by\n  exact rnDeriv_withDensity_withDensity_rnDeriv_left \u03bc \u03bd hf\u03bc hf_ne_top", "annotated_tactic": ["have h2 : (\u03bc'.withDensity f).<a>rnDeriv</a> \u03bd =\u1d50[\u03bd] (\u03bc.withDensity f).<a>rnDeriv</a> \u03bd := by\n    exact <a>rnDeriv_withDensity_withDensity_rnDeriv_left</a> \u03bc \u03bd hf\u03bc hf_ne_top", [{"full_name": "MeasureTheory.Measure.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [80, 31], "def_end_pos": [80, 38]}, {"full_name": "MeasureTheory.Measure.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [80, 31], "def_end_pos": [80, 38]}, {"full_name": "MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [98, 7], "def_end_pos": [98, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\nh2 : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] (\u03bc.withDensity f).rnDeriv \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "filter_upwards [h, h1, h2] with x hx hx1 hx2", "annotated_tactic": ["filter_upwards [h, h1, h2] with x hx hx1 hx2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\nh2 : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] (\u03bc.withDensity f).rnDeriv \u03bd\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc.rnDeriv \u03bd x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\nh2 : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] (\u03bc.withDensity f).rnDeriv \u03bd\nx : \u03b1\nhx : (\u03bc'.withDensity f).rnDeriv \u03bd x = f x * \u03bc'.rnDeriv \u03bd x\nhx1 : \u03bc'.rnDeriv \u03bd x = \u03bc.rnDeriv \u03bd x\nhx2 : (\u03bc'.withDensity f).rnDeriv \u03bd x = (\u03bc.withDensity f).rnDeriv \u03bd x\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd x = f x * \u03bc.rnDeriv \u03bd x"}, {"tactic": "rw [\u2190 hx2, hx, hx1]", "annotated_tactic": ["rw [\u2190 hx2, hx, hx1]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\nh2 : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] (\u03bc.withDensity f).rnDeriv \u03bd\nx : \u03b1\nhx : (\u03bc'.withDensity f).rnDeriv \u03bd x = f x * \u03bc'.rnDeriv \u03bd x\nhx1 : \u03bc'.rnDeriv \u03bd x = \u03bc.rnDeriv \u03bd x\nhx2 : (\u03bc'.withDensity f).rnDeriv \u03bd x = (\u03bc.withDensity f).rnDeriv \u03bd x\n\u22a2 (\u03bc.withDensity f).rnDeriv \u03bd x = f x * \u03bc.rnDeriv \u03bd x", "state_after": "no goals"}, {"tactic": "exact rnDeriv_withDensity_withDensity_rnDeriv_left \u03bc \u03bd hf\u03bc hf_ne_top", "annotated_tactic": ["exact <a>rnDeriv_withDensity_withDensity_rnDeriv_left</a> \u03bc \u03bd hf\u03bc hf_ne_top", [{"full_name": "MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left", "def_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "def_pos": [98, 7], "def_end_pos": [98, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d\u00b9 \u03bd\u271d\u00b9 \u03bc\u271d \u03bd\u271d : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf\u03bc : AEMeasurable f \u03bc\nhf\u03bd : AEMeasurable f \u03bd\nhf_ne_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u03bc' : Measure \u03b1 := \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh\u03bc'\u03bd : \u03bc' \u226a \u03bd\nh : (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] fun x => f x * \u03bc'.rnDeriv \u03bd x\nh1 : \u03bc'.rnDeriv \u03bd =\u1da0[ae \u03bd] \u03bc.rnDeriv \u03bd\n\u22a2 (\u03bc'.withDensity f).rnDeriv \u03bd =\u1da0[ae \u03bd] (\u03bc.withDensity f).rnDeriv \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/PontryaginDual.lean", "full_name": "PontryaginDual.map_one", "start": [79, 1], "end": [80, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.norm_apply_le_norm", "start": [554, 1], "end": [561, 61], "traced_tactics": [{"tactic": "rcases eq_or_ne p \u221e with (rfl | hp')", "annotated_tactic": ["rcases <a>eq_or_ne</a> p \u221e with (rfl | hp')", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : \u21a5(lp E \u22a4)\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016"}, {"tactic": "have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp hp'", "annotated_tactic": ["have hp'' : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp hp'", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016"}, {"tactic": "have : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ p.toReal := fun i => Real.rpow_nonneg (norm_nonneg _) _", "annotated_tactic": ["have : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ p.toReal := fun i => <a>Real.rpow_nonneg</a> (<a>norm_nonneg</a> _) _", [{"full_name": "Real.rpow_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [169, 9], "def_end_pos": [169, 20]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ p.toReal\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016"}, {"tactic": "rw [\u2190 Real.rpow_le_rpow_iff (norm_nonneg _) (norm_nonneg' _) hp'']", "annotated_tactic": ["rw [\u2190 <a>Real.rpow_le_rpow_iff</a> (<a>norm_nonneg</a> _) (<a>norm_nonneg'</a> _) hp'']", [{"full_name": "Real.rpow_le_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [582, 9], "def_end_pos": [582, 25]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "lp.norm_nonneg'", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [430, 9], "def_end_pos": [430, 21]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ p.toReal\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ p.toReal\n\u22a2 \u2016\u2191f i\u2016 ^ p.toReal \u2264 \u2016f\u2016 ^ p.toReal"}, {"tactic": "convert le_hasSum (hasSum_norm hp'' f) i fun i _ => this i", "annotated_tactic": ["convert <a>le_hasSum</a> (<a>hasSum_norm</a> hp'' f) i fun i _ => this i", [{"full_name": "le_hasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "def_pos": [108, 3], "def_end_pos": [108, 14]}, {"full_name": "lp.hasSum_norm", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [424, 9], "def_end_pos": [424, 20]}]], "state_before": "case inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : \u21a5(lp E p)\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < p.toReal\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ p.toReal\n\u22a2 \u2016\u2191f i\u2016 ^ p.toReal \u2264 \u2016f\u2016 ^ p.toReal", "state_after": "no goals"}, {"tactic": "haveI : Nonempty \u03b1 := \u27e8i\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> \u03b1 := \u27e8i\u27e9", [{"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : \u21a5(lp E \u22a4)\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : \u21a5(lp E \u22a4)\nthis : Nonempty \u03b1\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016"}, {"tactic": "exact (isLUB_norm f).1 \u27e8i, rfl\u27e9", "annotated_tactic": ["exact (<a>isLUB_norm</a> f).1 \u27e8i, <a>rfl</a>\u27e9", [{"full_name": "lp.isLUB_norm", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [401, 9], "def_end_pos": [401, 19]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : \u21a5(lp E \u22a4)\nthis : Nonempty \u03b1\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.Iso.self_symm_id", "start": [197, 1], "end": [198, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Integer.lean", "full_name": "IsLocalization.isInteger_mul", "start": [59, 1], "end": [60, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.of_thinkN_terminates", "start": [417, 1], "end": [418, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Concept.lean", "full_name": "Concept.ext", "start": [181, 1], "end": [186, 6], "traced_tactics": [{"tactic": "obtain \u27e8\u27e8s\u2081, t\u2081\u27e9, h\u2081, _\u27e9 := c", "annotated_tactic": ["obtain \u27e8\u27e8s\u2081, t\u2081\u27e9, h\u2081, _\u27e9 := c", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nc d : Concept \u03b1 \u03b2 r\nh : c.toProd.1 = d.toProd.1\n\u22a2 c = d", "state_after": "case mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082 : Set \u03b2\nd : Concept \u03b1 \u03b2 r\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r (s\u2081, t\u2081).1 = (s\u2081, t\u2081).2\nclosure_snd\u271d : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\nh : { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d }.toProd.1 = d.toProd.1\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d } = d"}, {"tactic": "obtain \u27e8\u27e8s\u2082, t\u2082\u27e9, h\u2082, _\u27e9 := d", "annotated_tactic": ["obtain \u27e8\u27e8s\u2082, t\u2082\u27e9, h\u2082, _\u27e9 := d", []], "state_before": "case mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082 : Set \u03b2\nd : Concept \u03b1 \u03b2 r\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r (s\u2081, t\u2081).1 = (s\u2081, t\u2081).2\nclosure_snd\u271d : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\nh : { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d }.toProd.1 = d.toProd.1\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d } = d", "state_after": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r (s\u2081, t\u2081).1 = (s\u2081, t\u2081).2\nclosure_snd\u271d\u00b9 : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nh\u2082 : intentClosure r (s\u2082, t\u2082).1 = (s\u2082, t\u2082).2\nclosure_snd\u271d : extentClosure r (s\u2082, t\u2082).2 = (s\u2082, t\u2082).1\nh :\n  { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 }.toProd.1 =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }.toProd.1\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }"}, {"tactic": "dsimp at h\u2081 h\u2082 h", "annotated_tactic": ["dsimp at h\u2081 h\u2082 h", []], "state_before": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r (s\u2081, t\u2081).1 = (s\u2081, t\u2081).2\nclosure_snd\u271d\u00b9 : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nh\u2082 : intentClosure r (s\u2082, t\u2082).1 = (s\u2082, t\u2082).2\nclosure_snd\u271d : extentClosure r (s\u2082, t\u2082).2 = (s\u2082, t\u2082).1\nh :\n  { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 }.toProd.1 =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }.toProd.1\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }", "state_after": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r s\u2081 = t\u2081\nclosure_snd\u271d\u00b9 : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nh\u2082 : intentClosure r s\u2082 = t\u2082\nclosure_snd\u271d : extentClosure r (s\u2082, t\u2082).2 = (s\u2082, t\u2082).1\nh : s\u2081 = s\u2082\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }"}, {"tactic": "substs h h\u2081 h\u2082", "annotated_tactic": ["substs h h\u2081 h\u2082", []], "state_before": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nh\u2081 : intentClosure r s\u2081 = t\u2081\nclosure_snd\u271d\u00b9 : extentClosure r (s\u2081, t\u2081).2 = (s\u2081, t\u2081).1\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nh\u2082 : intentClosure r s\u2082 = t\u2082\nclosure_snd\u271d : extentClosure r (s\u2082, t\u2082).2 = (s\u2082, t\u2082).1\nh : s\u2081 = s\u2082\n\u22a2 { toProd := (s\u2081, t\u2081), closure_fst := h\u2081, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2082, t\u2082), closure_fst := h\u2082, closure_snd := closure_snd\u271d }", "state_after": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\ns\u2081 : Set \u03b1\nclosure_snd\u271d\u00b9 closure_snd\u271d : extentClosure r (s\u2081, intentClosure r s\u2081).2 = (s\u2081, intentClosure r s\u2081).1\n\u22a2 { toProd := (s\u2081, intentClosure r s\u2081), closure_fst := \u22ef, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2081, intentClosure r s\u2081), closure_fst := \u22ef, closure_snd := closure_snd\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Sort u_5\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns s\u2081\u271d s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\ns\u2081 : Set \u03b1\nclosure_snd\u271d\u00b9 closure_snd\u271d : extentClosure r (s\u2081, intentClosure r s\u2081).2 = (s\u2081, intentClosure r s\u2081).1\n\u22a2 { toProd := (s\u2081, intentClosure r s\u2081), closure_fst := \u22ef, closure_snd := closure_snd\u271d\u00b9 } =\n    { toProd := (s\u2081, intentClosure r s\u2081), closure_fst := \u22ef, closure_snd := closure_snd\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/SemidirectProduct.lean", "full_name": "SemidirectProduct.map_inr", "start": [300, 1], "end": [300, 76], "traced_tactics": [{"tactic": "simp [map]", "annotated_tactic": ["simp [<a>map</a>]", [{"full_name": "SemidirectProduct.map", "def_path": "Mathlib/GroupTheory/SemidirectProduct.lean", "def_pos": [266, 5], "def_end_pos": [266, 8]}]], "state_before": "N : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), f\u2081.comp (MulEquiv.toMonoidHom (\u03c6 g)) = (MulEquiv.toMonoidHom (\u03c6\u2081 (f\u2082 g))).comp f\u2081\ng : G\n\u22a2 (map f\u2081 f\u2082 h) (inr g) = inr (f\u2082 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_iff_ordConnected", "start": [635, 1], "end": [637, 85], "traced_tactics": [{"tactic": "simp_rw [convex_iff_segment_subset, segment_eq_uIcc, ordConnected_iff_uIcc_subset]", "annotated_tactic": ["simp_rw [<a>convex_iff_segment_subset</a>, <a>segment_eq_uIcc</a>, <a>ordConnected_iff_uIcc_subset</a>]", [{"full_name": "convex_iff_segment_subset", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 34]}, {"full_name": "segment_eq_uIcc", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [556, 9], "def_end_pos": [556, 24]}, {"full_name": "Set.ordConnected_iff_uIcc_subset", "def_path": "Mathlib/Order/Interval/Set/OrdConnected.lean", "def_pos": [335, 9], "def_end_pos": [335, 37]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\n\u22a2 Convex \ud835\udd5c s \u2194 s.OrdConnected", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Basis/Flag.lean", "full_name": "Basis.flag_le_ker_coord_iff", "start": [74, 1], "end": [76, 76], "traced_tactics": [{"tactic": "simp [flag_le_iff, Finsupp.single_apply_eq_zero, imp_false, imp_not_comm]", "annotated_tactic": ["simp [<a>flag_le_iff</a>, <a>Finsupp.single_apply_eq_zero</a>, <a>imp_false</a>, <a>imp_not_comm</a>]", [{"full_name": "Basis.flag_le_iff", "def_path": "Mathlib/LinearAlgebra/Basis/Flag.lean", "def_pos": [38, 9], "def_end_pos": [38, 20]}, {"full_name": "Finsupp.single_apply_eq_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [361, 9], "def_end_pos": [361, 29]}, {"full_name": "imp_false", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1420, 17], "def_end_pos": [1420, 26]}, {"full_name": "imp_not_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1424, 9], "def_end_pos": [1424, 21]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nn : \u2115\ninst\u271d : Nontrivial R\nb : Basis (Fin n) R M\nk : Fin (n + 1)\nl : Fin n\n\u22a2 b.flag k \u2264 LinearMap.ker (b.coord l) \u2194 k \u2264 l.castSucc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/ConjAct.lean", "full_name": "ConjAct.orbitRel_conjAct", "start": [311, 1], "end": [312, 63], "traced_tactics": [{"tactic": "rw [orbitRel_apply, mem_orbit_conjAct]", "annotated_tactic": ["rw [<a>orbitRel_apply</a>, <a>mem_orbit_conjAct</a>]", [{"full_name": "MulAction.orbitRel_apply", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [423, 9], "def_end_pos": [423, 23]}, {"full_name": "ConjAct.mem_orbit_conjAct", "def_path": "Mathlib/GroupTheory/GroupAction/ConjAct.lean", "def_pos": [307, 9], "def_end_pos": [307, 26]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\ng h : G\n\u22a2 (orbitRel (ConjAct G) G).Rel g h = IsConj g h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Lattice.lean", "full_name": "Multiset.sup_coe", "start": [34, 1], "end": [35, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Field.lean", "full_name": "IsPreconnected.eq_or_eq_neg_of_sq_eq", "start": [142, 1], "end": [149, 74], "traced_tactics": [{"tactic": "have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by\n  simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx", "annotated_tactic": ["have hsq : <a>EqOn</a> ((f / g) ^ 2) 1 S := fun x hx => by\n    simpa [<a>div_eq_one_iff_eq</a> (<a>pow_ne_zero</a> _ (hg_ne hx))] using hsq hx", [{"full_name": "Set.EqOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [263, 5], "def_end_pos": [263, 9]}, {"full_name": "div_eq_one_iff_eq", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [373, 7], "def_end_pos": [373, 24]}, {"full_name": "pow_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [200, 7], "def_end_pos": [200, 18]}]], "state_before": "K : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\n\u22a2 EqOn f g S \u2228 EqOn f (-g) S", "state_after": "K : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq\u271d : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\nhsq : EqOn ((f / g) ^ 2) 1 S\n\u22a2 EqOn f g S \u2228 EqOn f (-g) S"}, {"tactic": "simpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)]\n  using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq", "annotated_tactic": ["simpa (config := { contextual := <a>true</a> }) [<a>EqOn</a>, <a>div_eq_iff</a> (hg_ne _)]\n    using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Set.EqOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [263, 5], "def_end_pos": [263, 9]}, {"full_name": "div_eq_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [354, 22], "def_end_pos": [354, 32]}]], "state_before": "K : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq\u271d : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\nhsq : EqOn ((f / g) ^ 2) 1 S\n\u22a2 EqOn f g S \u2228 EqOn f (-g) S", "state_after": "no goals"}, {"tactic": "simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx", "annotated_tactic": ["simpa [<a>div_eq_one_iff_eq</a> (<a>pow_ne_zero</a> _ (hg_ne hx))] using hsq hx", [{"full_name": "div_eq_one_iff_eq", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [373, 7], "def_end_pos": [373, 24]}, {"full_name": "pow_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [200, 7], "def_end_pos": [200, 18]}]], "state_before": "K : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\nx : \u03b1\nhx : x \u2208 S\n\u22a2 ((f / g) ^ 2) x = 1 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "full_name": "IsSigmaCompact.image", "start": [95, 1], "end": [97, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean", "full_name": "MeasureTheory.snorm_le_snorm_mul_snorm'_of_norm", "start": [268, 1], "end": [272, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Codisjoint.dual", "start": [426, 1], "end": [428, 5], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.map_strictMono_of_injective", "start": [436, 1], "end": [437, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "iff_mpr_iff_true_intro", "start": [428, 1], "end": [428, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "full_name": "Polynomial.not_irreducible_C", "start": [598, 1], "end": [602, 84], "traced_tactics": [{"tactic": "by_cases H : x = 0", "annotated_tactic": ["by_cases H : x = 0", []], "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\n\u22a2 \u00acIrreducible (C x)", "state_after": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible (C x)\n\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : \u00acx = 0\n\u22a2 \u00acIrreducible (C x)"}, {"tactic": "rw [H, C_0]", "annotated_tactic": ["rw [H, <a>C_0</a>]", [{"full_name": "Polynomial.C_0", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 12]}]], "state_before": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible (C x)", "state_after": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible 0"}, {"tactic": "exact not_irreducible_zero", "annotated_tactic": ["exact <a>not_irreducible_zero</a>", [{"full_name": "not_irreducible_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [225, 9], "def_end_pos": [225, 29]}]], "state_before": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible 0", "state_after": "no goals"}, {"tactic": "exact fun hx => Irreducible.not_unit hx <| isUnit_C.2 <| isUnit_iff_ne_zero.2 H", "annotated_tactic": ["exact fun hx => <a>Irreducible.not_unit</a> hx <| <a>isUnit_C</a>.2 <| <a>isUnit_iff_ne_zero</a>.2 H", [{"full_name": "Irreducible.not_unit", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [193, 3], "def_end_pos": [193, 11]}, {"full_name": "Polynomial.isUnit_C", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [150, 9], "def_end_pos": [150, 17]}, {"full_name": "isUnit_iff_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 27]}]], "state_before": "case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : \u00acx = 0\n\u22a2 \u00acIrreducible (C x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.pow_sup_pow_eq_top", "start": [728, 1], "end": [729, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.on_preimage", "start": [416, 1], "end": [418, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean", "full_name": "EisensteinSeries.r1_eq", "start": [52, 1], "end": [53, 61], "traced_tactics": [{"tactic": "rw [div_pow, div_add_one (by positivity), one_div_div, r1]", "annotated_tactic": ["rw [<a>div_pow</a>, <a>div_add_one</a> (by positivity), <a>one_div_div</a>, <a>r1</a>]", [{"full_name": "div_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [845, 7], "def_end_pos": [845, 14]}, {"full_name": "div_add_one", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [47, 9], "def_end_pos": [47, 20]}, {"full_name": "one_div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 20]}, {"full_name": "EisensteinSeries.r1", "def_path": "Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean", "def_pos": [50, 5], "def_end_pos": [50, 7]}]], "state_before": "z : \u210d\n\u22a2 r1 z = 1 / ((z.re / z.im) ^ 2 + 1)", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "z : \u210d\n\u22a2 z.im ^ 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean", "full_name": "derivWithin_arctan", "start": [147, 1], "end": [149, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Grade.lean", "full_name": "IsMax.grade", "start": [146, 11], "end": [147, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Homotopy/Basic.lean", "full_name": "ContinuousMap.HomotopyWith.apply_zero", "start": [450, 1], "end": [451, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.Disjoint.conj", "start": [127, 1], "end": [128, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "IsStrictOrder.swap", "start": [101, 1], "end": [102, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.glueDist_swap", "start": [95, 1], "end": [100, 95], "traced_tactics": [{"tactic": "simp only [glueDist, Sum.swap_inl, Sum.swap_inr, dist_comm, add_comm]", "annotated_tactic": ["simp only [<a>glueDist</a>, <a>Sum.swap_inl</a>, <a>Sum.swap_inr</a>, <a>dist_comm</a>, <a>add_comm</a>]", [{"full_name": "Metric.glueDist", "def_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "def_pos": [65, 5], "def_end_pos": [65, 13]}, {"full_name": "Sum.swap_inl", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [102, 17], "def_end_pos": [102, 25]}, {"full_name": "Sum.swap_inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [104, 17], "def_end_pos": [104, 25]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "X : Type u\nY : Type v\nZ : Type w\ninst\u271d\u00b9 : MetricSpace X\ninst\u271d : MetricSpace Y\n\u03a6\u271d : Z \u2192 X\n\u03a8\u271d : Z \u2192 Y\n\u03b5\u271d : \u211d\n\u03a6 : Z \u2192 X\n\u03a8 : Z \u2192 Y\n\u03b5 : \u211d\nval\u271d\u00b9 : X\nval\u271d : Y\n\u22a2 glueDist \u03a8 \u03a6 \u03b5 (Sum.inl val\u271d\u00b9).swap (Sum.inr val\u271d).swap = glueDist \u03a6 \u03a8 \u03b5 (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d)", "state_after": "no goals"}, {"tactic": "simp only [glueDist, Sum.swap_inl, Sum.swap_inr, dist_comm, add_comm]", "annotated_tactic": ["simp only [<a>glueDist</a>, <a>Sum.swap_inl</a>, <a>Sum.swap_inr</a>, <a>dist_comm</a>, <a>add_comm</a>]", [{"full_name": "Metric.glueDist", "def_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "def_pos": [65, 5], "def_end_pos": [65, 13]}, {"full_name": "Sum.swap_inl", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [102, 17], "def_end_pos": [102, 25]}, {"full_name": "Sum.swap_inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [104, 17], "def_end_pos": [104, 25]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "X : Type u\nY : Type v\nZ : Type w\ninst\u271d\u00b9 : MetricSpace X\ninst\u271d : MetricSpace Y\n\u03a6\u271d : Z \u2192 X\n\u03a8\u271d : Z \u2192 Y\n\u03b5\u271d : \u211d\n\u03a6 : Z \u2192 X\n\u03a8 : Z \u2192 Y\n\u03b5 : \u211d\nval\u271d\u00b9 : Y\nval\u271d : X\n\u22a2 glueDist \u03a8 \u03a6 \u03b5 (Sum.inr val\u271d\u00b9).swap (Sum.inl val\u271d).swap = glueDist \u03a6 \u03a8 \u03b5 (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Alter.lean", "full_name": "Batteries.RBNode.Balanced.modify", "start": [290, 11], "end": [291, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/ConjExponents.lean", "full_name": "Real.isConjExponent_iff_eq_conjExponent", "start": [132, 1], "end": [133, 60], "traced_tactics": [{"tactic": "field_simp [h]", "annotated_tactic": ["field_simp [h]", []], "state_before": "a b p q : \u211d\nh\u271d : p.IsConjExponent q\nhp : 1 < p\nh : q = p / (p - 1)\n\u22a2 p\u207b\u00b9 + q\u207b\u00b9 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "full_name": "Profinite.NobelingProof.GoodProducts.linearIndependentEmpty", "start": [768, 1], "end": [769, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "NNReal.tendsto_sum_nat_add", "start": [1235, 1], "end": [1238, 12], "traced_tactics": [{"tactic": "rw [\u2190 tendsto_coe]", "annotated_tactic": ["rw [\u2190 <a>tendsto_coe</a>]", [{"full_name": "NNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 Tendsto (fun i => \u2211' (k : \u2115), f (k + i)) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 Tendsto (fun a => \u2191(\u2211' (k : \u2115), f (k + a))) atTop (\ud835\udcdd \u21910)"}, {"tactic": "convert _root_.tendsto_sum_nat_add fun i => (f i : \u211d)", "annotated_tactic": ["convert <a>_root_.tendsto_sum_nat_add</a> fun i => (f i : \u211d)", [{"full_name": "tendsto_sum_nat_add", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [252, 3], "def_end_pos": [252, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 Tendsto (fun a => \u2191(\u2211' (k : \u2115), f (k + a))) atTop (\ud835\udcdd \u21910)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nx\u271d : \u2115\n\u22a2 \u2191(\u2211' (k : \u2115), f (k + x\u271d)) = \u2211' (k : \u2115), \u2191(f (k + x\u271d))"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nx\u271d : \u2115\n\u22a2 \u2191(\u2211' (k : \u2115), f (k + x\u271d)) = \u2211' (k : \u2115), \u2191(f (k + x\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/List.lean", "full_name": "List.formPerm_apply_get_length", "start": [160, 1], "end": [162, 39], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx\u271d x : \u03b1\nxs : List \u03b1\n\u22a2 xs.length < (x :: xs).length", "state_after": "no goals"}, {"tactic": "simp [formPerm_apply_getElem_length]", "annotated_tactic": ["simp [<a>formPerm_apply_getElem_length</a>]", [{"full_name": "List.formPerm_apply_getElem_length", "def_path": "Mathlib/GroupTheory/Perm/List.lean", "def_pos": [156, 9], "def_end_pos": [156, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx\u271d x : \u03b1\nxs : List \u03b1\n\u22a2 (x :: xs).formPerm ((x :: xs).get \u27e8xs.length, \u22ef\u27e9) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.natDegree_C_add", "start": [757, 9], "end": [758, 22], "traced_tactics": [{"tactic": "simp [add_comm _ p]", "annotated_tactic": ["simp [<a>add_comm</a> _ p]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d p q : R[X]\n\u03b9 : Type u_1\na : R\n\u22a2 (C a + p).natDegree = p.natDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.exists", "start": [308, 1], "end": [310, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Set.Finite.eventuallyEq_iInter", "start": [1825, 1], "end": [1828, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Side.lean", "full_name": "AffineSubspace.wOppSide_iff_exists_left", "start": [474, 1], "end": [491, 32], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 s.WOppSide x y \u2194 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 s.WOppSide x y \u2192 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 (x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2192 s.WOppSide x y"}, {"tactic": "rintro \u27e8p\u2081', hp\u2081', p\u2082', hp\u2082', h0 | h0 | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, hr\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8p\u2081', hp\u2081', p\u2082', hp\u2082', h0 | h0 | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, hr\u27e9\u27e9", []], "state_before": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 s.WOppSide x y \u2192 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x -\u1d65 p\u2081' = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)"}, {"tactic": "rw [vsub_eq_zero_iff_eq] at h0", "annotated_tactic": ["rw [<a>vsub_eq_zero_iff_eq</a>] at h0", [{"full_name": "vsub_eq_zero_iff_eq", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [136, 9], "def_end_pos": [136, 28]}]], "state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x -\u1d65 p\u2081' = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)"}, {"tactic": "rw [h0]", "annotated_tactic": ["rw [h0]", []], "state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 p\u2081' \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (p\u2081' -\u1d65 p\u2081) (p\u2082 -\u1d65 y)"}, {"tactic": "exact Or.inl hp\u2081'", "annotated_tactic": ["exact <a>Or.inl</a> hp\u2081'", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 p\u2081' \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (p\u2081' -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "no goals"}, {"tactic": "refine Or.inr \u27e8p\u2082', hp\u2082', ?_\u27e9", "annotated_tactic": ["refine <a>Or.inr</a> \u27e8p\u2082', hp\u2082', ?_\u27e9", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) (p\u2082' -\u1d65 y)"}, {"tactic": "rw [h0]", "annotated_tactic": ["rw [h0]", []], "state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) (p\u2082' -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) 0"}, {"tactic": "exact SameRay.zero_right _", "annotated_tactic": ["exact <a>SameRay.zero_right</a> _", [{"full_name": "SameRay.zero_right", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [56, 9], "def_end_pos": [56, 19]}]], "state_before": "case mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) 0", "state_after": "no goals"}, {"tactic": "refine Or.inr \u27e8(-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082', s.smul_vsub_vadd_mem _ h hp\u2081' hp\u2082',\n  Or.inr (Or.inr \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, ?_\u27e9)\u27e9", "annotated_tactic": ["refine <a>Or.inr</a> \u27e8(-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082', s.smul_vsub_vadd_mem _ h hp\u2081' hp\u2082',\n        <a>Or.inr</a> (<a>Or.inr</a> \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, ?_\u27e9)\u27e9", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "state_after": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 ((-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082' -\u1d65 y)"}, {"tactic": "rw [vadd_vsub_assoc, smul_add, \u2190 hr, smul_smul, neg_div, mul_neg,\n  mul_div_cancel\u2080 _ hr\u2082.ne.symm, neg_smul, neg_add_eq_sub, \u2190 smul_sub,\n  vsub_sub_vsub_cancel_right]", "annotated_tactic": ["rw [<a>vadd_vsub_assoc</a>, <a>smul_add</a>, \u2190 hr, <a>smul_smul</a>, <a>neg_div</a>, <a>mul_neg</a>,\n        <a>mul_div_cancel\u2080</a> _ hr\u2082.ne.symm, <a>neg_smul</a>, <a>neg_add_eq_sub</a>, \u2190 <a>smul_sub</a>,\n        <a>vsub_sub_vsub_cancel_right</a>]", [{"full_name": "vadd_vsub_assoc", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [117, 9], "def_end_pos": [117, 24]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "smul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [446, 7], "def_end_pos": [446, 16]}, {"full_name": "neg_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [117, 9], "def_end_pos": [117, 16]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "mul_div_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [516, 7], "def_end_pos": [516, 22]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [253, 9], "def_end_pos": [253, 17]}, {"full_name": "neg_add_eq_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [740, 3], "def_end_pos": [740, 14]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}, {"full_name": "vsub_sub_vsub_cancel_right", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [172, 9], "def_end_pos": [172, 35]}]], "state_before": "case mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 ((-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082' -\u1d65 y)", "state_after": "no goals"}, {"tactic": "rintro (h' | \u27e8h\u2081, h\u2082, h\u2083\u27e9)", "annotated_tactic": ["rintro (h' | \u27e8h\u2081, h\u2082, h\u2083\u27e9)", []], "state_before": "case mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 (x \u2208 s \u2228 \u2203 p\u2082 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2192 s.WOppSide x y", "state_after": "case mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh' : x \u2208 s\n\u22a2 s.WOppSide x y\n\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh\u2081 : P\nh\u2082 : h\u2081 \u2208 s\nh\u2083 : SameRay R (x -\u1d65 p\u2081) (h\u2081 -\u1d65 y)\n\u22a2 s.WOppSide x y"}, {"tactic": "exact wOppSide_of_left_mem y h'", "annotated_tactic": ["exact <a>wOppSide_of_left_mem</a> y h'", [{"full_name": "AffineSubspace.wOppSide_of_left_mem", "def_path": "Mathlib/Analysis/Convex/Side.lean", "def_pos": [260, 9], "def_end_pos": [260, 29]}]], "state_before": "case mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh' : x \u2208 s\n\u22a2 s.WOppSide x y", "state_after": "no goals"}, {"tactic": "exact \u27e8p\u2081, h, h\u2081, h\u2082, h\u2083\u27e9", "annotated_tactic": ["exact \u27e8p\u2081, h, h\u2081, h\u2082, h\u2083\u27e9", []], "state_before": "case mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh\u2081 : P\nh\u2082 : h\u2081 \u2208 s\nh\u2083 : SameRay R (x -\u1d65 p\u2081) (h\u2081 -\u1d65 y)\n\u22a2 s.WOppSide x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Circular.lean", "full_name": "Set.right_mem_cIcc", "start": [365, 1], "end": [366, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "HasFDerivAtFilter.prod", "start": [58, 1], "end": [61, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/Convergence.lean", "full_name": "LSeries.abscissaOfAbsConv_congr'", "start": [34, 1], "end": [38, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Operations.lean", "full_name": "ENNReal.coe_sub", "start": [360, 9], "end": [360, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Defs.lean", "full_name": "Int.neg_pred", "start": [219, 1], "end": [220, 57], "traced_tactics": [{"tactic": "rw [\u2190 Int.neg_eq_comm.mp (neg_succ (-a)), Int.neg_neg]", "annotated_tactic": ["rw [\u2190 Int.neg_eq_comm.mp (<a>neg_succ</a> (-a)), <a>Int.neg_neg</a>]", [{"full_name": "Int.neg_succ", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [213, 7], "def_end_pos": [213, 15]}, {"full_name": "Int.neg_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [75, 27], "def_end_pos": [75, 34]}]], "state_before": "a\u271d b c d m n a : \u2124\n\u22a2 -a.pred = (-a).succ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "ArithmeticFunction.add_apply", "start": [230, 1], "end": [231, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "full_name": "FractionalIdeal.coe_spanSingleton", "start": [629, 1], "end": [631, 6], "traced_tactics": [{"tactic": "rw [spanSingleton]", "annotated_tactic": ["rw [<a>spanSingleton</a>]", [{"full_name": "FractionalIdeal.spanSingleton", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [623, 17], "def_end_pos": [623, 30]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u2075 : CommRing P\ninst\u271d\u2074 : Algebra R P\nloc : IsLocalization S P\nR\u2081 : Type u_3\ninst\u271d\u00b3 : CommRing R\u2081\nK : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R\u2081 K\ninst\u271d : IsFractionRing R\u2081 K\nx : P\n\u22a2 \u2191(spanSingleton S x) = span R {x}", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u2075 : CommRing P\ninst\u271d\u2074 : Algebra R P\nloc : IsLocalization S P\nR\u2081 : Type u_3\ninst\u271d\u00b3 : CommRing R\u2081\nK : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R\u2081 K\ninst\u271d : IsFractionRing R\u2081 K\nx : P\n\u22a2 \u2191\u27e8span R {x}, \u22ef\u27e9 = span R {x}"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u2075 : CommRing P\ninst\u271d\u2074 : Algebra R P\nloc : IsLocalization S P\nR\u2081 : Type u_3\ninst\u271d\u00b3 : CommRing R\u2081\nK : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R\u2081 K\ninst\u271d : IsFractionRing R\u2081 K\nx : P\n\u22a2 \u2191\u27e8span R {x}, \u22ef\u27e9 = span R {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartialHomeomorph.lean", "full_name": "PartialHomeomorph.nhdsWithin_source_inter", "start": [278, 1], "end": [279, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.ext", "start": [75, 1], "end": [76, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.LpToLpRestrictCLM_coeFn", "start": [1112, 1], "end": [1114, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "full_name": "MeasureTheory.setLIntegral_tilted", "start": [165, 1], "end": [180, 9], "traced_tactics": [{"tactic": "by_cases hf : AEMeasurable f \u03bc", "annotated_tactic": ["by_cases hf : <a>AEMeasurable</a> f \u03bc", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [418, 5], "def_end_pos": [418, 17]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "rw [Measure.tilted, setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080']", "annotated_tactic": ["rw [<a>Measure.tilted</a>, <a>setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080'</a>]", [{"full_name": "MeasureTheory.Measure.tilted", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [38, 5], "def_end_pos": [38, 19]}, {"full_name": "MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080'", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [532, 9], "def_end_pos": [532, 70]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1) in s, ((fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) * g) a \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc\n\ncase pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) (\u03bc.restrict s)\n\ncase pos.h'f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4"}, {"tactic": "simp only [Pi.mul_apply]", "annotated_tactic": ["simp only [<a>Pi.mul_apply</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1) in s, ((fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) * g) a \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine AEMeasurable.restrict ?_", "annotated_tactic": ["refine <a>AEMeasurable.restrict</a> ?_", [{"full_name": "AEMeasurable.restrict", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [313, 9], "def_end_pos": [313, 30]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) (\u03bc.restrict s)", "state_after": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) \u03bc"}, {"tactic": "exact ((measurable_exp.comp_aemeasurable hf).div_const _).ennreal_ofReal", "annotated_tactic": ["exact ((measurable_exp.comp_aemeasurable hf).<a>div_const</a> _).<a>ennreal_ofReal</a>", [{"full_name": "AEMeasurable.div_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [319, 9], "def_end_pos": [319, 31]}, {"full_name": "AEMeasurable.ennreal_ofReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "def_pos": [199, 7], "def_end_pos": [199, 34]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) \u03bc", "state_after": "no goals"}, {"tactic": "filter_upwards", "annotated_tactic": ["filter_upwards", []], "state_before": "case pos.h'f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4", "state_after": "case pos.h'f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200 (a : \u03b1), ENNReal.ofReal (rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4"}, {"tactic": "simp only [ENNReal.ofReal_lt_top, implies_true]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_lt_top</a>, <a>implies_true</a>]", [{"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [337, 17], "def_end_pos": [337, 30]}, {"full_name": "implies_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [132, 17], "def_end_pos": [132, 29]}]], "state_before": "case pos.h'f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200 (a : \u03b1), ENNReal.ofReal (rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4", "state_after": "no goals"}, {"tactic": "have hf' : \u00ac Integrable (fun x \u21a6 exp (f x)) \u03bc := by\n  exact fun h \u21a6 hf (aemeasurable_of_aemeasurable_exp h.1.aemeasurable)", "annotated_tactic": ["have hf' : \u00ac <a>Integrable</a> (fun x \u21a6 <a>exp</a> (f x)) \u03bc := by\n      exact fun h \u21a6 hf (<a>aemeasurable_of_aemeasurable_exp</a> h.1.<a>aemeasurable</a>)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [438, 5], "def_end_pos": [438, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.aemeasurable_of_aemeasurable_exp", "def_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "def_pos": [47, 7], "def_end_pos": [47, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1246, 19], "def_end_pos": [1246, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "simp only [hf, not_false_eq_true, tilted_of_not_aemeasurable, Measure.restrict_zero,\n  lintegral_zero_measure]", "annotated_tactic": ["simp only [hf, <a>not_false_eq_true</a>, <a>tilted_of_not_aemeasurable</a>, <a>Measure.restrict_zero</a>,\n      <a>lintegral_zero_measure</a>]", [{"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}, {"full_name": "MeasureTheory.tilted_of_not_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [47, 7], "def_end_pos": [47, 33]}, {"full_name": "MeasureTheory.Measure.restrict_zero", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [165, 9], "def_end_pos": [165, 22]}, {"full_name": "MeasureTheory.lintegral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [666, 9], "def_end_pos": [666, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "rw [integral_undef hf']", "annotated_tactic": ["rw [<a>integral_undef</a> hf']", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [833, 9], "def_end_pos": [833, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / 0) * g x \u2202\u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / 0) * g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact fun h \u21a6 hf (aemeasurable_of_aemeasurable_exp h.1.aemeasurable)", "annotated_tactic": ["exact fun h \u21a6 hf (<a>aemeasurable_of_aemeasurable_exp</a> h.1.<a>aemeasurable</a>)", [{"full_name": "Real.aemeasurable_of_aemeasurable_exp", "def_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "def_pos": [47, 7], "def_end_pos": [47, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1246, 19], "def_end_pos": [1246, 31]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u00acIntegrable (fun x => rexp (f x)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "isPathConnected_iff_eq", "start": [958, 1], "end": [963, 22], "traced_tactics": [{"tactic": "constructor <;> rintro \u27e8x, x_in, h\u27e9 <;> use x, x_in", "annotated_tactic": ["constructor <;> rintro \u27e8x, x_in, h\u27e9 <;> use x, x_in", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\nF : Set X\n\u22a2 IsPathConnected F \u2194 \u2203 x \u2208 F, pathComponentIn x F = F", "state_after": "case right\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\nF : Set X\nx : X\nx_in : x \u2208 F\nh : \u2200 {y : X}, y \u2208 F \u2192 JoinedIn F x y\n\u22a2 pathComponentIn x F = F\n\ncase right\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\nF : Set X\nx : X\nx_in : x \u2208 F\nh : pathComponentIn x F = F\n\u22a2 \u2200 {y : X}, y \u2208 F \u2192 JoinedIn F x y"}, {"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "case right\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\nF : Set X\nx : X\nx_in : x \u2208 F\nh : \u2200 {y : X}, y \u2208 F \u2192 JoinedIn F x y\n\u22a2 pathComponentIn x F = F", "state_after": "case right.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y\u271d z : X\n\u03b9 : Type u_3\nF : Set X\nx : X\nx_in : x \u2208 F\nh : \u2200 {y : X}, y \u2208 F \u2192 JoinedIn F x y\ny : X\n\u22a2 y \u2208 pathComponentIn x F \u2194 y \u2208 F"}, {"tactic": "exact \u27e8fun hy => hy.mem.2, h\u27e9", "annotated_tactic": ["exact \u27e8fun hy => hy.mem.2, h\u27e9", []], "state_before": "case right.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y\u271d z : X\n\u03b9 : Type u_3\nF : Set X\nx : X\nx_in : x \u2208 F\nh : \u2200 {y : X}, y \u2208 F \u2192 JoinedIn F x y\ny : X\n\u22a2 y \u2208 pathComponentIn x F \u2194 y \u2208 F", "state_after": "no goals"}, {"tactic": "intro y y_in", "annotated_tactic": 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goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.mul_subset_iff_left", "start": [431, 1], "end": [432, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.mem_Lp_of_ae_le_mul", "start": [406, 1], "end": [408, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.Sigma.fst_eq_of_dist_lt_one", "start": [355, 1], "end": [358, 28], "traced_tactics": [{"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\nx y : (i : \u03b9) \u00d7 E i\nh : dist x y < 1\n\u22a2 x.fst = y.fst", "state_after": "case mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\ny : (i : \u03b9) \u00d7 E i\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : dist \u27e8fst\u271d, snd\u271d\u27e9 y < 1\n\u22a2 \u27e8fst\u271d, snd\u271d\u27e9.fst = y.fst"}, {"tactic": "cases y", "annotated_tactic": ["cases y", []], "state_before": "case mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\ny : (i : \u03b9) \u00d7 E i\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : dist \u27e8fst\u271d, snd\u271d\u27e9 y < 1\n\u22a2 \u27e8fst\u271d, snd\u271d\u27e9.fst = y.fst", "state_after": "case mk.mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\nfst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : E fst\u271d\u00b9\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : dist \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9 \u27e8fst\u271d, snd\u271d\u27e9 < 1\n\u22a2 \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9.fst = \u27e8fst\u271d, snd\u271d\u27e9.fst"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case mk.mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\nfst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : E fst\u271d\u00b9\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : dist \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9 \u27e8fst\u271d, snd\u271d\u27e9 < 1\n\u22a2 \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9.fst = \u27e8fst\u271d, snd\u271d\u27e9.fst", "state_after": "case mk.mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\nfst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : E fst\u271d\u00b9\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9.fst \u2260 \u27e8fst\u271d, snd\u271d\u27e9.fst\n\u22a2 1 \u2264 dist \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9 \u27e8fst\u271d, snd\u271d\u27e9"}, {"tactic": "apply one_le_dist_of_ne h", "annotated_tactic": ["apply <a>one_le_dist_of_ne</a> h", [{"full_name": "Metric.Sigma.one_le_dist_of_ne", "def_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "def_pos": [349, 9], "def_end_pos": [349, 26]}]], "state_before": "case mk.mk\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\nfst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : E fst\u271d\u00b9\nfst\u271d : \u03b9\nsnd\u271d : E fst\u271d\nh : \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9.fst \u2260 \u27e8fst\u271d, snd\u271d\u27e9.fst\n\u22a2 1 \u2264 dist \u27e8fst\u271d\u00b9, snd\u271d\u00b9\u27e9 \u27e8fst\u271d, snd\u271d\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "full_name": "Polynomial.reverse_natDegree", "start": [295, 1], "end": [296, 85], "traced_tactics": [{"tactic": "rw [f.natDegree_eq_reverse_natDegree_add_natTrailingDegree, add_tsub_cancel_right]", "annotated_tactic": ["rw [f.natDegree_eq_reverse_natDegree_add_natTrailingDegree, <a>add_tsub_cancel_right</a>]", [{"full_name": "add_tsub_cancel_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [356, 9], "def_end_pos": [356, 30]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\n\u22a2 f.reverse.natDegree = f.natDegree - f.natTrailingDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "full_name": "IsometryEquiv.image_closedBall", "start": [637, 1], "end": [639, 64], "traced_tactics": [{"tactic": "rw [\u2190 h.preimage_symm, h.symm.preimage_closedBall, symm_symm]", "annotated_tactic": ["rw [\u2190 h.preimage_symm, h.symm.preimage_closedBall, <a>symm_symm</a>]", [{"full_name": "IsometryEquiv.symm_symm", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [418, 9], "def_end_pos": [418, 18]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nh\u271d h : \u03b1 \u2243\u1d62 \u03b2\nx : \u03b1\nr : \u211d\n\u22a2 \u21d1h '' Metric.closedBall x r = Metric.closedBall (h x) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Cones.lean", "full_name": "CategoryTheory.Limits.CoconeMorphism.ext", "start": [525, 1], "end": [528, 8], "traced_tactics": [{"tactic": "cases f", "annotated_tactic": ["cases f", []], "state_before": "J : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : J \u2964 C\nc c' : Cocone F\nf g : c \u27f6 c'\nw : f.hom = g.hom\n\u22a2 f = g", "state_after": "case mk\nJ : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : J \u2964 C\nc c' : Cocone F\ng : c \u27f6 c'\nhom\u271d : c.pt \u27f6 c'.pt\nw\u271d : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d = c'.\u03b9.app j\nw : { hom := hom\u271d, w := w\u271d }.hom = g.hom\n\u22a2 { hom := hom\u271d, w := w\u271d } = g"}, {"tactic": "cases g", "annotated_tactic": ["cases g", []], "state_before": "case mk\nJ : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : J \u2964 C\nc c' : Cocone F\ng : c \u27f6 c'\nhom\u271d : c.pt \u27f6 c'.pt\nw\u271d : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d = c'.\u03b9.app j\nw : { hom := hom\u271d, w := w\u271d }.hom = g.hom\n\u22a2 { hom := hom\u271d, w := w\u271d } = g", "state_after": "case mk.mk\nJ : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : J \u2964 C\nc c' : Cocone F\nhom\u271d\u00b9 : c.pt \u27f6 c'.pt\nw\u271d\u00b9 : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d\u00b9 = c'.\u03b9.app j\nhom\u271d : c.pt \u27f6 c'.pt\nw\u271d : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d = c'.\u03b9.app j\nw : { hom := hom\u271d\u00b9, w := w\u271d\u00b9 }.hom = { hom := hom\u271d, w := w\u271d }.hom\n\u22a2 { hom := hom\u271d\u00b9, w := w\u271d\u00b9 } = { hom := hom\u271d, w := w\u271d }"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk\nJ : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : J \u2964 C\nc c' : Cocone F\nhom\u271d\u00b9 : c.pt \u27f6 c'.pt\nw\u271d\u00b9 : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d\u00b9 = c'.\u03b9.app j\nhom\u271d : c.pt \u27f6 c'.pt\nw\u271d : \u2200 (j : J), c.\u03b9.app j \u226b hom\u271d = c'.\u03b9.app j\nw : { hom := hom\u271d\u00b9, w := w\u271d\u00b9 }.hom = { hom := hom\u271d, w := w\u271d }.hom\n\u22a2 { hom := hom\u271d\u00b9, w := w\u271d\u00b9 } = { hom := hom\u271d, w := w\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.const_sub", "start": [186, 1], "end": [188, 34], "traced_tactics": [{"tactic": "simp only [\u2190 sub_sub, h.sub_eq]", "annotated_tactic": ["simp only [\u2190 <a>sub_sub</a>, h.sub_eq]", [{"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d : AddCommGroup \u03b1\nh : Periodic f c\na x : \u03b1\n\u22a2 (fun x => f (a - x)) (x + c) = (fun x => f (a - x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "full_name": "Finset.card_le_mul_card_image", "start": [290, 1], "end": [292, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Basic.lean", "full_name": "ENNReal.toReal_eq_toReal_iff", "start": [438, 1], "end": [440, 71], "traced_tactics": [{"tactic": "simp only [ENNReal.toReal, NNReal.coe_inj, toNNReal_eq_toNNReal_iff]", "annotated_tactic": ["simp only [<a>ENNReal.toReal</a>, <a>NNReal.coe_inj</a>, <a>toNNReal_eq_toNNReal_iff</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "NNReal.coe_inj", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [167, 26], "def_end_pos": [167, 33]}, {"full_name": "ENNReal.toNNReal_eq_toNNReal_iff", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 33]}]], "state_before": "\u03b1 : Type u_1\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\n\u22a2 x.toReal = y.toReal \u2194 x = y \u2228 x = 0 \u2227 y = \u22a4 \u2228 x = \u22a4 \u2227 y = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithBot.map_one", "start": [527, 11], "end": [528, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.smul_re", "start": [263, 1], "end": [264, 43], "traced_tactics": [{"tactic": "rw [real_smul_eq_coe_mul, re_ofReal_mul]", "annotated_tactic": ["rw [<a>real_smul_eq_coe_mul</a>, <a>re_ofReal_mul</a>]", [{"full_name": "RCLike.real_smul_eq_coe_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 29]}, {"full_name": "RCLike.re_ofReal_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nr : \u211d\nz : K\n\u22a2 re (r \u2022 z) = r * re z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_eq_two", "start": [315, 1], "end": [322, 55], "traced_tactics": [{"tactic": "refine \u27e8fun h \u21a6 ?_, fun \u27e8x, y, hne, hs\u27e9 \u21a6 by rw [hs, encard_pair hne]\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 ?_, fun \u27e8x, y, hne, hs\u27e9 \u21a6 by rw [hs, <a>encard_pair</a> hne]\u27e9", [{"full_name": "Set.encard_pair", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\n\u22a2 s.encard = 2 \u2194 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "obtain \u27e8x, hx\u27e9 := nonempty_of_encard_ne_zero (s := s) (by rw [h]; simp)", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 := <a>nonempty_of_encard_ne_zero</a> (s := s) (by rw [h]; simp)", [{"full_name": "Set.nonempty_of_encard_ne_zero", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [98, 9], "def_end_pos": [98, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "rw [\u2190 insert_eq_of_mem hx, \u2190 insert_diff_singleton, encard_insert_of_not_mem (fun h \u21a6 h.2 rfl),\n  \u2190 one_add_one_eq_two, WithTop.add_right_cancel_iff (WithTop.one_ne_top), encard_eq_one] at h", "annotated_tactic": ["rw [\u2190 <a>insert_eq_of_mem</a> hx, \u2190 <a>insert_diff_singleton</a>, <a>encard_insert_of_not_mem</a> (fun h \u21a6 h.2 <a>rfl</a>),\n    \u2190 <a>one_add_one_eq_two</a>, <a>WithTop.add_right_cancel_iff</a> (<a>WithTop.one_ne_top</a>), <a>encard_eq_one</a>] at h", [{"full_name": "Set.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 25]}, {"full_name": "Set.insert_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2020, 9], "def_end_pos": [2020, 30]}, {"full_name": "Set.encard_insert_of_not_mem", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [231, 9], "def_end_pos": [231, 27]}, {"full_name": "WithTop.add_right_cancel_iff", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [164, 9], "def_end_pos": [164, 29]}, {"full_name": "WithTop.one_ne_top", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [57, 37], "def_end_pos": [57, 47]}, {"full_name": "Set.encard_eq_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [290, 9], "def_end_pos": [290, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nh : \u2203 x_1, s \\ {x} = {x_1}\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "obtain \u27e8y, h\u27e9 := h", "annotated_tactic": ["obtain \u27e8y, h\u27e9 := h", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nh : \u2203 x_1, s \\ {x} = {x_1}\nhx : x \u2208 s\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}"}, {"tactic": "refine \u27e8x, y, by rintro rfl; exact (h.symm.subset rfl).2 rfl, ?_\u27e9", "annotated_tactic": ["refine \u27e8x, y, by rintro rfl; exact (h.symm.subset <a>rfl</a>).2 <a>rfl</a>, ?_\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y}", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 s = {x, y}"}, {"tactic": "rw [\u2190 h, insert_diff_singleton, insert_eq_of_mem hx]", "annotated_tactic": ["rw [\u2190 h, <a>insert_diff_singleton</a>, <a>insert_eq_of_mem</a> hx]", [{"full_name": "Set.insert_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2020, 9], "def_end_pos": [2020, 30]}, {"full_name": "Set.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 s = {x, y}", "state_after": "no goals"}, {"tactic": "rw [hs, encard_pair hne]", "annotated_tactic": ["rw [hs, <a>encard_pair</a> hne]", [{"full_name": "Set.encard_pair", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx\u271d : \u2203 x y, x \u2260 y \u2227 s = {x, y}\nx y : \u03b1\nhne : x \u2260 y\nhs : s = {x, y}\n\u22a2 s.encard = 2", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\n\u22a2 s.encard \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\n\u22a2 2 \u2260 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : s.encard = 2\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nh : s \\ {x} = {y}\n\u22a2 x \u2260 y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : s \\ {x} = {x}\n\u22a2 False"}, {"tactic": "exact (h.symm.subset rfl).2 rfl", "annotated_tactic": ["exact (h.symm.subset <a>rfl</a>).2 <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : s \\ {x} = {x}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "Inducing.continuousAt_iff", "start": [126, 1], "end": [128, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean", "full_name": "Nat.sum_nil", "start": [178, 9], "end": [178, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.injective_toWeakDualBCNN", "start": [567, 1], "end": [578, 46], "traced_tactics": [{"tactic": "intro \u03bc \u03bd h\u03bc\u03bd", "annotated_tactic": ["intro \u03bc \u03bd h\u03bc\u03bd", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u22a2 Injective toWeakDualBCNN", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\n\u22a2 \u03bc = \u03bd"}, {"tactic": "apply ext_of_forall_lintegral_eq", "annotated_tactic": ["apply <a>ext_of_forall_lintegral_eq</a>", [{"full_name": "MeasureTheory.FiniteMeasure.ext_of_forall_lintegral_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [325, 9], "def_end_pos": [325, 35]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\n\u22a2 \u03bc = \u03bd", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd"}, {"tactic": "have key := congr_fun (congrArg DFunLike.coe h\u03bc\u03bd) f", "annotated_tactic": ["have key := <a>congr_fun</a> (<a>congrArg</a> <a>DFunLike.coe</a> h\u03bc\u03bd) f", [{"full_name": "congr_fun", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [78, 7], "def_end_pos": [78, 16]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "DFunLike.coe", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [147, 3], "def_end_pos": [147, 6]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd"}, {"tactic": "apply (ENNReal.toNNReal_eq_toNNReal_iff' ?_ ?_).mp key", "annotated_tactic": ["apply (<a>ENNReal.toNNReal_eq_toNNReal_iff'</a> ?_ ?_).<a>mp</a> key", [{"full_name": "ENNReal.toNNReal_eq_toNNReal_iff'", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [443, 9], "def_end_pos": [443, 34]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u2191\u03bd", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u2191\u03bc \u2260 \u22a4\n\n\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u2191\u03bd \u2260 \u22a4"}, {"tactic": "exact (lintegral_lt_top_of_nnreal \u03bc f).ne", "annotated_tactic": ["exact (<a>lintegral_lt_top_of_nnreal</a> \u03bc f).<a>ne</a>", [{"full_name": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [42, 9], "def_end_pos": [42, 35]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u2191\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact (lintegral_lt_top_of_nnreal \u03bd f).ne", "annotated_tactic": ["exact (<a>lintegral_lt_top_of_nnreal</a> \u03bd f).<a>ne</a>", [{"full_name": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean", "def_pos": [42, 9], "def_end_pos": [42, 35]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2076 : SMul R \u211d\u22650\ninst\u271d\u2075 : SMul R \u211d\u22650\u221e\ninst\u271d\u2074 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : HasOuterApproxClosed \u03a9\ninst\u271d : BorelSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh\u03bc\u03bd : \u03bc.toWeakDualBCNN = \u03bd.toWeakDualBCNN\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey : \u03bc.toWeakDualBCNN f = \u03bd.toWeakDualBCNN f\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u2191\u03bd \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.neg_le_of_neg_le", "start": [994, 11], "end": [994, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Finset.lean", "full_name": "Finset.Nonempty.csSup_eq_max'", "start": [25, 1], "end": [26, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.digits_lt_base", "start": [421, 1], "end": [423, 27], "traced_tactics": [{"tactic": "rcases b with (_ | _ | b) <;> try simp_all", "annotated_tactic": ["rcases b with (_ | _ | b) <;> try simp_all", []], "state_before": "n b m d : \u2115\nhb : 1 < b\nhd : d \u2208 b.digits m\n\u22a2 d < b", "state_after": "case succ.succ\nn m d b : \u2115\nhd : d \u2208 (b + 1 + 1).digits m\n\u22a2 d < b + 1 + 1"}, {"tactic": "exact digits_lt_base' hd", "annotated_tactic": ["exact <a>digits_lt_base'</a> hd", [{"full_name": "Nat.digits_lt_base'", "def_path": "Mathlib/Data/Nat/Digits.lean", "def_pos": [403, 9], "def_end_pos": [403, 24]}]], "state_before": "case succ.succ\nn m d b : \u2115\nhd : d \u2208 (b + 1 + 1).digits m\n\u22a2 d < b + 1 + 1", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case succ.succ\nn m d b : \u2115\nhb : 1 < b + 1 + 1\nhd : d \u2208 (b + 1 + 1).digits m\n\u22a2 d < b + 1 + 1", "state_after": "case succ.succ\nn m d b : \u2115\nhd : d \u2208 (b + 1 + 1).digits m\n\u22a2 d < b + 1 + 1"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean", "full_name": "Matrix.exp_neg", "start": [182, 1], "end": [187, 36], "traced_tactics": [{"tactic": "rw [nonsing_inv_eq_ring_inverse]", "annotated_tactic": ["rw [<a>nonsing_inv_eq_ring_inverse</a>]", [{"full_name": "Matrix.nonsing_inv_eq_ring_inverse", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [245, 9], "def_end_pos": [245, 36]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (-A) = (exp \ud835\udd42 A)\u207b\u00b9", "state_after": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)"}, {"tactic": "letI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing", "annotated_tactic": ["letI : <a>SeminormedRing</a> (<a>Matrix</a> m m \ud835\udd38) := <a>Matrix.linftyOpSemiNormedRing</a>", [{"full_name": "SeminormedRing", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [44, 7], "def_end_pos": [44, 21]}, {"full_name": "Matrix", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Matrix.linftyOpSemiNormedRing", "def_path": "Mathlib/Analysis/Matrix.lean", "def_pos": [401, 15], "def_end_pos": [401, 37]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)", "state_after": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)"}, {"tactic": "letI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing", "annotated_tactic": ["letI : <a>NormedRing</a> (<a>Matrix</a> m m \ud835\udd38) := <a>Matrix.linftyOpNormedRing</a>", [{"full_name": "NormedRing", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [75, 7], "def_end_pos": [75, 17]}, {"full_name": "Matrix", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Matrix.linftyOpNormedRing", "def_path": "Mathlib/Analysis/Matrix.lean", "def_pos": [420, 15], "def_end_pos": [420, 33]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)", "state_after": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)"}, {"tactic": "letI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra", "annotated_tactic": ["letI : <a>NormedAlgebra</a> \ud835\udd42 (<a>Matrix</a> m m \ud835\udd38) := <a>Matrix.linftyOpNormedAlgebra</a>", [{"full_name": "NormedAlgebra", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [262, 7], "def_end_pos": [262, 20]}, {"full_name": "Matrix", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [57, 5], "def_end_pos": [57, 11]}, {"full_name": "Matrix.linftyOpNormedAlgebra", "def_path": "Mathlib/Analysis/Matrix.lean", "def_pos": [429, 15], "def_end_pos": [429, 36]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)", "state_after": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)"}, {"tactic": "exact (Ring.inverse_exp _ A).symm", "annotated_tactic": ["exact (<a>Ring.inverse_exp</a> _ A).<a>symm</a>", [{"full_name": "Ring.inverse_exp", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [511, 9], "def_end_pos": [511, 32]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "Nat.le_div_two_iff_mul_two_le", "start": [1162, 1], "end": [1163, 86], "traced_tactics": [{"tactic": "rw [Nat.le_div_iff_mul_le zero_lt_two, \u2190 Int.ofNat_le, Int.ofNat_mul, Nat.cast_two]", "annotated_tactic": ["rw [<a>Nat.le_div_iff_mul_le</a> <a>zero_lt_two</a>, \u2190 <a>Int.ofNat_le</a>, <a>Int.ofNat_mul</a>, <a>Nat.cast_two</a>]", [{"full_name": "Nat.le_div_iff_mul_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [216, 9], "def_end_pos": [216, 26]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"full_name": "Int.ofNat_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [51, 28], "def_end_pos": [51, 36]}, {"full_name": "Int.ofNat_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [26, 22], "def_end_pos": [26, 31]}, {"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [208, 9], "def_end_pos": [208, 17]}]], "state_before": "n m : \u2115\n\u22a2 m \u2264 n / 2 \u2194 \u2191m * 2 \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.eqOn_union", "start": [236, 1], "end": [237, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Support.lean", "full_name": "mulTSupport_mul", "start": [111, 1], "end": [115, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.floor_neg", "start": [1203, 1], "end": [1204, 87], "traced_tactics": [{"tactic": "rw [le_neg, ceil_le, le_floor, Int.cast_neg, le_neg]", "annotated_tactic": ["rw [<a>le_neg</a>, <a>ceil_le</a>, <a>le_floor</a>, <a>Int.cast_neg</a>, <a>le_neg</a>]", [{"full_name": "le_neg", "def_path": "Mathlib/Algebra/Order/Group/OrderIso.lean", "def_pos": [60, 15], "def_end_pos": [60, 21]}, {"full_name": "Int.ceil_le", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1199, 9], "def_end_pos": [1199, 16]}, {"full_name": "Int.le_floor", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [694, 9], "def_end_pos": [694, 17]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "le_neg", "def_path": "Mathlib/Algebra/Order/Group/OrderIso.lean", "def_pos": [60, 15], "def_end_pos": [60, 21]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 z \u2264 \u230a-a\u230b \u2194 z \u2264 -\u2308a\u2309", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.prod_sum", "start": [599, 1], "end": [603, 51], "traced_tactics": [{"tactic": "simp_rw [prod_sum_left, prod_sum_right, sum_sum]", "annotated_tactic": ["simp_rw [<a>prod_sum_left</a>, <a>prod_sum_right</a>, <a>sum_sum</a>]", [{"full_name": "MeasureTheory.Measure.prod_sum_left", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [583, 7], "def_end_pos": [583, 20]}, {"full_name": "MeasureTheory.Measure.prod_sum_right", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [589, 7], "def_end_pos": [589, 21]}, {"full_name": "MeasureTheory.Measure.sum_sum", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SFinite \u03bd\n\u03b9 : Type u_7\n\u03b9' : Type u_8\ninst\u271d\u00b9 : Countable \u03b9'\nm : \u03b9 \u2192 Measure \u03b1\nm' : \u03b9' \u2192 Measure \u03b2\ninst\u271d : \u2200 (n : \u03b9'), SFinite (m' n)\n\u22a2 (sum m).prod (sum m') = sum fun p => (m p.1).prod (m' p.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "divp_inv", "start": [523, 1], "end": [524, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/Rat.lean", "full_name": "Nat.dist_cast_rat", "start": [57, 1], "end": [58, 52], "traced_tactics": [{"tactic": "rw [\u2190 Nat.dist_cast_real, \u2190 Rat.dist_cast]", "annotated_tactic": ["rw [\u2190 <a>Nat.dist_cast_real</a>, \u2190 <a>Rat.dist_cast</a>]", [{"full_name": "Nat.dist_cast_real", "def_path": "Mathlib/Topology/Instances/Nat.lean", "def_pos": [32, 9], "def_end_pos": [32, 23]}, {"full_name": "Rat.dist_cast", "def_path": "Mathlib/Topology/Instances/Rat.lean", "def_pos": [30, 9], "def_end_pos": [30, 18]}]], "state_before": "x y : \u2115\n\u22a2 dist \u2191x \u2191y = dist x y", "state_after": "x y : \u2115\n\u22a2 dist \u2191\u2191x \u2191\u2191y = dist \u2191x \u2191y"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "x y : \u2115\n\u22a2 dist \u2191\u2191x \u2191\u2191y = dist \u2191x \u2191y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Limits.lean", "full_name": "Profinite.finiteCoproduct.\u03b9_injective", "start": [206, 1], "end": [208, 43], "traced_tactics": [{"tactic": "intro x y hxy", "annotated_tactic": ["intro x y hxy", []], "state_before": "\u03b1 : Type w\ninst\u271d : Finite \u03b1\nX : \u03b1 \u2192 Profinite\na : \u03b1\n\u22a2 Function.Injective \u21d1(\u03b9 X a)", "state_after": "\u03b1 : Type w\ninst\u271d : Finite \u03b1\nX : \u03b1 \u2192 Profinite\na : \u03b1\nx y : (forget Profinite).obj (X a)\nhxy : (\u03b9 X a) x = (\u03b9 X a) y\n\u22a2 x = y"}, {"tactic": "exact eq_of_heq (Sigma.ext_iff.mp hxy).2", "annotated_tactic": ["exact <a>eq_of_heq</a> (Sigma.ext_iff.mp hxy).2", [{"full_name": "eq_of_heq", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [464, 9], "def_end_pos": [464, 18]}]], "state_before": "\u03b1 : Type w\ninst\u271d : Finite \u03b1\nX : \u03b1 \u2192 Profinite\na : \u03b1\nx y : (forget Profinite).obj (X a)\nhxy : (\u03b9 X a) x = (\u03b9 X a) y\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Submodule.mem_toSubalgebra", "start": [552, 1], "end": [553, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.ListBlank.ext", "start": [321, 1], "end": [336, 81], "traced_tactics": [{"tactic": "refine ListBlank.induction_on L\u2081 fun l\u2081 \u21a6 ListBlank.induction_on L\u2082 fun l\u2082 H \u21a6 ?_", "annotated_tactic": ["refine <a>ListBlank.induction_on</a> L\u2081 fun l\u2081 \u21a6 <a>ListBlank.induction_on</a> L\u2082 fun l\u2082 H \u21a6 ?_", [{"full_name": "Turing.ListBlank.induction_on", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [211, 19], "def_end_pos": [211, 41]}, {"full_name": "Turing.ListBlank.induction_on", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [211, 19], "def_end_pos": [211, 41]}]], "state_before": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\n\u22a2 (\u2200 (i_1 : \u2115), L\u2081.nth i_1 = L\u2082.nth i_1) \u2192 L\u2081 = L\u2082", "state_after": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\n\u22a2 mk l\u2081 = mk l\u2082"}, {"tactic": "wlog h : l\u2081.length \u2264 l\u2082.length", "annotated_tactic": ["wlog h : l\u2081.length \u2264 l\u2082.length", []], "state_before": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\n\u22a2 mk l\u2081 = mk l\u2082", "state_after": "case inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\n\u22a2 mk l\u2081 = mk l\u2082\n\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 mk l\u2081 = mk l\u2082"}, {"tactic": "refine Quotient.sound' (Or.inl \u27e8l\u2082.length - l\u2081.length, ?_\u27e9)", "annotated_tactic": ["refine <a>Quotient.sound'</a> (<a>Or.inl</a> \u27e8l\u2082.length - l\u2081.length, ?_\u27e9)", [{"full_name": "Quotient.sound'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [778, 9], "def_end_pos": [778, 15]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 mk l\u2081 = mk l\u2082", "state_after": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 l\u2082 = l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default"}, {"tactic": "refine List.ext_get ?_ fun i h h\u2082 \u21a6 Eq.symm ?_", "annotated_tactic": ["refine <a>List.ext_get</a> ?_ fun i h h\u2082 \u21a6 <a>Eq.symm</a> ?_", [{"full_name": "List.ext_get", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [267, 9], "def_end_pos": [267, 16]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 l\u2082 = l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default", "state_after": "case refine_1\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 l\u2082.length = (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\n\ncase refine_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), (mk l\u2081).nth i = (mk l\u2082).nth i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9"}, {"tactic": "simp only [ListBlank.nth_mk] at H", "annotated_tactic": ["simp only [<a>ListBlank.nth_mk</a>] at H", [{"full_name": "Turing.ListBlank.nth_mk", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [302, 9], "def_end_pos": [302, 25]}]], "state_before": "case refine_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), (mk l\u2081).nth i = (mk l\u2082).nth i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9", "state_after": "case refine_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9"}, {"tactic": "cases' lt_or_le i l\u2081.length with h' h'", "annotated_tactic": ["cases' <a>lt_or_le</a> i l\u2081.length with h' h'", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [338, 9], "def_end_pos": [338, 17]}]], "state_before": "case refine_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9", "state_after": "case refine_2.inl\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\nh' : i < l\u2081.length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9\n\ncase refine_2.inr\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\nh' : l\u2081.length \u2264 i\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9"}, {"tactic": "cases le_total l\u2081.length l\u2082.length <;> [skip; symm] <;> apply this <;> try assumption", "annotated_tactic": ["cases <a>le_total</a> l\u2081.length l\u2082.length <;> [skip; symm] <;> apply this <;> try assumption", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [297, 9], "def_end_pos": [297, 17]}]], "state_before": "case inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\n\u22a2 mk l\u2081 = mk l\u2082", "state_after": "case inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\nh\u271d : l\u2082.length \u2264 l\u2081.length\n\u22a2 \u2200 (i_1 : \u2115), (mk l\u2082).nth i_1 = (mk l\u2081).nth i_1"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\nh\u271d : l\u2082.length \u2264 l\u2081.length\n\u22a2 \u2200 (i_1 : \u2115), (mk l\u2082).nth i_1 = (mk l\u2081).nth i_1", "state_after": "case inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\nh\u271d : l\u2082.length \u2264 l\u2081.length\ni\u271d : \u2115\n\u22a2 (mk l\u2082).nth i\u271d = (mk l\u2081).nth i\u271d"}, {"tactic": "rw [H]", "annotated_tactic": ["rw [H]", []], "state_before": "case inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\nh\u271d : l\u2082.length \u2264 l\u2081.length\ni\u271d : \u2115\n\u22a2 (mk l\u2082).nth i\u271d = (mk l\u2081).nth i\u271d", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case inr.inr.h\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nthis :\n  \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n    (\u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1) \u2192 l\u2081.length \u2264 l\u2082.length \u2192 mk l\u2081 = mk l\u2082\nh : \u00acl\u2081.length \u2264 l\u2082.length\nh\u271d : l\u2082.length \u2264 l\u2081.length\n\u22a2 l\u2082.length \u2264 l\u2081.length", "state_after": "no goals"}, {"tactic": "simp only [Nat.add_sub_cancel' h, List.length_append, List.length_replicate]", "annotated_tactic": ["simp only [<a>Nat.add_sub_cancel'</a> h, <a>List.length_append</a>, <a>List.length_replicate</a>]", [{"full_name": "Nat.add_sub_cancel'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [113, 27], "def_end_pos": [113, 42]}, {"full_name": "List.length_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [496, 17], "def_end_pos": [496, 30]}, {"full_name": "List.length_replicate", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [575, 17], "def_end_pos": [575, 33]}]], "state_before": "case refine_1\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), (mk l\u2081).nth i_1 = (mk l\u2082).nth i_1\nh : l\u2081.length \u2264 l\u2082.length\n\u22a2 l\u2082.length = (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length", "state_after": "no goals"}, {"tactic": "simp only [List.get_append _ h', List.get?_eq_get h, List.get?_eq_get h',\n  \u2190 List.getI_eq_get _ h, \u2190 List.getI_eq_get _ h', H]", "annotated_tactic": ["simp only [<a>List.get_append</a> _ h', <a>List.get?_eq_get</a> h, <a>List.get?_eq_get</a> h',\n      \u2190 <a>List.getI_eq_get</a> _ h, \u2190 <a>List.getI_eq_get</a> _ h', H]", [{"full_name": "List.get_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 19]}, {"full_name": "List.get?_eq_get", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 20]}, {"full_name": "List.get?_eq_get", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 20]}, {"full_name": "List.getI_eq_get", "def_path": "Mathlib/Data/List/GetD.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}, {"full_name": "List.getI_eq_get", "def_path": "Mathlib/Data/List/GetD.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}]], "state_before": "case refine_2.inl\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\nh' : i < l\u2081.length\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9", "state_after": "no goals"}, {"tactic": "simp only [List.get_append_right' h', List.get_replicate, List.get?_eq_get h,\n  List.get?_len_le h', \u2190 List.getI_eq_default _ h', H, List.getI_eq_get _ h]", "annotated_tactic": ["simp only [<a>List.get_append_right'</a> h', <a>List.get_replicate</a>, <a>List.get?_eq_get</a> h,\n      <a>List.get?_len_le</a> h', \u2190 <a>List.getI_eq_default</a> _ h', H, <a>List.getI_eq_get</a> _ h]", [{"full_name": "List.get_append_right'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1046, 9], "def_end_pos": [1046, 26]}, {"full_name": "List.get_replicate", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1316, 9], "def_end_pos": [1316, 22]}, {"full_name": "List.get?_eq_get", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [143, 9], "def_end_pos": [143, 20]}, {"full_name": "List.get?_len_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [139, 9], "def_end_pos": [139, 20]}, {"full_name": "List.getI_eq_default", "def_path": "Mathlib/Data/List/GetD.lean", "def_pos": [134, 9], "def_end_pos": [134, 24]}, {"full_name": "List.getI_eq_get", "def_path": "Mathlib/Data/List/GetD.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}]], "state_before": "case refine_2.inr\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), l\u2081.getI i = l\u2082.getI i\nh\u271d : l\u2081.length \u2264 l\u2082.length\ni : \u2115\nh : i < l\u2082.length\nh\u2082 : i < (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).length\nh' : l\u2081.length \u2264 i\n\u22a2 (l\u2081 ++ List.replicate (l\u2082.length - l\u2081.length) default).get \u27e8i, h\u2082\u27e9 = l\u2082.get \u27e8i, h\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "full_name": "contMDiff_of_locally_contMDiffOn", "start": [913, 1], "end": [915, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Choose/Factorization.lean", "full_name": "Nat.pow_factorization_choose_le", "start": [49, 1], "end": [50, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "full_name": "Surreal.Multiplication.P1_of_ih", "start": [259, 1], "end": [281, 56], "traced_tactics": [{"tactic": "have ihxy := ih1 ih", "annotated_tactic": ["have ihxy := <a>ih1</a> ih", [{"full_name": "Surreal.Multiplication.ih1", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [220, 7], "def_end_pos": [220, 10]}]], "state_before": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\n\u22a2 (x * y).Numeric", "state_after": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\n\u22a2 (x * y).Numeric"}, {"tactic": "have ihyx := ih1_swap ih", "annotated_tactic": ["have ihyx := <a>ih1_swap</a> ih", [{"full_name": "Surreal.Multiplication.ih1_swap", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [225, 7], "def_end_pos": [225, 15]}]], "state_before": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\n\u22a2 (x * y).Numeric", "state_after": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\n\u22a2 (x * y).Numeric"}, {"tactic": "have ihxyn := ih1_neg_left (ih1_neg_right ihxy)", "annotated_tactic": ["have ihxyn := <a>ih1_neg_left</a> (<a>ih1_neg_right</a> ihxy)", [{"full_name": "Surreal.Multiplication.ih1_neg_left", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [197, 7], "def_end_pos": [197, 19]}, {"full_name": "Surreal.Multiplication.ih1_neg_right", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [202, 7], "def_end_pos": [202, 20]}]], "state_before": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\n\u22a2 (x * y).Numeric", "state_after": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\n\u22a2 (x * y).Numeric"}, {"tactic": "have ihyxn := ih1_neg_left (ih1_neg_right ihyx)", "annotated_tactic": ["have ihyxn := <a>ih1_neg_left</a> (<a>ih1_neg_right</a> ihyx)", [{"full_name": "Surreal.Multiplication.ih1_neg_left", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [197, 7], "def_end_pos": [197, 19]}, {"full_name": "Surreal.Multiplication.ih1_neg_right", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [202, 7], "def_end_pos": [202, 20]}]], "state_before": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\n\u22a2 (x * y).Numeric", "state_after": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 (x * y).Numeric"}, {"tactic": "refine numeric_def.mpr \u27e8?_, ?_, ?_\u27e9", "annotated_tactic": ["refine numeric_def.mpr \u27e8?_, ?_, ?_\u27e9", []], "state_before": "x x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 (x * y).Numeric", "state_after": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j\n\ncase refine_2\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric\n\ncase refine_3\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric"}, {"tactic": "all_goals\n  cases x; cases y\n  rintro (\u27e8i,j\u27e9|\u27e8i,j\u27e9) <;>\n  refine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub\n    (numeric_option_mul_option ih ?_ ?_) <;>\n  solve_by_elim [IsOption.mk_left, IsOption.mk_right]", "annotated_tactic": ["all_goals\n    cases x; cases y\n    rintro (\u27e8i,j\u27e9|\u27e8i,j\u27e9) <;>\n    refine ((<a>numeric_option_mul</a> ih ?_).<a>add</a> <| <a>numeric_mul_option</a> ih ?_).<a>sub</a>\n      (<a>numeric_option_mul_option</a> ih ?_ ?_) <;>\n    solve_by_elim [<a>IsOption.mk_left</a>, <a>IsOption.mk_right</a>]", [{"full_name": "Surreal.Multiplication.numeric_option_mul", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [211, 7], "def_end_pos": [211, 25]}, {"full_name": "SetTheory.PGame.Numeric.add", "def_path": "Mathlib/SetTheory/Surreal/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 12]}, {"full_name": "Surreal.Multiplication.numeric_mul_option", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [214, 7], "def_end_pos": [214, 25]}, {"full_name": "SetTheory.PGame.Numeric.sub", "def_path": "Mathlib/SetTheory/Surreal/Basic.lean", "def_pos": [256, 9], "def_end_pos": [256, 12]}, {"full_name": "Surreal.Multiplication.numeric_option_mul_option", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [217, 7], "def_end_pos": [217, 32]}, {"full_name": "SetTheory.PGame.IsOption.mk_left", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [225, 9], "def_end_pos": [225, 25]}, {"full_name": "SetTheory.PGame.IsOption.mk_right", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [230, 9], "def_end_pos": [230, 26]}]], "state_before": "case refine_2\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric\n\ncase refine_3\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric", "state_after": "no goals"}, {"tactic": "simp_rw [lt_iff_game_lt]", "annotated_tactic": ["simp_rw [<a>lt_iff_game_lt</a>]", [{"full_name": "SetTheory.Game.PGame.lt_iff_game_lt", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 29]}]], "state_before": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j", "state_after": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves) (j : (x * y).RightMoves), \u27e6(x * y).moveLeft i\u27e7 < \u27e6(x * y).moveRight j\u27e7"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (i : (x * y).LeftMoves) (j : (x * y).RightMoves), \u27e6(x * y).moveLeft i\u27e7 < \u27e6(x * y).moveRight j\u27e7", "state_after": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\n\u22a2 \u2200 (j : (x * y).RightMoves), \u27e6(x * y).moveLeft i\u27e7 < \u27e6(x * y).moveRight j\u27e7"}, {"tactic": "rw [rightMoves_mul_iff]", "annotated_tactic": ["rw [<a>rightMoves_mul_iff</a>]", [{"full_name": "SetTheory.PGame.rightMoves_mul_iff", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [894, 7], "def_end_pos": [894, 25]}]], "state_before": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\n\u22a2 \u2200 (j : (x * y).RightMoves), \u27e6(x * y).moveLeft i\u27e7 < \u27e6(x * y).moveRight j\u27e7", "state_after": "case refine_1\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\n\u22a2 (\u2200 (i_1 : x.LeftMoves) (j : (-y).LeftMoves), \u27e6(x * y).moveLeft i\u27e7 < -\u27e6x.mulOption (-y) i_1 j\u27e7) \u2227\n    \u2200 (i_1 : (-x).LeftMoves) (j : y.LeftMoves), \u27e6(x * y).moveLeft i\u27e7 < -\u27e6(-x).mulOption y i_1 j\u27e7"}, {"tactic": "intro j l", "annotated_tactic": ["intro j l", []], "state_before": "case refine_1.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\n\u22a2 \u2200 (i_1 : (-x).LeftMoves) (j : y.LeftMoves), \u27e6(x * y).moveLeft i\u27e7 < -\u27e6(-x).mulOption y i_1 j\u27e7", "state_after": "case refine_1.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\nj : (-x).LeftMoves\nl : y.LeftMoves\n\u22a2 \u27e6(x * y).moveLeft i\u27e7 < -\u27e6(-x).mulOption y j l\u27e7"}, {"tactic": "revert i", "annotated_tactic": ["revert i", []], "state_before": "case refine_1.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\ni : (x * y).LeftMoves\nj : (-x).LeftMoves\nl : y.LeftMoves\n\u22a2 \u27e6(x * y).moveLeft i\u27e7 < -\u27e6(-x).mulOption y j l\u27e7", "state_after": "case refine_1.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\n\u22a2 \u2200 (i : (x * y).LeftMoves), \u27e6(x * y).moveLeft i\u27e7 < -\u27e6(-x).mulOption y j l\u27e7"}, {"tactic": "constructor <;> intro i k", "annotated_tactic": ["constructor <;> intro i k", []], "state_before": "case refine_1.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\n\u22a2 (\u2200 (i : x.LeftMoves) (j_1 : y.LeftMoves), -\u27e6(-x).mulOption y j l\u27e7 > \u27e6x.mulOption y i j_1\u27e7) \u2227\n    \u2200 (i : (-x).LeftMoves) (j_1 : (-y).LeftMoves), -\u27e6(-x).mulOption y j l\u27e7 > \u27e6(-x).mulOption (-y) i j_1\u27e7", "state_after": "case refine_1.right.left\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : x.LeftMoves\nk : y.LeftMoves\n\u22a2 -\u27e6(-x).mulOption y j l\u27e7 > \u27e6x.mulOption y i k\u27e7\n\ncase refine_1.right.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-x).mulOption y j l\u27e7 > \u27e6(-x).mulOption (-y) i k\u27e7"}, {"tactic": "apply mulOption_lt hx hy ihxy ihyx", "annotated_tactic": ["apply <a>mulOption_lt</a> hx hy ihxy ihyx", [{"full_name": "Surreal.Multiplication.mulOption_lt", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [247, 7], "def_end_pos": [247, 19]}]], "state_before": "case refine_1.left.left\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : x.LeftMoves\nl : (-y).LeftMoves\ni : x.LeftMoves\nk : y.LeftMoves\n\u22a2 -\u27e6x.mulOption (-y) j l\u27e7 > \u27e6x.mulOption y i k\u27e7", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 mulOption_symm (-y), mulOption_neg_neg x]", "annotated_tactic": ["simp_rw [\u2190 <a>mulOption_symm</a> (-y), <a>mulOption_neg_neg</a> x]", [{"full_name": "SetTheory.PGame.mulOption_symm", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [863, 7], "def_end_pos": [863, 21]}, {"full_name": "SetTheory.PGame.mulOption_neg_neg", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [855, 7], "def_end_pos": [855, 24]}]], "state_before": "case refine_1.left.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : x.LeftMoves\nl : (-y).LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6x.mulOption (-y) j l\u27e7 > \u27e6(-x).mulOption (-y) i k\u27e7", "state_after": "case refine_1.left.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : x.LeftMoves\nl : (-y).LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))\u27e7 > \u27e6(-y).mulOption (-x) k i\u27e7"}, {"tactic": "apply mulOption_lt hy.neg hx.neg ihyxn ihxyn", "annotated_tactic": ["apply <a>mulOption_lt</a> hy.neg hx.neg ihyxn ihxyn", [{"full_name": "Surreal.Multiplication.mulOption_lt", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [247, 7], "def_end_pos": [247, 19]}]], "state_before": "case refine_1.left.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : x.LeftMoves\nl : (-y).LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))\u27e7 > \u27e6(-y).mulOption (-x) k i\u27e7", "state_after": "no goals"}, {"tactic": "simp only [\u2190 mulOption_symm y]", "annotated_tactic": ["simp only [\u2190 <a>mulOption_symm</a> y]", [{"full_name": "SetTheory.PGame.mulOption_symm", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [863, 7], "def_end_pos": [863, 21]}]], "state_before": "case refine_1.right.left\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : x.LeftMoves\nk : y.LeftMoves\n\u22a2 -\u27e6(-x).mulOption y j l\u27e7 > \u27e6x.mulOption y i k\u27e7", "state_after": "case refine_1.right.left\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : x.LeftMoves\nk : y.LeftMoves\n\u22a2 -\u27e6y.mulOption (-x) l j\u27e7 > \u27e6y.mulOption x k i\u27e7"}, {"tactic": "apply mulOption_lt hy hx ihyx ihxy", "annotated_tactic": ["apply <a>mulOption_lt</a> hy hx ihyx ihxy", [{"full_name": "Surreal.Multiplication.mulOption_lt", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [247, 7], "def_end_pos": [247, 19]}]], "state_before": "case refine_1.right.left\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : x.LeftMoves\nk : y.LeftMoves\n\u22a2 -\u27e6y.mulOption (-x) l j\u27e7 > \u27e6y.mulOption x k i\u27e7", "state_after": "no goals"}, {"tactic": "rw [mulOption_neg_neg y]", "annotated_tactic": ["rw [<a>mulOption_neg_neg</a> y]", [{"full_name": "SetTheory.PGame.mulOption_neg_neg", "def_path": "Mathlib/SetTheory/Game/Basic.lean", "def_pos": [855, 7], "def_end_pos": [855, 24]}]], "state_before": "case refine_1.right.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-x).mulOption y j l\u27e7 > \u27e6(-x).mulOption (-y) i k\u27e7", "state_after": "case refine_1.right.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))\u27e7 > \u27e6(-x).mulOption (-y) i k\u27e7"}, {"tactic": "apply mulOption_lt hx.neg hy.neg ihxyn ihyxn", "annotated_tactic": ["apply <a>mulOption_lt</a> hx.neg hy.neg ihxyn ihyxn", [{"full_name": "Surreal.Multiplication.mulOption_lt", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [247, 7], "def_end_pos": [247, 19]}]], "state_before": "case refine_1.right.right\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\nj : (-x).LeftMoves\nl : y.LeftMoves\ni : (-x).LeftMoves\nk : (-y).LeftMoves\n\u22a2 -\u27e6(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))\u27e7 > \u27e6(-x).mulOption (-y) i k\u27e7", "state_after": "no goals"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case refine_3\nx x\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 x y) \u2192 P124 a\nhx : x.Numeric\nhy : y.Numeric\nihxy : IH1 x y\nihyx : IH1 y x\nihxyn : IH1 (-x) (-y)\nihyxn : IH1 (-y) (-x)\n\u22a2 \u2200 (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric", "state_after": "case refine_3.mk\nx\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nhy : y.Numeric\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) y) \u2192 P124 a\nhx : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).Numeric\nihxy : IH1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) y\nihyx : IH1 y (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihxyn : IH1 (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (-y)\nihyxn : IH1 (-y) (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 \u2200 (j : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y).RightMoves), ((PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y).moveRight j).Numeric"}, {"tactic": "cases y", "annotated_tactic": ["cases y", []], "state_before": "case refine_3.mk\nx\u2081 x\u2082 x\u2083 x' y y\u2081 y\u2082 y\u2083 y' : PGame\nhy : y.Numeric\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nih : \u2200 (a : Args), ArgsRel a (Args.P1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) y) \u2192 P124 a\nhx : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).Numeric\nihxy : IH1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) y\nihyx : IH1 y (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihxyn : IH1 (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (-y)\nihyxn : IH1 (-y) (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 \u2200 (j : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y).RightMoves), ((PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y).moveRight j).Numeric", "state_after": "case refine_3.mk.mk\nx\u2081 x\u2082 x\u2083 x' y\u2081 y\u2082 y\u2083 y' : PGame\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\nhx : (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2).Numeric\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nhy : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).Numeric\nih : \u2200 (a : Args), ArgsRel a (Args.P1 (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)) \u2192 P124 a\nihxy : IH1 (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihyx : IH1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\nihxyn : IH1 (-PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihyxn : IH1 (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (-PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u22a2 \u2200 (j : (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).RightMoves),\n    ((PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).moveRight j).Numeric"}, {"tactic": "rintro (\u27e8i,j\u27e9|\u27e8i,j\u27e9) <;>\nrefine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub\n  (numeric_option_mul_option ih ?_ ?_) <;>\nsolve_by_elim [IsOption.mk_left, IsOption.mk_right]", "annotated_tactic": ["rintro (\u27e8i,j\u27e9|\u27e8i,j\u27e9) <;>\n    refine ((<a>numeric_option_mul</a> ih ?_).<a>add</a> <| <a>numeric_mul_option</a> ih ?_).<a>sub</a>\n      (<a>numeric_option_mul_option</a> ih ?_ ?_) <;>\n    solve_by_elim [<a>IsOption.mk_left</a>, <a>IsOption.mk_right</a>]", [{"full_name": "Surreal.Multiplication.numeric_option_mul", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [211, 7], "def_end_pos": [211, 25]}, {"full_name": "SetTheory.PGame.Numeric.add", "def_path": "Mathlib/SetTheory/Surreal/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 12]}, {"full_name": "Surreal.Multiplication.numeric_mul_option", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [214, 7], "def_end_pos": [214, 25]}, {"full_name": "SetTheory.PGame.Numeric.sub", "def_path": "Mathlib/SetTheory/Surreal/Basic.lean", "def_pos": [256, 9], "def_end_pos": [256, 12]}, {"full_name": "Surreal.Multiplication.numeric_option_mul_option", "def_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "def_pos": [217, 7], "def_end_pos": [217, 32]}, {"full_name": "SetTheory.PGame.IsOption.mk_left", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [225, 9], "def_end_pos": [225, 25]}, {"full_name": "SetTheory.PGame.IsOption.mk_right", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [230, 9], "def_end_pos": [230, 26]}]], "state_before": "case refine_3.mk.mk\nx\u2081 x\u2082 x\u2083 x' y\u2081 y\u2082 y\u2083 y' : PGame\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\nhx : (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2).Numeric\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nhy : (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).Numeric\nih : \u2200 (a : Args), ArgsRel a (Args.P1 (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)) \u2192 P124 a\nihxy : IH1 (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihyx : IH1 (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\nihxyn : IH1 (-PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\nihyxn : IH1 (-PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (-PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u22a2 \u2200 (j : (PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).RightMoves),\n    ((PGame.mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d).moveRight j).Numeric", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.isSplittingField_iSup", "start": [771, 1], "end": [783, 58], "traced_tactics": [{"tactic": "let F : IntermediateField K L := \u2a06 i \u2208 s, t i", "annotated_tactic": ["let F : <a>IntermediateField</a> K L := \u2a06 i \u2208 s, t i", [{"full_name": "IntermediateField", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [49, 11], "def_end_pos": [49, 28]}]], "state_before": "F : Type u_1\ninst\u271d\u2075 : Field F\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nh : \u2200 i \u2208 s, IsSplittingField K (\u21a5(t i)) (p i)\n\u22a2 IsSplittingField K (\u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i)", "state_after": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nh : \u2200 i \u2208 s, IsSplittingField K (\u21a5(t i)) (p i)\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\n\u22a2 IsSplittingField K (\u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i)"}, {"tactic": "have hF : \u2200 i \u2208 s, t i \u2264 F := fun i hi \u21a6 le_iSup_of_le i (le_iSup (fun _ \u21a6 t i) hi)", "annotated_tactic": ["have hF : \u2200 i \u2208 s, t i \u2264 F := fun i hi \u21a6 <a>le_iSup_of_le</a> i (<a>le_iSup</a> (fun _ \u21a6 t i) hi)", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [734, 9], "def_end_pos": [734, 22]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [702, 9], "def_end_pos": [702, 16]}]], "state_before": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nh : \u2200 i \u2208 s, IsSplittingField K (\u21a5(t i)) (p i)\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\n\u22a2 IsSplittingField K (\u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i)", "state_after": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nh : \u2200 i \u2208 s, IsSplittingField K (\u21a5(t i)) (p i)\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\n\u22a2 IsSplittingField K (\u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i)"}, {"tactic": "simp only [isSplittingField_iff] at h \u22a2", "annotated_tactic": ["simp only [<a>isSplittingField_iff</a>] at h \u22a2", [{"full_name": "IntermediateField.isSplittingField_iff", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [653, 9], "def_end_pos": [653, 29]}]], "state_before": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nh : \u2200 i \u2208 s, IsSplittingField K (\u21a5(t i)) (p i)\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\n\u22a2 IsSplittingField K (\u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i)", "state_after": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 Splits (algebraMap K \u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i) \u2227 \u2a06 i \u2208 s, t i = adjoin K ((\u220f i \u2208 s, p i).rootSet L)"}, {"tactic": "refine\n  \u27e8splits_prod (algebraMap K F) fun i hi \u21a6\n      splits_comp_of_splits (algebraMap K (t i)) (inclusion (hF i hi)).toRingHom\n        (h i hi).1,\n    ?_\u27e9", "annotated_tactic": ["refine\n    \u27e8<a>splits_prod</a> (<a>algebraMap</a> K F) fun i hi \u21a6\n        <a>splits_comp_of_splits</a> (<a>algebraMap</a> K (t i)) (<a>inclusion</a> (hF i hi)).<a>toRingHom</a>\n          (h i hi).1,\n      ?_\u27e9", [{"full_name": "Polynomial.splits_prod", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [146, 9], "def_end_pos": [146, 20]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [440, 9], "def_end_pos": [440, 30]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IntermediateField.inclusion", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [570, 5], "def_end_pos": [570, 14]}, {"full_name": "AlgHom.toRingHom", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [36, 14], "def_end_pos": [36, 30]}]], "state_before": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 Splits (algebraMap K \u21a5(\u2a06 i \u2208 s, t i)) (\u220f i \u2208 s, p i) \u2227 \u2a06 i \u2208 s, t i = adjoin K ((\u220f i \u2208 s, p i).rootSet L)", "state_after": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 \u2a06 i \u2208 s, t i = adjoin K ((\u220f i \u2208 s, p i).rootSet L)"}, {"tactic": "simp only [rootSet_prod p s h0, \u2190 Set.iSup_eq_iUnion, (@gc K _ L _ _).l_iSup\u2082]", "annotated_tactic": ["simp only [<a>rootSet_prod</a> p s h0, \u2190 <a>Set.iSup_eq_iUnion</a>, (@<a>gc</a> K _ L _ _).<a>l_iSup\u2082</a>]", [{"full_name": "Polynomial.rootSet_prod", "def_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"full_name": "Set.iSup_eq_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [290, 9], "def_end_pos": [290, 23]}, {"full_name": "IntermediateField.gc", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [82, 9], "def_end_pos": [82, 11]}, {"full_name": "GaloisConnection.l_iSup\u2082", "def_path": "Mathlib/Order/GaloisConnection.lean", "def_pos": [283, 9], "def_end_pos": [283, 16]}]], "state_before": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 \u2a06 i \u2208 s, t i = adjoin K ((\u220f i \u2208 s, p i).rootSet L)", "state_after": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 \u2a06 i \u2208 s, t i = \u2a06 i \u2208 s, adjoin K ((p i).rootSet L)"}, {"tactic": "exact iSup_congr fun i \u21a6 iSup_congr fun hi \u21a6 (h i hi).2", "annotated_tactic": ["exact <a>iSup_congr</a> fun i \u21a6 <a>iSup_congr</a> fun hi \u21a6 (h i hi).2", [{"full_name": "iSup_congr", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [568, 9], "def_end_pos": [568, 19]}, {"full_name": "iSup_congr", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [568, 9], "def_end_pos": [568, 19]}]], "state_before": "F\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nE : Type u_2\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F\u271d E\nS : Set E\n\u03b1 : E\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nE1 E2 : IntermediateField K L\n\u03b9 : Type u_5\nt : \u03b9 \u2192 IntermediateField K L\np : \u03b9 \u2192 K[X]\ns : Finset \u03b9\nh0 : \u220f i \u2208 s, p i \u2260 0\nF : IntermediateField K L := \u2a06 i \u2208 s, t i\nhF : \u2200 i \u2208 s, t i \u2264 F\nh : \u2200 i \u2208 s, Splits (algebraMap K \u21a5(t i)) (p i) \u2227 t i = adjoin K ((p i).rootSet L)\n\u22a2 \u2a06 i \u2208 s, t i = \u2a06 i \u2208 s, adjoin K ((p i).rootSet L)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.repr_unitsSMul", "start": [1244, 1], "end": [1246, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.disjoint_comm", "start": [59, 1], "end": [60, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean", "full_name": "Action.tensor_hom", "start": [66, 1], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "full_name": "MeasureTheory.Mem\u2112p.of_bound", "start": [595, 1], "end": [597, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "ModelWithCorners.toPartialEquiv_coe", "start": [200, 1], "end": [201, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Archimedean.lean", "full_name": "Real.sInf_le_sSup", "start": [302, 1], "end": [305, 35], "traced_tactics": [{"tactic": "rcases s.eq_empty_or_nonempty with (rfl | hne)", "annotated_tactic": ["rcases s.eq_empty_or_nonempty with (rfl | hne)", []], "state_before": "s : Set \u211d\nh\u2081 : BddBelow s\nh\u2082 : BddAbove s\n\u22a2 sInf s \u2264 sSup s", "state_after": "case inl\nh\u2081 : BddBelow \u2205\nh\u2082 : BddAbove \u2205\n\u22a2 sInf \u2205 \u2264 sSup \u2205\n\ncase inr\ns : Set \u211d\nh\u2081 : BddBelow s\nh\u2082 : BddAbove s\nhne : s.Nonempty\n\u22a2 sInf s \u2264 sSup s"}, {"tactic": "rw [sInf_empty, sSup_empty]", "annotated_tactic": ["rw [<a>sInf_empty</a>, <a>sSup_empty</a>]", [{"full_name": "Real.sInf_empty", "def_path": "Mathlib/Data/Real/Archimedean.lean", "def_pos": [215, 9], "def_end_pos": [215, 19]}, {"full_name": "Real.sSup_empty", "def_path": "Mathlib/Data/Real/Archimedean.lean", "def_pos": [184, 9], "def_end_pos": [184, 19]}]], "state_before": "case inl\nh\u2081 : BddBelow \u2205\nh\u2082 : BddAbove \u2205\n\u22a2 sInf \u2205 \u2264 sSup \u2205", "state_after": "no goals"}, {"tactic": "exact csInf_le_csSup h\u2081 h\u2082 hne", "annotated_tactic": ["exact <a>csInf_le_csSup</a> h\u2081 h\u2082 hne", [{"full_name": "csInf_le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [690, 9], "def_end_pos": [690, 23]}]], "state_before": "case inr\ns : Set \u211d\nh\u2081 : BddBelow s\nh\u2082 : BddAbove s\nhne : s.Nonempty\n\u22a2 sInf s \u2264 sSup s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "HasFDerivWithinAt.sub", "start": [511, 8], "end": [513, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "full_name": "LieModule.isCompl_weightSpace_zero_posFittingComp", "start": [608, 1], "end": [622, 59], "traced_tactics": [{"tactic": "let P : LieSubmodule R L M \u2192 Prop := fun N \u21a6 IsCompl (weightSpace N 0) (posFittingComp R L N)", "annotated_tactic": ["let P : <a>LieSubmodule</a> R L M \u2192 Prop := fun N \u21a6 <a>IsCompl</a> (<a>weightSpace</a> N 0) (<a>posFittingComp</a> R L N)", [{"full_name": "LieSubmodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [46, 11], "def_end_pos": [46, 23]}, {"full_name": "IsCompl", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [469, 11], "def_end_pos": [469, 18]}, {"full_name": "LieModule.weightSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [177, 5], "def_end_pos": [177, 16]}, {"full_name": "LieModule.posFittingComp", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [432, 5], "def_end_pos": [432, 19]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)"}, {"tactic": "suffices P \u22a4 by\n  let e := LieModuleEquiv.ofTop R L M\n  rw [\u2190 map_weightSpace_eq e, \u2190 map_posFittingComp_eq e]\n  exact (LieSubmodule.orderIsoMapComap e).isCompl_iff.mp this", "annotated_tactic": ["suffices P \u22a4 by\n    let e := <a>LieModuleEquiv.ofTop</a> R L M\n    rw [\u2190 <a>map_weightSpace_eq</a> e, \u2190 <a>map_posFittingComp_eq</a> e]\n    exact (<a>LieSubmodule.orderIsoMapComap</a> e).isCompl_iff.mp this", [{"full_name": "LieModuleEquiv.ofTop", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1540, 5], "def_end_pos": [1540, 25]}, {"full_name": "LieModule.map_weightSpace_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [528, 7], "def_end_pos": [528, 25]}, {"full_name": "LieModule.map_posFittingComp_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [532, 7], "def_end_pos": [532, 28]}, {"full_name": "LieSubmodule.orderIsoMapComap", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [971, 14], "def_end_pos": [971, 30]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\n\u22a2 P \u22a4"}, {"tactic": "refine (LieSubmodule.wellFounded_of_isArtinian R L M).induction (C := P) _ fun N hN \u21a6 ?_", "annotated_tactic": ["refine (<a>LieSubmodule.wellFounded_of_isArtinian</a> R L M).<a>induction</a> (C := P) _ fun N hN \u21a6 ?_", [{"full_name": "LieSubmodule.wellFounded_of_isArtinian", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [623, 9], "def_end_pos": [623, 34]}, {"full_name": "WellFounded.induction", "def_path": ".lake/packages/lean4/src/lean/Init/WF.lean", "def_pos": [71, 9], "def_end_pos": [71, 18]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\n\u22a2 P \u22a4", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\n\u22a2 P N"}, {"tactic": "refine isCompl_weightSpace_zero_posFittingComp_aux R L N fun N' hN' \u21a6 ?_", "annotated_tactic": ["refine <a>isCompl_weightSpace_zero_posFittingComp_aux</a> R L N fun N' hN' \u21a6 ?_", [{"full_name": "_private.Mathlib.Algebra.Lie.Weights.Basic.0.LieModule.isCompl_weightSpace_zero_posFittingComp_aux", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [577, 15], "def_end_pos": [577, 58]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\n\u22a2 P N", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')"}, {"tactic": "suffices IsCompl (weightSpace (N'.map N.incl) 0) (posFittingComp R L (N'.map N.incl)) by\n  let e := LieSubmodule.equivMapOfInjective N' N.injective_incl\n  rw [\u2190 map_weightSpace_eq e, \u2190 map_posFittingComp_eq e] at this\n  exact (LieSubmodule.orderIsoMapComap e).isCompl_iff.mpr this", "annotated_tactic": ["suffices <a>IsCompl</a> (<a>weightSpace</a> (N'.map N.incl) 0) (<a>posFittingComp</a> R L (N'.map N.incl)) by\n    let e := <a>LieSubmodule.equivMapOfInjective</a> N' N.injective_incl\n    rw [\u2190 <a>map_weightSpace_eq</a> e, \u2190 <a>map_posFittingComp_eq</a> e] at this\n    exact (<a>LieSubmodule.orderIsoMapComap</a> e).isCompl_iff.mpr this", [{"full_name": "IsCompl", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [469, 11], "def_end_pos": [469, 18]}, {"full_name": "LieModule.weightSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [177, 5], "def_end_pos": [177, 16]}, {"full_name": "LieModule.posFittingComp", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [432, 5], "def_end_pos": [432, 19]}, {"full_name": "LieSubmodule.equivMapOfInjective", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [962, 19], "def_end_pos": [962, 38]}, {"full_name": "LieModule.map_weightSpace_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [528, 7], "def_end_pos": [528, 25]}, {"full_name": "LieModule.map_posFittingComp_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [532, 7], "def_end_pos": [532, 28]}, {"full_name": "LieSubmodule.orderIsoMapComap", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [971, 14], "def_end_pos": [971, 30]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\n\u22a2 IsCompl (weightSpace (\u21a5\u2191(LieSubmodule.map N.incl N')) 0) (posFittingComp R L \u21a5\u2191(LieSubmodule.map N.incl N'))"}, {"tactic": "exact hN _ (LieSubmodule.map_incl_lt_iff_lt_top.mpr hN')", "annotated_tactic": ["exact hN _ (LieSubmodule.map_incl_lt_iff_lt_top.mpr hN')", []], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\n\u22a2 IsCompl (weightSpace (\u21a5\u2191(LieSubmodule.map N.incl N')) 0) (posFittingComp R L \u21a5\u2191(LieSubmodule.map N.incl N'))", "state_after": "no goals"}, {"tactic": "let e := LieModuleEquiv.ofTop R L M", "annotated_tactic": ["let e := <a>LieModuleEquiv.ofTop</a> R L M", [{"full_name": "LieModuleEquiv.ofTop", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1540, 5], "def_end_pos": [1540, 25]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nthis : P \u22a4\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nthis : P \u22a4\ne : \u21a5\u2191\u22a4 \u2243\u2097\u2045R,L\u2046 M := LieModuleEquiv.ofTop R L M\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)"}, {"tactic": "rw [\u2190 map_weightSpace_eq e, \u2190 map_posFittingComp_eq e]", "annotated_tactic": ["rw [\u2190 <a>map_weightSpace_eq</a> e, \u2190 <a>map_posFittingComp_eq</a> e]", [{"full_name": "LieModule.map_weightSpace_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [528, 7], "def_end_pos": [528, 25]}, {"full_name": "LieModule.map_posFittingComp_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [532, 7], "def_end_pos": [532, 28]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nthis : P \u22a4\ne : \u21a5\u2191\u22a4 \u2243\u2097\u2045R,L\u2046 M := LieModuleEquiv.ofTop R L M\n\u22a2 IsCompl (weightSpace M 0) (posFittingComp R L M)", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nthis : P \u22a4\ne : \u21a5\u2191\u22a4 \u2243\u2097\u2045R,L\u2046 M := LieModuleEquiv.ofTop R L M\n\u22a2 IsCompl (LieSubmodule.map e.toLieModuleHom (weightSpace (\u21a5\u2191\u22a4) 0))\n    (LieSubmodule.map e.toLieModuleHom (posFittingComp R L \u21a5\u2191\u22a4))"}, {"tactic": "exact (LieSubmodule.orderIsoMapComap e).isCompl_iff.mp this", "annotated_tactic": ["exact (<a>LieSubmodule.orderIsoMapComap</a> e).isCompl_iff.mp this", [{"full_name": "LieSubmodule.orderIsoMapComap", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [971, 14], "def_end_pos": [971, 30]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nthis : P \u22a4\ne : \u21a5\u2191\u22a4 \u2243\u2097\u2045R,L\u2046 M := LieModuleEquiv.ofTop R L M\n\u22a2 IsCompl (LieSubmodule.map e.toLieModuleHom (weightSpace (\u21a5\u2191\u22a4) 0))\n    (LieSubmodule.map e.toLieModuleHom (posFittingComp R L \u21a5\u2191\u22a4))", "state_after": "no goals"}, {"tactic": "let e := LieSubmodule.equivMapOfInjective N' N.injective_incl", "annotated_tactic": ["let e := <a>LieSubmodule.equivMapOfInjective</a> N' N.injective_incl", [{"full_name": "LieSubmodule.equivMapOfInjective", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [962, 19], "def_end_pos": [962, 38]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\nthis : IsCompl (weightSpace (\u21a5\u2191(LieSubmodule.map N.incl N')) 0) (posFittingComp R L \u21a5\u2191(LieSubmodule.map N.incl N'))\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\nthis : IsCompl (weightSpace (\u21a5\u2191(LieSubmodule.map N.incl N')) 0) (posFittingComp R L \u21a5\u2191(LieSubmodule.map N.incl N'))\ne : \u21a5\u2191N' \u2243\u2097\u2045R,L\u2046 \u21a5\u2191(LieSubmodule.map N.incl N') := N'.equivMapOfInjective \u22ef\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')"}, {"tactic": "rw [\u2190 map_weightSpace_eq e, \u2190 map_posFittingComp_eq e] at this", "annotated_tactic": ["rw [\u2190 <a>map_weightSpace_eq</a> e, \u2190 <a>map_posFittingComp_eq</a> e] at this", [{"full_name": "LieModule.map_weightSpace_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [528, 7], "def_end_pos": [528, 25]}, {"full_name": "LieModule.map_posFittingComp_eq", "def_path": "Mathlib/Algebra/Lie/Weights/Basic.lean", "def_pos": [532, 7], "def_end_pos": [532, 28]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\nthis : IsCompl (weightSpace (\u21a5\u2191(LieSubmodule.map N.incl N')) 0) (posFittingComp R L \u21a5\u2191(LieSubmodule.map N.incl N'))\ne : \u21a5\u2191N' \u2243\u2097\u2045R,L\u2046 \u21a5\u2191(LieSubmodule.map N.incl N') := N'.equivMapOfInjective \u22ef\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')", "state_after": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\ne : \u21a5\u2191N' \u2243\u2097\u2045R,L\u2046 \u21a5\u2191(LieSubmodule.map N.incl N') := N'.equivMapOfInjective \u22ef\nthis :\n  IsCompl (LieSubmodule.map e.toLieModuleHom (weightSpace (\u21a5\u2191N') 0))\n    (LieSubmodule.map e.toLieModuleHom (posFittingComp R L \u21a5\u2191N'))\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')"}, {"tactic": "exact (LieSubmodule.orderIsoMapComap e).isCompl_iff.mpr this", "annotated_tactic": ["exact (<a>LieSubmodule.orderIsoMapComap</a> e).isCompl_iff.mpr this", [{"full_name": "LieSubmodule.orderIsoMapComap", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [971, 14], "def_end_pos": [971, 30]}]], "state_before": "K : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : IsNoetherian R M\ninst\u271d : IsArtinian R M\nP : LieSubmodule R L M \u2192 Prop := fun N => IsCompl (weightSpace (\u21a5\u2191N) 0) (posFittingComp R L \u21a5\u2191N)\nN : LieSubmodule R L M\nhN : \u2200 y < N, P y\nN' : LieSubmodule R L \u21a5\u2191N\nhN' : N' < \u22a4\ne : \u21a5\u2191N' \u2243\u2097\u2045R,L\u2046 \u21a5\u2191(LieSubmodule.map N.incl N') := N'.equivMapOfInjective \u22ef\nthis :\n  IsCompl (LieSubmodule.map e.toLieModuleHom (weightSpace (\u21a5\u2191N') 0))\n    (LieSubmodule.map e.toLieModuleHom (posFittingComp R L \u21a5\u2191N'))\n\u22a2 IsCompl (weightSpace (\u21a5\u2191N') 0) (posFittingComp R L \u21a5\u2191N')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fintype/Pi.lean", "full_name": "Fintype.mem_piFinset", "start": [34, 1], "end": [42, 46], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 f \u2208 piFinset t \u2194 \u2200 (a : \u03b1), f a \u2208 t a", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 f \u2208 piFinset t \u2192 \u2200 (a : \u03b1), f a \u2208 t a\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192 f \u2208 piFinset t"}, {"tactic": "simp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp,\n  mem_pi]", "annotated_tactic": ["simp only [<a>piFinset</a>, <a>mem_map</a>, <a>and_imp</a>, <a>forall_prop_of_true</a>, <a>exists_prop</a>, <a>mem_univ</a>, <a>exists_imp</a>,\n      <a>mem_pi</a>]", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.mem_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [75, 9], "def_end_pos": [75, 16]}, {"full_name": "and_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [115, 17], "def_end_pos": [115, 24]}, {"full_name": "forall_prop_of_true", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [312, 9], "def_end_pos": [312, 28]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "exists_imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [200, 9], "def_end_pos": [200, 19]}, {"full_name": "Finset.mem_pi", "def_path": "Mathlib/Data/Finset/Pi.lean", "def_pos": [49, 9], "def_end_pos": [49, 15]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 f \u2208 piFinset t \u2192 \u2200 (a : \u03b1), f a \u2208 t a", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 \u2200 (x : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n    (\u2200 (a : \u03b1), x a \u22ef \u2208 t a) \u2192 { toFun := fun f a => f a \u22ef, inj' := \u22ef } x = f \u2192 \u2200 (a : \u03b1), f a \u2208 t a"}, {"tactic": "rintro g hg hgf a", "annotated_tactic": ["rintro g hg hgf a", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 \u2200 (x : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n    (\u2200 (a : \u03b1), x a \u22ef \u2208 t a) \u2192 { toFun := fun f a => f a \u22ef, inj' := \u22ef } x = f \u2192 \u2200 (a : \u03b1), f a \u2208 t a", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a \u22ef \u2208 t a\nhgf : { toFun := fun f a => f a \u22ef, inj' := \u22ef } g = f\na : \u03b1\n\u22a2 f a \u2208 t a"}, {"tactic": "rw [\u2190 hgf]", "annotated_tactic": ["rw [\u2190 hgf]", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a \u22ef \u2208 t a\nhgf : { toFun := fun f a => f a \u22ef, inj' := \u22ef } g = f\na : \u03b1\n\u22a2 f a \u2208 t a", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a \u22ef \u2208 t a\nhgf : { toFun := fun f a => f a \u22ef, inj' := \u22ef } g = f\na : \u03b1\n\u22a2 { toFun := fun f a => f a \u22ef, inj' := \u22ef } g a \u2208 t a"}, {"tactic": "exact hg a", "annotated_tactic": ["exact hg a", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a \u22ef \u2208 t a\nhgf : { toFun := fun f a => f a \u22ef, inj' := \u22ef } g = f\na : \u03b1\n\u22a2 { toFun := fun f a => f a \u22ef, inj' := \u22ef } g a \u2208 t a", "state_after": "no goals"}, {"tactic": "simp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]", "annotated_tactic": ["simp only [<a>piFinset</a>, <a>mem_map</a>, <a>forall_prop_of_true</a>, <a>exists_prop</a>, <a>mem_univ</a>, <a>mem_pi</a>]", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.mem_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [75, 9], "def_end_pos": [75, 16]}, {"full_name": "forall_prop_of_true", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [312, 9], "def_end_pos": [312, 28]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "Finset.mem_pi", "def_path": "Mathlib/Data/Finset/Pi.lean", "def_pos": [49, 9], "def_end_pos": [49, 15]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192 f \u2208 piFinset t", "state_after": "case mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192 \u2203 a, (\u2200 (a_1 : \u03b1), a a_1 \u22ef \u2208 t a_1) \u2227 { toFun := fun f a => f a \u22ef, inj' := \u22ef } a = f"}, {"tactic": "exact fun hf => \u27e8fun a _ => f a, hf, rfl\u27e9", "annotated_tactic": ["exact fun hf => \u27e8fun a _ => f a, hf, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b3 : \u03b1 \u2192 Type u_2\n\u03b4 : \u03b1 \u2192 Type u_3\ns : (a : \u03b1) \u2192 Finset (\u03b3 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192 \u2203 a, (\u2200 (a_1 : \u03b1), a a_1 \u22ef \u2208 t a_1) \u2227 { toFun := fun f a => f a \u22ef, inj' := \u22ef } a = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "OrderHomClass.to_lattice_hom_apply", "start": [1191, 1], "end": [1192, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.ceil_le", "start": [1199, 1], "end": [1200, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_id_eq", "start": [367, 1], "end": [367, 62], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\n\u22a2 image id = id", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t x\u271d\u00b9 : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 id '' x\u271d\u00b9 \u2194 x\u271d \u2208 id x\u271d\u00b9"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t x\u271d\u00b9 : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 id '' x\u271d\u00b9 \u2194 x\u271d \u2208 id x\u271d\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "full_name": "Complex.Gamma_integrand_deriv_integrable_A", "start": [174, 9], "end": [178, 41], "traced_tactics": [{"tactic": "convert (Gamma_integrand_interval_integrable (s + 1) _ hX).neg", "annotated_tactic": ["convert (<a>Gamma_integrand_interval_integrable</a> (s + 1) _ hX).<a>neg</a>", [{"full_name": "_private.Mathlib.Analysis.SpecialFunctions.Gamma.Basic.0.Complex.Gamma_integrand_interval_integrable", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [169, 17], "def_end_pos": [169, 52]}, {"full_name": "IntervalIntegrable.neg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [187, 9], "def_end_pos": [187, 12]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 IntervalIntegrable (fun x => -(\u2191(rexp (-x)) * \u2191x ^ s)) volume 0 X", "state_after": "case h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(\u2191(rexp (-x\u271d)) * \u2191x\u271d ^ s) = (-fun x => \u2191(rexp (-x)) * \u2191x ^ (s + 1 - 1)) x\u271d\n\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < (s + 1).re"}, {"tactic": "simp only [ofReal_exp, ofReal_neg, add_sub_cancel_right]", "annotated_tactic": ["simp only [<a>ofReal_exp</a>, <a>ofReal_neg</a>, <a>add_sub_cancel_right</a>]", [{"full_name": "Complex.ofReal_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [270, 9], "def_end_pos": [270, 19]}, {"full_name": "Complex.ofReal_neg", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 19]}, {"full_name": "add_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1008, 3], "def_end_pos": [1008, 14]}]], "state_before": "case h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(\u2191(rexp (-x\u271d)) * \u2191x\u271d ^ s) = (-fun x => \u2191(rexp (-x)) * \u2191x ^ (s + 1 - 1)) x\u271d", "state_after": "case h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(cexp (-\u2191x\u271d) * \u2191x\u271d ^ s) = (-fun x => cexp (-\u2191x) * \u2191x ^ s) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(cexp (-\u2191x\u271d) * \u2191x\u271d ^ s) = (-fun x => cexp (-\u2191x) * \u2191x ^ s) x\u271d", "state_after": "no goals"}, {"tactic": "simp only [add_re, one_re]", "annotated_tactic": ["simp only [<a>add_re</a>, <a>one_re</a>]", [{"full_name": "Complex.add_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 15]}, {"full_name": "Complex.one_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < (s + 1).re", "state_after": "s : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < s.re + 1"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "s : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < s.re + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.zeroLocus_singleton_zero", "start": [272, 1], "end": [273, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_set_not_mem", "start": [1214, 1], "end": [1216, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Fraisse.lean", "full_name": "FirstOrder.Language.IsFraisseLimit.isFraisse", "start": [319, 1], "end": [321, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.exists_subset_bijOn", "start": [1464, 1], "end": [1465, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldl_toList", "start": [356, 1], "end": [360, 30], "traced_tactics": [{"tactic": "rw [\u2190 FreeMonoid.toList_ofList (toList xs), \u2190 foldl.unop_ofFreeMonoid]", "annotated_tactic": ["rw [\u2190 <a>FreeMonoid.toList_ofList</a> (<a>toList</a> xs), \u2190 <a>foldl.unop_ofFreeMonoid</a>]", [{"full_name": "FreeMonoid.toList_ofList", "def_path": "Mathlib/Algebra/FreeMonoid/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 22]}, {"full_name": "Traversable.toList", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [228, 5], "def_end_pos": [228, 11]}, {"full_name": "Traversable.foldl.unop_ofFreeMonoid", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [265, 9], "def_end_pos": [265, 32]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nt : Type u \u2192 Type u\ninst\u271d\u00b9 : Traversable t\ninst\u271d : LawfulTraversable t\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\nxs : t \u03b2\nx : \u03b1\n\u22a2 foldl f x xs = List.foldl f x (toList xs)", "state_after": "\u03b1 \u03b2 \u03b3 : Type u\nt : Type u \u2192 Type u\ninst\u271d\u00b9 : Traversable t\ninst\u271d : LawfulTraversable t\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\nxs : t \u03b2\nx : \u03b1\n\u22a2 foldl f x xs = unop ((Foldl.ofFreeMonoid f) (FreeMonoid.ofList (toList xs))) 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"a : EReal\nh : 0 < a\nh' : a \u2260 \u22a4\n\u22a2 0 < a\u207b\u00b9", "state_after": "no goals"}, {"tactic": "exact (not_lt_bot h).rec", "annotated_tactic": ["exact (<a>not_lt_bot</a> h).<a>rec</a>", [{"full_name": "not_lt_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [284, 9], "def_end_pos": [284, 19]}, {"full_name": "False.rec", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [202, 11], "def_end_pos": [202, 16]}]], "state_before": "case h_bot\nh : 0 < \u22a5\nh' : \u22a5 \u2260 \u22a4\n\u22a2 0 < \u22a5\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rw [\u2190 coe_inv a]", "annotated_tactic": ["rw [\u2190 <a>coe_inv</a> a]", [{"full_name": "EReal.coe_inv", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1493, 7], "def_end_pos": [1493, 14]}]], "state_before": "case h_real\na : \u211d\nh : 0 < \u2191a\nh' : \u2191a \u2260 \u22a4\n\u22a2 0 < (\u2191a)\u207b\u00b9", "state_after": "case h_real\na : \u211d\nh : 0 < \u2191a\nh' : \u2191a \u2260 \u22a4\n\u22a2 0 < \u2191a\u207b\u00b9"}, {"tactic": "norm_cast at *", "annotated_tactic": ["norm_cast at *", []], "state_before": "case h_real\na : \u211d\nh : 0 < \u2191a\nh' : \u2191a \u2260 \u22a4\n\u22a2 0 < \u2191a\u207b\u00b9", "state_after": "case h_real\na : \u211d\nh' : \u00ac\u2191a = \u22a4\nh : 0 < a\n\u22a2 0 < a\u207b\u00b9"}, {"tactic": "exact inv_pos_of_pos h", "annotated_tactic": ["exact <a>inv_pos_of_pos</a> h", [{"full_name": "inv_pos_of_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [52, 11], "def_end_pos": [52, 25]}]], "state_before": "case h_real\na : \u211d\nh' : \u00ac\u2191a = \u22a4\nh : 0 < a\n\u22a2 0 < a\u207b\u00b9", "state_after": "no goals"}, {"tactic": "exact (h' (Eq.refl \u22a4)).rec", "annotated_tactic": ["exact (h' (<a>Eq.refl</a> \u22a4)).<a>rec</a>", [{"full_name": "Eq.refl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [279, 5], "def_end_pos": [279, 9]}, {"full_name": "False.rec", 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"def_pos": [122, 17], "def_end_pos": [122, 42]}, {"full_name": "FiniteDimensional.finrank_eq_card_chooseBasisIndex", "def_path": "Mathlib/LinearAlgebra/Dimension/Free.lean", "def_pos": [88, 9], "def_end_pos": [88, 66]}]], "state_before": "R : Type u\nM : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Nontrivial R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module.Free R M\ninst\u271d\u00b9 : Module.Finite R M\nf : M \u2192\u2097[R] M\ninst\u271d : StrongRankCondition R\n\u22a2 f.charpoly.natDegree = finrank R M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.BijOn.bijective", "start": [1127, 1], "end": [1130, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean", "full_name": "Asymptotics.IsEquivalent.mul", "start": [290, 1], "end": [291, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean", "full_name": "Batteries.BinomialHeap.Imp.Heap.WF.merge'", "start": [370, 1], "end": [404, 37], "traced_tactics": [{"tactic": "unfold merge", "annotated_tactic": ["unfold <a>merge</a>", [{"full_name": "Batteries.BinomialHeap.Imp.Heap.merge", "def_path": ".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean", "def_pos": [127, 19], "def_end_pos": [127, 29]}]], "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WF le n s\u2081\nh\u2082 : WF le n s\u2082\n\u22a2 WF le n (merge le s\u2081 s\u2082) \u2227 ((s\u2081.rankGT n \u2194 s\u2082.rankGT n) \u2192 (merge le s\u2081 s\u2082).rankGT n)", "state_after": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WF le n s\u2081\nh\u2082 : WF le n s\u2082\n\u22a2 WF le n\n      (match s\u2081, s\u2082 with\n      | Heap.nil, h => h\n      | h, Heap.nil => h\n      | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n        if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((s\u2081.rankGT n \u2194 s\u2082.rankGT n) \u2192\n      (match s\u2081, s\u2082 with\n          | Heap.nil, h => h\n          | h, Heap.nil => h\n          | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n            if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n            else\n              if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n              else\n                match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n                | (a, n) =>\n                  let r := r\u2081 + 1;\n                  if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                  else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WF le n s\u2081\nh\u2082 : WF le n s\u2082\n\u22a2 WF le n\n      (match s\u2081, s\u2082 with\n      | Heap.nil, h => h\n      | h, Heap.nil => h\n      | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n        if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((s\u2081.rankGT n \u2194 s\u2082.rankGT n) \u2192\n      (match s\u2081, s\u2082 with\n          | Heap.nil, h => h\n          | h, Heap.nil => h\n          | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n            if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n            else\n              if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n              else\n                match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n                | (a, n) =>\n                  let r := r\u2081 + 1;\n                  if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                  else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_1\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2082 : WF le n s\u2082\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nh\u2081 : WF le n Heap.nil\n\u22a2 WF le n s\u2082 \u2227 ((Heap.nil.rankGT n \u2194 s\u2082.rankGT n) \u2192 s\u2082.rankGT n)\n\ncase h_2\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WF le n s\u2081\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nx\u271d : s\u2081 = Heap.nil \u2192 False\nh\u2082 : WF le n Heap.nil\n\u22a2 WF le n s\u2081 \u2227 ((s\u2081.rankGT n \u2194 Heap.nil.rankGT n) \u2192 s\u2081.rankGT n)\n\ncase h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u2082 : WF le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\n\u22a2 WF le n\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n      else\n        if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n        else\n          match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n          | (a, n) =>\n            let r := r\u2081\u271d + 1;\n            if t\u2081\u271d.rankGT r then if t\u2082\u271d.rankGT r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n            else if t\u2082\u271d.rankGT r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)) \u2227\n    (((cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d).rankGT n \u2194 (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d).rankGT n) \u2192\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n          else\n            if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n            else\n              match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n              | (a, n) =>\n                let r := r\u2081\u271d + 1;\n                if t\u2081\u271d.rankGT r then\n                  if t\u2082\u271d.rankGT r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n                else if t\u2082\u271d.rankGT r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)).rankGT\n        n)"}, {"tactic": "exact \u27e8h\u2082, fun h => h.1 h\u2081\u27e9", "annotated_tactic": ["exact \u27e8h\u2082, fun h => h.1 h\u2081\u27e9", []], "state_before": "case h_1\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2082 : WF le n s\u2082\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nh\u2081 : WF le n Heap.nil\n\u22a2 WF le n s\u2082 \u2227 ((Heap.nil.rankGT n \u2194 s\u2082.rankGT n) \u2192 s\u2082.rankGT n)", "state_after": "no goals"}, {"tactic": "exact \u27e8h\u2081, fun h => h.2 h\u2082\u27e9", "annotated_tactic": ["exact \u27e8h\u2081, fun h => h.2 h\u2082\u27e9", []], "state_before": "case h_2\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WF le n s\u2081\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nx\u271d : s\u2081 = Heap.nil \u2192 False\nh\u2082 : WF le n Heap.nil\n\u22a2 WF le n s\u2081 \u2227 ((s\u2081.rankGT n \u2194 Heap.nil.rankGT n) \u2192 s\u2081.rankGT n)", "state_after": "no goals"}, {"tactic": "rename_i r\u2081 a\u2081 n\u2081 t\u2081 r\u2082 a\u2082 n\u2082 t\u2082", "annotated_tactic": ["rename_i r\u2081 a\u2081 n\u2081 t\u2081 r\u2082 a\u2082 n\u2082 t\u2082", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u2082 : WF le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\n\u22a2 WF le n\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n      else\n        if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n        else\n          match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n          | (a, n) =>\n            let r := r\u2081\u271d + 1;\n            if t\u2081\u271d.rankGT r then if t\u2082\u271d.rankGT r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n            else if t\u2082\u271d.rankGT r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)) \u2227\n    (((cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d).rankGT n \u2194 (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d).rankGT n) \u2192\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n          else\n            if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n            else\n              match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n              | (a, n) =>\n                let r := r\u2081\u271d + 1;\n                if t\u2081\u271d.rankGT r then\n                  if t\u2082\u271d.rankGT r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n                else if t\u2082\u271d.rankGT r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)).rankGT\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "let \u27e8hr\u2081, hn\u2081, ht\u2081\u27e9 := h\u2081", "annotated_tactic": ["let \u27e8hr\u2081, hn\u2081, ht\u2081\u27e9 := h\u2081", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "let \u27e8hr\u2082, hn\u2082, ht\u2082\u27e9 := h\u2082", "annotated_tactic": ["let \u27e8hr\u2082, hn\u2082, ht\u2082\u27e9 := h\u2082", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "split <;> rename_i lt\u2081", "annotated_tactic": ["split <;> rename_i lt\u2081", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\n\u22a2 WF le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\n\u22a2 WF le n (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))).rankGT n)\n\ncase h_3.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\n\u22a2 WF le n\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n      else\n        match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "split <;> rename_i lt\u2082", "annotated_tactic": ["split <;> rename_i lt\u2082", []], "state_before": "case h_3.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\n\u22a2 WF le n\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n      else\n        match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\n\u22a2 WF le n (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)).rankGT n)\n\ncase h_3.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : \u00acr\u2082 < r\u2081\n\u22a2 WF le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "cases Nat.le_antisymm (Nat.ge_of_not_lt lt\u2082) (Nat.ge_of_not_lt lt\u2081)", "annotated_tactic": ["cases <a>Nat.le_antisymm</a> (<a>Nat.ge_of_not_lt</a> lt\u2082) (<a>Nat.ge_of_not_lt</a> lt\u2081)", [{"full_name": "Nat.le_antisymm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1750, 19], "def_end_pos": [1750, 34]}, {"full_name": "Nat.ge_of_not_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [450, 9], "def_end_pos": [450, 21]}, {"full_name": "Nat.ge_of_not_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [450, 9], "def_end_pos": [450, 21]}]], "state_before": "case h_3.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : \u00acr\u2082 < r\u2081\n\u22a2 WF le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\n\u22a2 WF le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\n\u22a2 WF le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nheq\u271d : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a\u271d, n\u271d)\n\u22a2 WF le n\n      (let r := r\u2081 + 1;\n      if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n      else if t\u2082.rankGT r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)).rankGT\n        n)"}, {"tactic": "rename_i a n eq", "annotated_tactic": ["rename_i a n eq", []], "state_before": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nheq\u271d : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a\u271d, n\u271d)\n\u22a2 WF le n\n      (let r := r\u2081 + 1;\n      if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n      else if t\u2082.rankGT r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)).rankGT\n        n)", "state_after": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\n\u22a2 WF le n\u271d\n      (let r := r\u2081 + 1;\n      if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n      else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192\n      (let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n\u271d)"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\n\u22a2 WF le n\u271d\n      (let r := r\u2081 + 1;\n      if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n      else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192\n      (let r := r\u2081 + 1;\n          if t\u2081.rankGT r then if t\u2082.rankGT r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if t\u2082.rankGT r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)).rankGT\n        n\u271d)", "state_after": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\n\u22a2 WF le n\u271d\n      (if t\u2081.rankGT (r\u2081 + 1) then\n        if t\u2082.rankGT (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n      else if t\u2082.rankGT (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192\n      (if t\u2081.rankGT (r\u2081 + 1) then\n            if t\u2082.rankGT (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n          else\n            if t\u2082.rankGT (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)\n            else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT\n        n\u271d)"}, {"tactic": "split <;> split <;> rename_i hl\u2081 hl\u2082", "annotated_tactic": ["split <;> split <;> rename_i hl\u2081 hl\u2082", []], "state_before": "case h_3.isFalse.isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\n\u22a2 WF le n\u271d\n      (if t\u2081.rankGT (r\u2081 + 1) then\n        if t\u2082.rankGT (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n      else if t\u2082.rankGT (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192\n      (if t\u2081.rankGT (r\u2081 + 1) then\n            if t\u2082.rankGT (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n          else\n            if t\u2082.rankGT (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)\n            else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT\n        n\u271d)", "state_after": "case h_3.isFalse.isFalse.refl.isTrue.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)\n\ncase h_3.isFalse.isFalse.refl.isTrue.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082).rankGT n\u271d)\n\ncase h_3.isFalse.isFalse.refl.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)).rankGT n\u271d)\n\ncase h_3.isFalse.isFalse.refl.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)"}, {"tactic": "refine \u27e8\u27e8hr\u2081, hn\u2081, And.left (merge' ht\u2081 \u27e8lt\u2081, hn\u2082, ht\u2082\u27e9)\u27e9, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8hr\u2081, hn\u2081, <a>And.left</a> (merge' ht\u2081 \u27e8lt\u2081, hn\u2082, ht\u2082\u27e9)\u27e9, fun h => ?_\u27e9", [{"full_name": "And.left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [521, 3], "def_end_pos": [521, 7]}]], "state_before": "case h_3.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\n\u22a2 WF le n (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))).rankGT n)", "state_after": "case h_3.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\nh : (cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n\n\u22a2 (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))).rankGT n"}, {"tactic": "exact h.2 <| Nat.lt_of_le_of_lt hr\u2081 lt\u2081", "annotated_tactic": ["exact h.2 <| <a>Nat.lt_of_le_of_lt</a> hr\u2081 lt\u2081", [{"full_name": "Nat.lt_of_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1747, 19], "def_end_pos": [1747, 37]}]], "state_before": "case h_3.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\nh : (cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n\n\u22a2 (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))).rankGT n", "state_after": "no goals"}, {"tactic": "refine \u27e8\u27e8hr\u2082, hn\u2082, And.left (merge' \u27e8lt\u2082, hn\u2081, ht\u2081\u27e9 ht\u2082)\u27e9, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8hr\u2082, hn\u2082, <a>And.left</a> (merge' \u27e8lt\u2082, hn\u2081, ht\u2081\u27e9 ht\u2082)\u27e9, fun h => ?_\u27e9", [{"full_name": "And.left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [521, 3], "def_end_pos": [521, 7]}]], "state_before": "case h_3.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\n\u22a2 WF le n (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n) \u2192\n      (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)).rankGT n)", "state_after": "case h_3.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\nh : (cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n\n\u22a2 (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)).rankGT n"}, {"tactic": "exact h.1 <| Nat.lt_of_le_of_lt hr\u2082 lt\u2082", "annotated_tactic": ["exact h.1 <| <a>Nat.lt_of_le_of_lt</a> hr\u2082 lt\u2082", [{"full_name": "Nat.lt_of_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1747, 19], "def_end_pos": [1747, 37]}]], "state_before": "case h_3.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WF le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2082\nht\u2082 : WF le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\nh : (cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n \u2194 (cons r\u2082 a\u2082 n\u2082 t\u2082).rankGT n\n\u22a2 (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)).rankGT n", "state_after": "no goals"}, {"tactic": "unfold combine at eq", "annotated_tactic": ["unfold <a>combine</a> at eq", [{"full_name": "Batteries.BinomialHeap.Imp.combine", "def_path": ".lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean", "def_pos": [120, 15], "def_end_pos": [120, 22]}]], "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\n\u22a2 HeapNode.WF le a n (r\u2081 + 1)", "state_after": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : (if le a\u2081 a\u2082 = true then (a\u2081, HeapNode.node a\u2082 n\u2082 n\u2081) else (a\u2082, HeapNode.node a\u2081 n\u2081 n\u2082)) = (a, n)\n\u22a2 HeapNode.WF le a n (r\u2081 + 1)"}, {"tactic": "split at eq <;> cases eq <;> rename_i h", "annotated_tactic": ["split at eq <;> cases eq <;> rename_i h", []], "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : (if le a\u2081 a\u2082 = true then (a\u2081, HeapNode.node a\u2082 n\u2082 n\u2081) else (a\u2082, HeapNode.node a\u2081 n\u2081 n\u2082)) = (a, n)\n\u22a2 HeapNode.WF le a n (r\u2081 + 1)", "state_after": "case isTrue.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : le a\u2081 a\u2082 = true\n\u22a2 HeapNode.WF le a\u2081 (HeapNode.node a\u2082 n\u2082 n\u2081) (r\u2081 + 1)\n\ncase isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : \u00acle a\u2081 a\u2082 = true\n\u22a2 HeapNode.WF le a\u2082 (HeapNode.node a\u2081 n\u2081 n\u2082) (r\u2081 + 1)"}, {"tactic": "exact \u27e8r\u2081, rfl, h, hn\u2082, hn\u2081\u27e9", "annotated_tactic": ["exact \u27e8r\u2081, <a>rfl</a>, h, hn\u2082, hn\u2081\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case isTrue.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : le a\u2081 a\u2082 = true\n\u22a2 HeapNode.WF le a\u2081 (HeapNode.node a\u2082 n\u2082 n\u2081) (r\u2081 + 1)", "state_after": "no goals"}, {"tactic": "exact \u27e8r\u2081, rfl, TotalBLE.total.resolve_left h, hn\u2081, hn\u2082\u27e9", "annotated_tactic": ["exact \u27e8r\u2081, <a>rfl</a>, TotalBLE.total.resolve_left h, hn\u2081, hn\u2082\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case isFalse.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : \u00acle a\u2081 a\u2082 = true\n\u22a2 HeapNode.WF le a\u2082 (HeapNode.node a\u2081 n\u2081 n\u2082) (r\u2081 + 1)", "state_after": "no goals"}, {"tactic": "exact \u27e8\u27e8Nat.le_succ_of_le hr\u2081, this,\n  (merge' (ht\u2081.of_rankGT hl\u2081) (ht\u2082.of_rankGT hl\u2082)).1\u27e9,\n  fun _ => Nat.lt_succ_of_le hr\u2081\u27e9", "annotated_tactic": ["exact \u27e8\u27e8<a>Nat.le_succ_of_le</a> hr\u2081, this,\n        (merge' (ht\u2081.of_rankGT hl\u2081) (ht\u2082.of_rankGT hl\u2082)).1\u27e9,\n        fun _ => <a>Nat.lt_succ_of_le</a> hr\u2081\u27e9", [{"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}, {"full_name": "Nat.lt_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 22]}]], "state_before": "case h_3.isFalse.isFalse.refl.isTrue.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2081 := .cons ..)\n  \u27e8Nat.le_succ_of_le hr\u2081, this, ht\u2081.of_rankGT hl\u2081\u27e9\n  (ht\u2082.of_le (Nat.le_succ_of_le hr\u2081))", "annotated_tactic": ["let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2081 := .cons ..)\n        \u27e8<a>Nat.le_succ_of_le</a> hr\u2081, this, ht\u2081.of_rankGT hl\u2081\u27e9\n        (ht\u2082.of_le (<a>Nat.le_succ_of_le</a> hr\u2081))", [{"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}, {"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}]], "state_before": "case h_3.isFalse.isFalse.refl.isTrue.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082).rankGT n\u271d)", "state_after": "case h_3.isFalse.isFalse.refl.isTrue.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le n\u271d (merge le (cons r\u2081.succ a n t\u2081) t\u2082)\nih\u2082 : ((cons r\u2081.succ a n t\u2081).rankGT n\u271d \u2194 t\u2082.rankGT n\u271d) \u2192 (merge le (cons r\u2081.succ a n t\u2081) t\u2082).rankGT n\u271d\n\u22a2 WF le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082).rankGT n\u271d)"}, {"tactic": "exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun _ => ht\u2082.rankGT.of_le hr\u2081, fun _ => Nat.lt_succ_of_le hr\u2081\u27e9\u27e9", "annotated_tactic": ["exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun _ => ht\u2082.rankGT.of_le hr\u2081, fun _ => <a>Nat.lt_succ_of_le</a> hr\u2081\u27e9\u27e9", [{"full_name": "Nat.lt_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 22]}]], "state_before": "case h_3.isFalse.isFalse.refl.isTrue.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : t\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le n\u271d (merge le (cons r\u2081.succ a n t\u2081) t\u2082)\nih\u2082 : ((cons r\u2081.succ a n t\u2081).rankGT n\u271d \u2194 t\u2082.rankGT n\u271d) \u2192 (merge le (cons r\u2081.succ a n t\u2081) t\u2082).rankGT n\u271d\n\u22a2 WF le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082).rankGT n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2082 := .cons ..) (ht\u2081.of_le (Nat.le_succ_of_le hr\u2081))\n  \u27e8Nat.le_succ_of_le hr\u2081, this, ht\u2082.of_rankGT hl\u2082\u27e9", "annotated_tactic": ["let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2082 := .cons ..) (ht\u2081.of_le (<a>Nat.le_succ_of_le</a> hr\u2081))\n        \u27e8<a>Nat.le_succ_of_le</a> hr\u2081, this, ht\u2082.of_rankGT hl\u2082\u27e9", [{"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}, {"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}]], "state_before": "case h_3.isFalse.isFalse.refl.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)).rankGT n\u271d)", "state_after": "case h_3.isFalse.isFalse.refl.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le n\u271d (merge le t\u2081 (cons r\u2081.succ a n t\u2082))\nih\u2082 : (t\u2081.rankGT n\u271d \u2194 (cons r\u2081.succ a n t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons r\u2081.succ a n t\u2082)).rankGT n\u271d\n\u22a2 WF le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)).rankGT n\u271d)"}, {"tactic": "exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun _ => Nat.lt_succ_of_le hr\u2081, fun _ => ht\u2081.rankGT.of_le hr\u2081\u27e9\u27e9", "annotated_tactic": ["exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun _ => <a>Nat.lt_succ_of_le</a> hr\u2081, fun _ => ht\u2081.rankGT.of_le hr\u2081\u27e9\u27e9", [{"full_name": "Nat.lt_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 22]}]], "state_before": "case h_3.isFalse.isFalse.refl.isFalse.isTrue\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : t\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le n\u271d (merge le t\u2081 (cons r\u2081.succ a n t\u2082))\nih\u2082 : (t\u2081.rankGT n\u271d \u2194 (cons r\u2081.succ a n t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons r\u2081.succ a n t\u2082)).rankGT n\u271d\n\u22a2 WF le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)).rankGT n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' ht\u2081 ht\u2082", "annotated_tactic": ["let \u27e8ih\u2081, ih\u2082\u27e9 := merge' ht\u2081 ht\u2082", []], "state_before": "case h_3.isFalse.isFalse.refl.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)", "state_after": "case h_3.isFalse.isFalse.refl.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le (r\u2081 + 1) (merge le t\u2081 t\u2082)\nih\u2082 : (t\u2081.rankGT (r\u2081 + 1) \u2194 t\u2082.rankGT (r\u2081 + 1)) \u2192 (merge le t\u2081 t\u2082).rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)"}, {"tactic": "exact \u27e8\u27e8Nat.le_succ_of_le hr\u2081, this, ih\u2081.of_rankGT (ih\u2082 (iff_of_false hl\u2081 hl\u2082))\u27e9,\n  fun _ => Nat.lt_succ_of_le hr\u2081\u27e9", "annotated_tactic": ["exact \u27e8\u27e8<a>Nat.le_succ_of_le</a> hr\u2081, this, ih\u2081.of_rankGT (ih\u2082 (<a>iff_of_false</a> hl\u2081 hl\u2082))\u27e9,\n        fun _ => <a>Nat.lt_succ_of_le</a> hr\u2081\u27e9", [{"full_name": "Nat.le_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1687, 9], "def_end_pos": [1687, 26]}, {"full_name": "iff_of_false", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1354, 9], "def_end_pos": [1354, 21]}, {"full_name": "Nat.lt_succ_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 22]}]], "state_before": "case h_3.isFalse.isFalse.refl.isFalse.isFalse\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WF le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WF le a\u2081 n\u2081 r\u2081\nht\u2081 : WF le (r\u2081 + 1) t\u2081\nh\u2082 : WF le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WF le a\u2082 n\u2082 r\u2081\nht\u2082 : WF le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WF le a n (r\u2081 + 1)\nhl\u2081 : \u00act\u2081.rankGT (r\u2081 + 1)\nhl\u2082 : \u00act\u2082.rankGT (r\u2081 + 1)\nih\u2081 : WF le (r\u2081 + 1) (merge le t\u2081 t\u2082)\nih\u2082 : (t\u2081.rankGT (r\u2081 + 1) \u2194 t\u2082.rankGT (r\u2081 + 1)) \u2192 (merge le t\u2081 t\u2082).rankGT (r\u2081 + 1)\n\u22a2 WF le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    (((cons r\u2081 a\u2081 n\u2081 t\u2081).rankGT n\u271d \u2194 (cons r\u2081 a\u2082 n\u2082 t\u2082).rankGT n\u271d) \u2192 (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)).rankGT n\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Comp.lean", "full_name": "derivWithin.scomp", "start": [138, 1], "end": [141, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "HasFDerivAtFilter.const_smul", "start": [59, 1], "end": [61, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "Filter.HasBasis.equicontinuousAt_iff", "start": [663, 1], "end": [670, 6], "traced_tactics": [{"tactic": "rw [equicontinuousAt_iff_continuousAt, ContinuousAt,\n  hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis \u03b9 \u03b1 _ h\u03b1)]", "annotated_tactic": ["rw [<a>equicontinuousAt_iff_continuousAt</a>, <a>ContinuousAt</a>,\n    hX.tendsto_iff (<a>UniformFun.hasBasis_nhds_of_basis</a> \u03b9 \u03b1 _ h\u03b1)]", [{"full_name": "equicontinuousAt_iff_continuousAt", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [509, 9], "def_end_pos": [509, 42]}, {"full_name": "ContinuousAt", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [156, 5], "def_end_pos": [156, 17]}, {"full_name": "UniformFun.hasBasis_nhds_of_basis", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [329, 19], "def_end_pos": [329, 41]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\n\u03ba\u2081 : Type u_13\n\u03ba\u2082 : Type u_14\np\u2081 : \u03ba\u2081 \u2192 Prop\ns\u2081 : \u03ba\u2081 \u2192 Set X\np\u2082 : \u03ba\u2082 \u2192 Prop\ns\u2082 : \u03ba\u2082 \u2192 Set (\u03b1 \u00d7 \u03b1)\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : (\ud835\udcdd x\u2080).HasBasis p\u2081 s\u2081\nh\u03b1 : (\ud835\udce4 \u03b1).HasBasis p\u2082 s\u2082\n\u22a2 EquicontinuousAt F x\u2080 \u2194 \u2200 (k\u2082 : \u03ba\u2082), p\u2082 k\u2082 \u2192 \u2203 k\u2081, p\u2081 k\u2081 \u2227 \u2200 x \u2208 s\u2081 k\u2081, \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 s\u2082 k\u2082", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\n\u03ba\u2081 : Type u_13\n\u03ba\u2082 : Type u_14\np\u2081 : \u03ba\u2081 \u2192 Prop\ns\u2081 : \u03ba\u2081 \u2192 Set X\np\u2082 : \u03ba\u2082 \u2192 Prop\ns\u2082 : \u03ba\u2082 \u2192 Set (\u03b1 \u00d7 \u03b1)\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : (\ud835\udcdd x\u2080).HasBasis p\u2081 s\u2081\nh\u03b1 : (\ud835\udce4 \u03b1).HasBasis p\u2082 s\u2082\n\u22a2 (\u2200 (ib : \u03ba\u2082),\n      p\u2082 ib \u2192\n        \u2203 ia,\n          p\u2081 ia \u2227\n            \u2200 x \u2208 s\u2081 ia,\n              (\u21d1UniformFun.ofFun \u2218 swap F) x \u2208\n                {g | ((\u21d1UniformFun.ofFun \u2218 swap F) x\u2080, g) \u2208 UniformFun.gen \u03b9 \u03b1 (s\u2082 ib)}) \u2194\n    \u2200 (k\u2082 : \u03ba\u2082), p\u2082 k\u2082 \u2192 \u2203 k\u2081, p\u2081 k\u2081 \u2227 \u2200 x \u2208 s\u2081 k\u2081, \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 s\u2082 k\u2082"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\n\u03ba\u2081 : Type u_13\n\u03ba\u2082 : Type u_14\np\u2081 : \u03ba\u2081 \u2192 Prop\ns\u2081 : \u03ba\u2081 \u2192 Set X\np\u2082 : \u03ba\u2082 \u2192 Prop\ns\u2082 : \u03ba\u2082 \u2192 Set (\u03b1 \u00d7 \u03b1)\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : (\ud835\udcdd x\u2080).HasBasis p\u2081 s\u2081\nh\u03b1 : (\ud835\udce4 \u03b1).HasBasis p\u2082 s\u2082\n\u22a2 (\u2200 (ib : \u03ba\u2082),\n      p\u2082 ib \u2192\n        \u2203 ia,\n          p\u2081 ia \u2227\n            \u2200 x \u2208 s\u2081 ia,\n              (\u21d1UniformFun.ofFun \u2218 swap F) x \u2208\n                {g | ((\u21d1UniformFun.ofFun \u2218 swap F) x\u2080, g) \u2208 UniformFun.gen \u03b9 \u03b1 (s\u2082 ib)}) \u2194\n    \u2200 (k\u2082 : \u03ba\u2082), p\u2082 k\u2082 \u2192 \u2203 k\u2081, p\u2081 k\u2081 \u2227 \u2200 x \u2208 s\u2081 k\u2081, \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 s\u2082 k\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.takeD_length", "start": [1805, 1], "end": [1807, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "bddBelow_closure", "start": [408, 1], "end": [409, 42], "traced_tactics": [{"tactic": "simp_rw [BddBelow, lowerBounds_closure]", "annotated_tactic": ["simp_rw [<a>BddBelow</a>, <a>lowerBounds_closure</a>]", [{"full_name": "BddBelow", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "lowerBounds_closure", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [404, 15], "def_end_pos": [404, 34]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\na b : \u03b1\ns : Set \u03b1\n\u22a2 BddBelow (closure s) \u2194 BddBelow s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "full_name": "Subring.closure_induction'", "start": [883, 1], "end": [897, 44], "traced_tactics": [{"tactic": "refine Exists.elim ?_ fun (ha : a \u2208 closure s) (hc : p a ha) => hc", "annotated_tactic": ["refine <a>Exists.elim</a> ?_ fun (ha : a \u2208 <a>closure</a> s) (hc : p a ha) => hc", [{"full_name": "Exists.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [827, 9], "def_end_pos": [827, 20]}, {"full_name": "Subring.closure", "def_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "def_pos": [838, 5], "def_end_pos": [838, 12]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 p a ha", "state_after": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2203 (x : a \u2208 closure s), p a x"}, {"tactic": "refine\n  closure_induction ha (fun m hm => \u27e8subset_closure hm, mem m hm\u27e9) \u27e8zero_mem _, zero\u27e9\n    \u27e8one_mem _, one\u27e9 ?_ (fun x hx => hx.elim fun hx' hx => \u27e8neg_mem hx', neg _ _ hx\u27e9) ?_", "annotated_tactic": ["refine\n    <a>closure_induction</a> ha (fun m hm => \u27e8<a>subset_closure</a> hm, mem m hm\u27e9) \u27e8<a>zero_mem</a> _, zero\u27e9\n      \u27e8<a>one_mem</a> _, one\u27e9 ?_ (fun x hx => hx.elim fun hx' hx => \u27e8<a>neg_mem</a> hx', neg _ _ hx\u27e9) ?_", [{"full_name": "Subring.closure_induction", "def_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "def_pos": [876, 9], "def_end_pos": [876, 26]}, {"full_name": "Subring.subset_closure", "def_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "def_pos": [848, 9], "def_end_pos": [848, 23]}, {"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}, {"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}, {"full_name": "NegMemClass.neg_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [108, 3], "def_end_pos": [108, 10]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2203 (x : a \u2208 closure s), p a x", "state_after": "case refine_1\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2200 (x y : R),\n    (\u2203 (x_1 : x \u2208 closure s), p x x_1) \u2192 (\u2203 (x : y \u2208 closure s), p y x) \u2192 \u2203 (x_1 : x + y \u2208 closure s), p (x + y) x_1\n\ncase refine_2\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2200 (x y : R),\n    (\u2203 (x_1 : x \u2208 closure s), p x x_1) \u2192 (\u2203 (x : y \u2208 closure s), p y x) \u2192 \u2203 (x_1 : x * y \u2208 closure s), p (x * y) x_1"}, {"tactic": "exact (fun x y hx hy => hx.elim fun hx' hx => hy.elim fun hy' hy =>\n  \u27e8add_mem hx' hy', add _ _ _ _ hx hy\u27e9)", "annotated_tactic": ["exact (fun x y hx hy => hx.elim fun hx' hx => hy.elim fun hy' hy =>\n      \u27e8<a>add_mem</a> hx' hy', add _ _ _ _ hx hy\u27e9)", [{"full_name": "AddMemClass.add_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}]], "state_before": "case refine_1\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2200 (x y : R),\n    (\u2203 (x_1 : x \u2208 closure s), p x x_1) \u2192 (\u2203 (x : y \u2208 closure s), p y x) \u2192 \u2203 (x_1 : x + y \u2208 closure s), p (x + y) x_1", "state_after": "no goals"}, {"tactic": "exact (fun x y hx hy => hx.elim fun hx' hx => hy.elim fun hy' hy =>\n  \u27e8mul_mem hx' hy', mul _ _ _ _ hx hy\u27e9)", "annotated_tactic": ["exact (fun x y hx hy => hx.elim fun hx' hx => hy.elim fun hy' hy =>\n      \u27e8<a>mul_mem</a> hx' hy', mul _ _ _ _ hx hy\u27e9)", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}]], "state_before": "case refine_2\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns : Set R\np : (x : R) \u2192 x \u2208 closure s \u2192 Prop\nmem : \u2200 (x : R) (h : x \u2208 s), p x \u22ef\nzero : p 0 \u22ef\none : p 1 \u22ef\nadd : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x + y) \u22ef\nneg : \u2200 (x : R) (hx : x \u2208 closure s), p x hx \u2192 p (-x) \u22ef\nmul : \u2200 (x : R) (hx : x \u2208 closure s) (y : R) (hy : y \u2208 closure s), p x hx \u2192 p y hy \u2192 p (x * y) \u22ef\na : R\nha : a \u2208 closure s\n\u22a2 \u2200 (x y : R),\n    (\u2203 (x_1 : x \u2208 closure s), p x x_1) \u2192 (\u2203 (x : y \u2208 closure s), p y x) \u2192 \u2203 (x_1 : x * y \u2208 closure s), p (x * y) x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "IsAtom.le_iff_eq", "start": [96, 1], "end": [97, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CompleteField.lean", "full_name": "LinearOrderedField.inducedMap_zero", "start": [203, 1], "end": [203, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsGreatest.lt_iff", "start": [314, 1], "end": [315, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupBotHom.coe_mk", "start": [752, 1], "end": [752, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "full_name": "MvPowerSeries.coeff_C_mul", "start": [431, 1], "end": [432, 95], "traced_tactics": [{"tactic": "simpa using coeff_add_monomial_mul 0 n \u03c6 a", "annotated_tactic": ["simpa using <a>coeff_add_monomial_mul</a> 0 n \u03c6 a", [{"full_name": "MvPowerSeries.coeff_add_monomial_mul", "def_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "def_pos": [237, 9], "def_end_pos": [237, 31]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nn : \u03c3 \u2192\u2080 \u2115\n\u03c6 : MvPowerSeries \u03c3 R\na : R\n\u22a2 (coeff R n) ((C \u03c3 R) a * \u03c6) = a * (coeff R n) \u03c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.star_apply", "start": [745, 11], "end": [746, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_apply_eq_zero_iff", "start": [196, 1], "end": [198, 62], "traced_tactics": [{"tactic": "rw [toOuterMeasure_apply, ENNReal.tsum_eq_zero]", "annotated_tactic": ["rw [<a>toOuterMeasure_apply</a>, <a>ENNReal.tsum_eq_zero</a>]", [{"full_name": "PMF.toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 29]}, {"full_name": "ENNReal.tsum_eq_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [893, 19], "def_end_pos": [893, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 p.toOuterMeasure s = 0 \u2194 Disjoint p.support s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 (\u2200 (i : \u03b1), s.indicator (\u21d1p) i = 0) \u2194 Disjoint p.support s"}, {"tactic": "exact Function.funext_iff.symm.trans Set.indicator_eq_zero'", "annotated_tactic": ["exact Function.funext_iff.symm.trans <a>Set.indicator_eq_zero'</a>", [{"full_name": "Set.indicator_eq_zero'", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [124, 3], "def_end_pos": [124, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 (\u2200 (i : \u03b1), s.indicator (\u21d1p) i = 0) \u2194 Disjoint p.support s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Finset.exists_not_mem", "start": [1359, 1], "end": [1359, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "full_name": "uniqueDiffWithinAt_inter'", "start": [282, 1], "end": [284, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.closure_univ", "start": [1228, 1], "end": [1229, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "full_name": "ModularForm.slash_action_eq'_iff", "start": [200, 1], "end": [208, 47], "traced_tactics": [{"tactic": "simp only [subgroup_slash, slash_def, ModularForm.slash]", "annotated_tactic": ["simp only [<a>subgroup_slash</a>, <a>slash_def</a>, <a>ModularForm.slash</a>]", [{"full_name": "ModularForm.subgroup_slash", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [167, 9], "def_end_pos": [167, 23]}, {"full_name": "ModularForm.slash_def", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "ModularForm.slash", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [92, 5], "def_end_pos": [92, 10]}]], "state_before": "\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 (f \u2223[k] \u03b3) z = f z \u2194 f (\u03b3 \u2022 z) = (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k * f z", "state_after": "\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1) * denom (\u2191\u2191\u03b3) z ^ (-k) = f z \u2194\n    f (\u03b3 \u2022 z) = (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k * f z"}, {"tactic": "convert inv_mul_eq_iff_eq_mul\u2080 (G\u2080 := \u2102) _ using 2", "annotated_tactic": ["convert <a>inv_mul_eq_iff_eq_mul\u2080</a> (G\u2080 := \u2102) _ using 2", [{"full_name": "inv_mul_eq_iff_eq_mul\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [306, 7], "def_end_pos": [306, 29]}]], "state_before": "\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1) * denom (\u2191\u2191\u03b3) z ^ (-k) = f z \u2194\n    f (\u03b3 \u2022 z) = (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k * f z", "state_after": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1) * denom (\u2191\u2191\u03b3) z ^ (-k) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u03b3 \u2022 z)\n\ncase convert_4\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k \u2260 0"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1) * denom (\u2191\u2191\u03b3) z ^ (-k) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u03b3 \u2022 z)", "state_after": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 denom (\u2191\u2191\u03b3) z ^ (-k) * (f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1)) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u03b3 \u2022 z)"}, {"tactic": "simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,\n  sl_moeb]", "annotated_tactic": ["simp only [<a>denom</a>, <a>zpow_neg</a>, <a>det_coe'</a>, <a>ofReal_one</a>, <a>one_zpow</a>, <a>mul_one</a>, <a>subgroup_to_sl_moeb</a>,\n      <a>sl_moeb</a>]", [{"full_name": "UpperHalfPlane.denom", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [201, 5], "def_end_pos": [201, 10]}, {"full_name": "zpow_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [593, 7], "def_end_pos": [593, 15]}, {"full_name": "UpperHalfPlane.ModularGroup.det_coe'", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [319, 9], "def_end_pos": [319, 17]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "one_zpow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [586, 7], "def_end_pos": [586, 15]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "UpperHalfPlane.ModularGroup.subgroup_to_sl_moeb", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 28]}, {"full_name": "UpperHalfPlane.ModularGroup.sl_moeb", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [426, 9], "def_end_pos": [426, 16]}]], "state_before": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 denom (\u2191\u2191\u03b3) z ^ (-k) * (f (\u2191\u2191\u03b3 \u2022 z) * \u2191(\u2191\u2191\u2191\u2191\u03b3).det ^ (k - 1)) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u03b3 \u2022 z)", "state_after": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 ((\u2191(\u2191\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u2191\u2191\u03b3 \u2022 z) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u2191\u2191\u03b3 \u2022 z)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_1.h.e'_2\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 ((\u2191(\u2191\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u2191\u2191\u03b3 \u2022 z) = ((\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k)\u207b\u00b9 * f (\u2191\u2191\u03b3 \u2022 z)", "state_after": "no goals"}, {"tactic": "convert zpow_ne_zero k (denom_ne_zero \u03b3 z)", "annotated_tactic": ["convert <a>zpow_ne_zero</a> k (<a>denom_ne_zero</a> \u03b3 z)", [{"full_name": "zpow_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [435, 7], "def_end_pos": [435, 19]}, {"full_name": "UpperHalfPlane.denom_ne_zero", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 22]}]], "state_before": "case convert_4\n\u0393\u271d : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : \u210d \u2192 \u2102\n\u03b3 : \u21a5\u0393\nz : \u210d\n\u22a2 (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Hom.lean", "full_name": "LinearMap.BilinForm.comp_apply", "start": [225, 1], "end": [226, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Permutation.lean", "full_name": "List.map_permutationsAux2'", "start": [90, 1], "end": [100, 23], "traced_tactics": [{"tactic": "induction' ys with ys_hd _ ys_ih generalizing f f'", "annotated_tactic": ["induction' ys with ys_hd _ ys_ih generalizing f f'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts ys : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r ys f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g ys) f').2", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r [] f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').2\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r (ys_hd :: tail\u271d) f).2 =\n    (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail\u271d)) f').2"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r [] f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').2", "state_after": "no goals"}, {"tactic": "simp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]", "annotated_tactic": ["simp only [<a>map</a>, <a>permutationsAux2_snd_cons</a>, <a>cons_append</a>, cons.injEq]", [{"full_name": "List.map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [361, 19], "def_end_pos": [361, 22]}, {"full_name": "List.permutationsAux2_snd_cons", "def_path": "Mathlib/Data/List/Permutation.lean", "def_pos": [69, 9], "def_end_pos": [69, 34]}, {"full_name": "List.cons_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [484, 17], "def_end_pos": [484, 28]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r (ys_hd :: tail\u271d) f).2 =\n    (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail\u271d)) f').2", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).1)) =\n      f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n    map g' (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).2 =\n      (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).2"}, {"tactic": "rw [ys_ih, permutationsAux2_fst]", "annotated_tactic": ["rw [ys_ih, <a>permutationsAux2_fst</a>]", [{"full_name": "List.permutationsAux2_fst", "def_path": "Mathlib/Data/List/Permutation.lean", "def_pos": [56, 9], "def_end_pos": [56, 29]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).1)) =\n      f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n    map g' (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).2 =\n      (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).2", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n    (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) ?cons.f').2 =\n      (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).2\n\ncase cons.f'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 List \u03b1' \u2192 \u03b2'\n\ncase cons.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 \u2200 (a : List \u03b1), g' (f (ys_hd :: a)) = ?cons.f' (map g a)"}, {"tactic": "refine \u27e8?_, rfl\u27e9", "annotated_tactic": ["refine \u27e8?_, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n    (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) ?cons.f').2 =\n      (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).2", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts))"}, {"tactic": "simp only [\u2190 map_cons, \u2190 map_append]", "annotated_tactic": ["simp only [\u2190 <a>map_cons</a>, \u2190 <a>map_append</a>]", [{"full_name": "List.map_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [366, 17], "def_end_pos": [366, 25]}, {"full_name": "List.map_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [839, 17], "def_end_pos": [839, 27]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts))", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (map g (t :: ys_hd :: (tail\u271d ++ ts)))"}, {"tactic": "apply H", "annotated_tactic": ["apply H", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (map g (t :: ys_hd :: (tail\u271d ++ ts)))", "state_after": "no goals"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "case cons.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 \u2200 (a : List \u03b1), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "state_after": "case cons.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\na : List \u03b1\n\u22a2 g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)"}, {"tactic": "apply H", "annotated_tactic": ["apply H", []], "state_before": "case cons.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b1' : Type u_3\n\u03b2' : Type u_4\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n  \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n    (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n      map g' (permutationsAux2 t ts r tail\u271d f).2 = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\na : List \u03b1\n\u22a2 g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.ae_restrict_union_iff", "start": [572, 1], "end": [573, 101], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (s \u222a t), p x) \u2194 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict s, p x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict t, p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.diam_singleton", "start": [409, 1], "end": [410, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/RadonNikodym.lean", "full_name": "ProbabilityTheory.kernel.rnDeriv_def'", "start": [242, 1], "end": [244, 82], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\n\u03b3 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba\u271d \u03b7\u271d : \u21a5(kernel \u03b1 \u03b3)\nh\u03b1\u03b3 : MeasurableSpace.CountableOrCountablyGenerated \u03b1 \u03b3\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b3)\n\u22a2 rnDeriv \u03ba \u03b7 = fun a x => ENNReal.ofReal (rnDerivAux \u03ba (\u03ba + \u03b7) a x) / ENNReal.ofReal (1 - rnDerivAux \u03ba (\u03ba + \u03b7) a x)", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b3 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba\u271d \u03b7\u271d : \u21a5(kernel \u03b1 \u03b3)\nh\u03b1\u03b3 : MeasurableSpace.CountableOrCountablyGenerated \u03b1 \u03b3\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b3)\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b3\n\u22a2 rnDeriv \u03ba \u03b7 x\u271d\u00b9 x\u271d = ENNReal.ofReal (rnDerivAux \u03ba (\u03ba + \u03b7) x\u271d\u00b9 x\u271d) / ENNReal.ofReal (1 - rnDerivAux \u03ba (\u03ba + \u03b7) x\u271d\u00b9 x\u271d)"}, {"tactic": "rw [rnDeriv_def]", "annotated_tactic": ["rw [<a>rnDeriv_def</a>]", [{"full_name": "ProbabilityTheory.kernel.rnDeriv_def", "def_path": "Mathlib/Probability/Kernel/RadonNikodym.lean", "def_pos": [239, 17], "def_end_pos": [239, 24]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b3 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba\u271d \u03b7\u271d : \u21a5(kernel \u03b1 \u03b3)\nh\u03b1\u03b3 : MeasurableSpace.CountableOrCountablyGenerated \u03b1 \u03b3\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b3)\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b3\n\u22a2 rnDeriv \u03ba \u03b7 x\u271d\u00b9 x\u271d = ENNReal.ofReal (rnDerivAux \u03ba (\u03ba + \u03b7) x\u271d\u00b9 x\u271d) / ENNReal.ofReal (1 - rnDerivAux \u03ba (\u03ba + \u03b7) x\u271d\u00b9 x\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Unique.lean", "full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd", "start": [81, 1], "end": [124, 50], "traced_tactics": [{"tactic": "let f := embeddingReal \u03a9", "annotated_tactic": ["let f := <a>embeddingReal</a> \u03a9", [{"full_name": "MeasureTheory.embeddingReal", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1105, 5], "def_end_pos": [1105, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x"}, {"tactic": "have hf := measurableEmbedding_embeddingReal \u03a9", "annotated_tactic": ["have hf := <a>measurableEmbedding_embeddingReal</a> \u03a9", [{"full_name": "MeasureTheory.measurableEmbedding_embeddingReal", "def_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "def_pos": [1108, 7], "def_end_pos": [1108, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x"}, {"tactic": "set \u03c1' : Measure (\u03b1 \u00d7 \u211d) := \u03c1.map (Prod.map id f) with h\u03c1'def", "annotated_tactic": ["set \u03c1' : <a>Measure</a> (\u03b1 \u00d7 \u211d) := \u03c1.map (<a>Prod.map</a> <a>id</a> f) with h\u03c1'def", [{"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [77, 11], "def_end_pos": [77, 18]}, {"full_name": "Prod.map", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1190, 5], "def_end_pos": [1190, 13]}, {"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x"}, {"tactic": "have h\u03c1' : \u03c1'.fst = \u03c1.fst := by\n  ext s hs\n  rw [h\u03c1'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply]\n  exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,\n    measurable_fst hs, hs, hs]", "annotated_tactic": ["have h\u03c1' : \u03c1'.fst = \u03c1.fst := by\n    ext s hs\n    rw [h\u03c1'def, <a>Measure.fst_apply</a>, <a>Measure.fst_apply</a>, <a>Measure.map_apply</a>]\n    exacts [<a>rfl</a>, <a>Measurable.prod</a> <a>measurable_fst</a> <| hf.measurable.comp <a>measurable_snd</a>,\n      <a>measurable_fst</a> hs, hs, hs]", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 18]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Measurable.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 23]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [711, 9], "def_end_pos": [711, 23]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x"}, {"tactic": "have h\u03c1'' : \u2200\u1d50 x \u2202\u03c1.fst, kernel.map \u03ba f hf.measurable x = \u03c1'.condKernel x := by\n  rw [\u2190 h\u03c1']\n  refine eq_condKernel_of_measure_eq_compProd_real (kernel.map \u03ba f hf.measurable) ?_\n  ext s hs\n  conv_lhs => rw [h\u03c1'def, h\u03ba]\n  rw [Measure.map_apply (measurable_id.prod_map hf.measurable) hs, h\u03c1',\n    Measure.compProd_apply hs, Measure.compProd_apply (measurable_id.prod_map hf.measurable hs)]\n  congr with a\n  rw [kernel.map_apply']\n  exacts [rfl, measurable_prod_mk_left hs]", "annotated_tactic": ["have h\u03c1'' : \u2200\u1d50 x \u2202\u03c1.fst, <a>kernel.map</a> \u03ba f hf.measurable x = \u03c1'.condKernel x := by\n    rw [\u2190 h\u03c1']\n    refine <a>eq_condKernel_of_measure_eq_compProd_real</a> (<a>kernel.map</a> \u03ba f hf.measurable) ?_\n    ext s hs\n    conv_lhs => rw [h\u03c1'def, h\u03ba]\n    rw [<a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) hs, h\u03c1',\n      <a>Measure.compProd_apply</a> hs, <a>Measure.compProd_apply</a> (measurable_id.prod_map hf.measurable hs)]\n    congr with a\n    rw [<a>kernel.map_apply'</a>]\n    exacts [<a>rfl</a>, <a>measurable_prod_mk_left</a> hs]", [{"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [627, 19], "def_end_pos": [627, 22]}, {"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration/Unique.lean", "def_pos": [60, 7], "def_end_pos": [60, 48]}, {"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [627, 19], "def_end_pos": [627, 22]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}, {"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}, {"full_name": "ProbabilityTheory.kernel.map_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [636, 9], "def_end_pos": [636, 19]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [752, 9], "def_end_pos": [752, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x"}, {"tactic": "suffices \u2200\u1d50 x \u2202\u03c1.fst, \u2200 s, MeasurableSet s \u2192 \u03c1'.condKernel x s = \u03c1.condKernel x (f \u207b\u00b9' s) by\n  filter_upwards [h\u03c1'', this] with x hx h\n  rw [kernel.map_apply] at hx\n  ext s hs\n  rw [\u2190 Set.preimage_image_eq s hf.injective,\n    \u2190 Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,\n    h _ <| hf.measurableSet_image.2 hs]", "annotated_tactic": ["suffices \u2200\u1d50 x \u2202\u03c1.fst, \u2200 s, <a>MeasurableSet</a> s \u2192 \u03c1'.condKernel x s = \u03c1.condKernel x (f \u207b\u00b9' s) by\n    filter_upwards [h\u03c1'', this] with x hx h\n    rw [<a>kernel.map_apply</a>] at hx\n    ext s hs\n    rw [\u2190 <a>Set.preimage_image_eq</a> s hf.injective,\n      \u2190 <a>Measure.map_apply</a> hf.measurable <| hf.measurableSet_image.2 hs, hx,\n      h _ <| hf.measurableSet_image.2 hs]", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [632, 9], "def_end_pos": [632, 18]}, {"full_name": "Set.preimage_image_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [494, 9], "def_end_pos": [494, 26]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)"}, {"tactic": "suffices \u03c1.map (Prod.map id f) = (\u03c1.fst \u2297\u2098 (kernel.map \u03c1.condKernel f hf.measurable)) by\n  rw [\u2190 h\u03c1'] at this\n  have heq := eq_condKernel_of_measure_eq_compProd_real _ this\n  rw [h\u03c1'] at heq\n  filter_upwards [heq] with x hx s hs\n  rw [\u2190 hx, kernel.map_apply, Measure.map_apply hf.measurable hs]", "annotated_tactic": ["suffices \u03c1.map (<a>Prod.map</a> <a>id</a> f) = (\u03c1.fst \u2297\u2098 (<a>kernel.map</a> \u03c1.condKernel f hf.measurable)) by\n    rw [\u2190 h\u03c1'] at this\n    have heq := <a>eq_condKernel_of_measure_eq_compProd_real</a> _ this\n    rw [h\u03c1'] at heq\n    filter_upwards [heq] with x hx s hs\n    rw [\u2190 hx, <a>kernel.map_apply</a>, <a>Measure.map_apply</a> hf.measurable hs]", [{"full_name": "Prod.map", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1190, 5], "def_end_pos": [1190, 13]}, {"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [627, 19], "def_end_pos": [627, 22]}, {"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration/Unique.lean", "def_pos": [60, 7], "def_end_pos": [60, 48]}, {"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [632, 9], "def_end_pos": [632, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 Measure.map (Prod.map id f) \u03c1 = \u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\n\u22a2 Measure.map (Prod.map id f) \u03c1 = \u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) \u03c1) s = (\u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef) s"}, {"tactic": "conv_lhs => rw [\u2190 \u03c1.compProd_fst_condKernel]", "annotated_tactic": ["conv_lhs => rw [\u2190 \u03c1.compProd_fst_condKernel]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) \u03c1) s = (\u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) (\u03c1.fst \u2297\u2098 \u03c1.condKernel)) s = (\u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef) s"}, {"tactic": "rw [Measure.compProd_apply hs, Measure.map_apply (measurable_id.prod_map hf.measurable) hs,\n  Measure.compProd_apply]", "annotated_tactic": ["rw [<a>Measure.compProd_apply</a> hs, <a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) hs,\n    <a>Measure.compProd_apply</a>]", [{"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) (\u03c1.fst \u2297\u2098 \u03c1.condKernel)) s = (\u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) \u2202\u03c1.fst =\n    \u222b\u207b (a : \u03b1), ((kernel.map \u03c1.condKernel f \u22ef) a) (Prod.mk a \u207b\u00b9' s) \u2202\u03c1.fst\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\n\u22a2 \u03c1'.fst = \u03c1.fst", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u03c1'.fst s = \u03c1.fst s"}, {"tactic": "rw [h\u03c1'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply]", "annotated_tactic": ["rw [h\u03c1'def, <a>Measure.fst_apply</a>, <a>Measure.fst_apply</a>, <a>Measure.map_apply</a>]", [{"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 18]}, {"full_name": "MeasureTheory.Measure.fst_apply", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u03c1'.fst s = \u03c1.fst s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' s)) = \u03c1 (Prod.fst \u207b\u00b9' s)\n\ncase h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable (Prod.map id f)\n\ncase h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s"}, {"tactic": "exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,\n  measurable_fst hs, hs, hs]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>Measurable.prod</a> <a>measurable_fst</a> <| hf.measurable.comp <a>measurable_snd</a>,\n      <a>measurable_fst</a> hs, hs, hs]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Measurable.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 23]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [711, 9], "def_end_pos": [711, 23]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' s)) = \u03c1 (Prod.fst \u207b\u00b9' s)\n\ncase h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable (Prod.map id f)\n\ncase h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s", "state_after": "no goals"}, {"tactic": "rw [\u2190 h\u03c1']", "annotated_tactic": ["rw [\u2190 h\u03c1']", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1'.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x"}, {"tactic": "refine eq_condKernel_of_measure_eq_compProd_real (kernel.map \u03ba f hf.measurable) ?_", "annotated_tactic": ["refine <a>eq_condKernel_of_measure_eq_compProd_real</a> (<a>kernel.map</a> \u03ba f hf.measurable) ?_", [{"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration/Unique.lean", "def_pos": [60, 7], "def_end_pos": [60, 48]}, {"full_name": "ProbabilityTheory.kernel.map", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [627, 19], "def_end_pos": [627, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1'.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u03c1' = \u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\n\u22a2 \u03c1' = \u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u03c1' s = (\u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef) s"}, {"tactic": "conv_lhs => rw [h\u03c1'def, h\u03ba]", "annotated_tactic": ["conv_lhs => rw [h\u03c1'def, h\u03ba]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u03c1' s = (\u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) (\u03c1.fst \u2297\u2098 \u03ba)) s = (\u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef) s"}, {"tactic": "rw [Measure.map_apply (measurable_id.prod_map hf.measurable) hs, h\u03c1',\n  Measure.compProd_apply hs, Measure.compProd_apply (measurable_id.prod_map hf.measurable hs)]", "annotated_tactic": ["rw [<a>Measure.map_apply</a> (measurable_id.prod_map hf.measurable) hs, h\u03c1',\n      <a>Measure.compProd_apply</a> hs, <a>Measure.compProd_apply</a> (measurable_id.prod_map hf.measurable hs)]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}, {"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 (Measure.map (Prod.map id f) (\u03c1.fst \u2297\u2098 \u03ba)) s = (\u03c1'.fst \u2297\u2098 kernel.map \u03ba f \u22ef) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) \u2202\u03c1.fst =\n    \u222b\u207b (a : \u03b1), ((kernel.map \u03ba f \u22ef) a) (Prod.mk a \u207b\u00b9' s) \u2202\u03c1.fst"}, {"tactic": "congr with a", "annotated_tactic": ["congr with a", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) \u2202\u03c1.fst =\n    \u222b\u207b (a : \u03b1), ((kernel.map \u03ba f \u22ef) a) (Prod.mk a \u207b\u00b9' s) \u2202\u03c1.fst", "state_after": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = ((kernel.map \u03ba f \u22ef) a) (Prod.mk a \u207b\u00b9' s)"}, {"tactic": "rw [kernel.map_apply']", "annotated_tactic": ["rw [<a>kernel.map_apply'</a>]", [{"full_name": "ProbabilityTheory.kernel.map_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [636, 9], "def_end_pos": [636, 19]}]], "state_before": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = ((kernel.map \u03ba f \u22ef) a) (Prod.mk a \u207b\u00b9' s)", "state_after": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = (\u03ba a) (f \u207b\u00b9' (Prod.mk a \u207b\u00b9' s))\n\ncase h.e_f.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 MeasurableSet (Prod.mk a \u207b\u00b9' s)"}, {"tactic": "exacts [rfl, measurable_prod_mk_left hs]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>measurable_prod_mk_left</a> hs]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [752, 9], "def_end_pos": [752, 32]}]], "state_before": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = (\u03ba a) (f \u207b\u00b9' (Prod.mk a \u207b\u00b9' s))\n\ncase h.e_f.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 MeasurableSet (Prod.mk a \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "filter_upwards [h\u03c1'', this] with x hx h", "annotated_tactic": ["filter_upwards [h\u03c1'', this] with x hx h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\n\u22a2 \u03ba x = \u03c1.condKernel x"}, {"tactic": "rw [kernel.map_apply] at hx", "annotated_tactic": ["rw [<a>kernel.map_apply</a>] at hx", [{"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [632, 9], "def_end_pos": [632, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\n\u22a2 \u03ba x = \u03c1.condKernel x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u03ba x) = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\n\u22a2 \u03ba x = \u03c1.condKernel x"}, {"tactic": "ext s hs", "annotated_tactic": ["ext s hs", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u03ba x) = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\n\u22a2 \u03ba x = \u03c1.condKernel x", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u03ba x) = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 (\u03ba x) s = (\u03c1.condKernel x) s"}, {"tactic": "rw [\u2190 Set.preimage_image_eq s hf.injective,\n  \u2190 Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,\n  h _ <| hf.measurableSet_image.2 hs]", "annotated_tactic": ["rw [\u2190 <a>Set.preimage_image_eq</a> s hf.injective,\n      \u2190 <a>Measure.map_apply</a> hf.measurable <| hf.measurableSet_image.2 hs, hx,\n      h _ <| hf.measurableSet_image.2 hs]", [{"full_name": "Set.preimage_image_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [494, 9], "def_end_pos": [494, 26]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\nx : \u03b1\nhx : Measure.map f (\u03ba x) = \u03c1'.condKernel x\nh : \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 (\u03ba x) s = (\u03c1.condKernel x) s", "state_after": "no goals"}, {"tactic": "rw [\u2190 h\u03c1'] at this", "annotated_tactic": ["rw [\u2190 h\u03c1'] at this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)"}, {"tactic": "have heq := eq_condKernel_of_measure_eq_compProd_real _ this", "annotated_tactic": ["have heq := <a>eq_condKernel_of_measure_eq_compProd_real</a> _ this", [{"full_name": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real", "def_path": "Mathlib/Probability/Kernel/Disintegration/Unique.lean", "def_pos": [60, 7], "def_end_pos": [60, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202(Measure.map (Prod.map id f) \u03c1).fst,\n    (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)"}, {"tactic": "rw [h\u03c1'] at heq", "annotated_tactic": ["rw [h\u03c1'] at heq", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq :\n  \u2200\u1d50 (x : \u03b1) \u2202(Measure.map (Prod.map id f) \u03c1).fst,\n    (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)"}, {"tactic": "filter_upwards [heq] with x hx s hs", "annotated_tactic": ["filter_upwards [heq] with x hx s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, \u2200 (s : Set \u211d), MeasurableSet s \u2192 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\nx : \u03b1\nhx : (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)"}, {"tactic": "rw [\u2190 hx, kernel.map_apply, Measure.map_apply hf.measurable hs]", "annotated_tactic": ["rw [\u2190 hx, <a>kernel.map_apply</a>, <a>Measure.map_apply</a> hf.measurable hs]", [{"full_name": "ProbabilityTheory.kernel.map_apply", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [632, 9], "def_end_pos": [632, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\nthis : Measure.map (Prod.map id f) \u03c1 = \u03c1'.fst \u2297\u2098 kernel.map \u03c1.condKernel f \u22ef\nheq : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\nx : \u03b1\nhx : (kernel.map \u03c1.condKernel f \u22ef) x = (Measure.map (Prod.map id f) \u03c1).condKernel x\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 (\u03c1'.condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "congr with a", "annotated_tactic": ["congr with a", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) \u2202\u03c1.fst =\n    \u222b\u207b (a : \u03b1), ((kernel.map \u03c1.condKernel f \u22ef) a) (Prod.mk a \u207b\u00b9' s) \u2202\u03c1.fst", "state_after": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) =\n    ((kernel.map \u03c1.condKernel f \u22ef) a) (Prod.mk a \u207b\u00b9' s)"}, {"tactic": "rw [kernel.map_apply']", "annotated_tactic": ["rw [<a>kernel.map_apply'</a>]", [{"full_name": "ProbabilityTheory.kernel.map_apply'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [636, 9], "def_end_pos": [636, 19]}]], "state_before": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) =\n    ((kernel.map \u03c1.condKernel f \u22ef) a) (Prod.mk a \u207b\u00b9' s)", "state_after": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = (\u03c1.condKernel a) (f \u207b\u00b9' (Prod.mk a \u207b\u00b9' s))\n\ncase h.e_f.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 MeasurableSet (Prod.mk a \u207b\u00b9' s)"}, {"tactic": "exacts [rfl, measurable_prod_mk_left hs]", "annotated_tactic": ["exacts [<a>rfl</a>, <a>measurable_prod_mk_left</a> hs]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [752, 9], "def_end_pos": [752, 32]}]], "state_before": "case h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 (\u03c1.condKernel a) (Prod.mk a \u207b\u00b9' (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)) = (\u03c1.condKernel a) (f \u207b\u00b9' (Prod.mk a \u207b\u00b9' s))\n\ncase h.e_f.h.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\na : \u03b1\n\u22a2 MeasurableSet (Prod.mk a \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "exact measurable_id.prod_map hf.measurable hs", "annotated_tactic": ["exact measurable_id.prod_map hf.measurable hs", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03a9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : StandardBorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\ninst\u271d : IsFiniteKernel \u03ba\nh\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba\nf : \u03a9 \u2192 \u211d := embeddingReal \u03a9\nhf : MeasurableEmbedding (embeddingReal \u03a9)\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\nh\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1\nh\u03c1' : \u03c1'.fst = \u03c1.fst\nh\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202\u03c1.fst, (kernel.map \u03ba f \u22ef) x = \u03c1'.condKernel x\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Prod.map id (embeddingReal \u03a9) \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.blsub_eq_zero_iff", "start": [1879, 1], "end": [1881, 30], "traced_tactics": [{"tactic": "rw [\u2190 lsub_eq_blsub, lsub_eq_zero_iff]", "annotated_tactic": ["rw [\u2190 <a>lsub_eq_blsub</a>, <a>lsub_eq_zero_iff</a>]", [{"full_name": "Ordinal.lsub_eq_blsub", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1770, 9], "def_end_pos": [1770, 22]}, {"full_name": "Ordinal.lsub_eq_zero_iff", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1669, 9], "def_end_pos": [1669, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) \u2192 a < o \u2192 Ordinal.{max u_5 u_4}\n\u22a2 o.blsub f = 0 \u2194 o = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) \u2192 a < o \u2192 Ordinal.{max u_5 u_4}\n\u22a2 IsEmpty (Quotient.out o).\u03b1 \u2194 o = 0"}, {"tactic": "exact out_empty_iff_eq_zero", "annotated_tactic": ["exact <a>out_empty_iff_eq_zero</a>", [{"full_name": "Ordinal.out_empty_iff_eq_zero", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [274, 9], "def_end_pos": [274, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) \u2192 a < o \u2192 Ordinal.{max u_5 u_4}\n\u22a2 IsEmpty (Quotient.out o).\u03b1 \u2194 o = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Separation.lean", "full_name": "t0Space_iff_uniformity'", "start": [155, 1], "end": [157, 69], "traced_tactics": [{"tactic": "simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity]", "annotated_tactic": ["simp [<a>t0Space_iff_not_inseparable</a>, <a>inseparable_iff_ker_uniformity</a>]", [{"full_name": "t0Space_iff_not_inseparable", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [271, 9], "def_end_pos": [271, 36]}, {"full_name": "inseparable_iff_ker_uniformity", "def_path": "Mathlib/Topology/UniformSpace/Separation.lean", "def_pos": [134, 9], "def_end_pos": [134, 39]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u22a2 T0Space \u03b1 \u2194 Pairwise fun x y => \u2203 r \u2208 \ud835\udce4 \u03b1, (x, y) \u2209 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Mul.lean", "full_name": "ConvexOn.smul''", "start": [69, 1], "end": [73, 12], "traced_tactics": [{"tactic": "rw [\u2190 neg_smul_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_smul_neg</a>]", [{"full_name": "neg_smul_neg", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [259, 9], "def_end_pos": [259, 21]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrderedCommRing \ud835\udd5c\ninst\u271d\u2079 : LinearOrderedCommRing E\ninst\u271d\u2078 : LinearOrderedAddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c F\ninst\u271d\u2075 : Module E F\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c E F\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c E F\ninst\u271d\u00b2 : OrderedSMul \ud835\udd5c E\ninst\u271d\u00b9 : OrderedSMul \ud835\udd5c F\ninst\u271d : OrderedSMul E F\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\ng : \ud835\udd5c \u2192 F\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nhf\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 f x \u2264 0\nhg\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 g x \u2264 0\nhfg : AntivaryOn f g s\n\u22a2 ConcaveOn \ud835\udd5c s (f \u2022 g)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrderedCommRing \ud835\udd5c\ninst\u271d\u2079 : LinearOrderedCommRing E\ninst\u271d\u2078 : LinearOrderedAddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c F\ninst\u271d\u2075 : Module E F\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c E F\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c E F\ninst\u271d\u00b2 : OrderedSMul \ud835\udd5c E\ninst\u271d\u00b9 : OrderedSMul \ud835\udd5c F\ninst\u271d : OrderedSMul E F\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\ng : \ud835\udd5c \u2192 F\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nhf\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 f x \u2264 0\nhg\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 g x \u2264 0\nhfg : AntivaryOn f g s\n\u22a2 ConcaveOn \ud835\udd5c s (-f \u2022 -g)"}, {"tactic": "exact hf.neg.smul' hg.neg (fun x hx \u21a6 neg_nonneg.2 <| hf\u2080 hx) (fun x hx \u21a6 neg_nonneg.2 <| hg\u2080 hx)\n  hfg.neg", "annotated_tactic": ["exact hf.neg.smul' hg.neg (fun x hx \u21a6 <a>neg_nonneg</a>.2 <| hf\u2080 hx) (fun x hx \u21a6 <a>neg_nonneg</a>.2 <| hg\u2080 hx)\n    hfg.neg", [{"full_name": "neg_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [671, 24], "def_end_pos": [671, 34]}, {"full_name": "neg_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [671, 24], "def_end_pos": [671, 34]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : LinearOrderedCommRing \ud835\udd5c\ninst\u271d\u2079 : LinearOrderedCommRing E\ninst\u271d\u2078 : LinearOrderedAddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c F\ninst\u271d\u2075 : Module E F\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c E F\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c E F\ninst\u271d\u00b2 : OrderedSMul \ud835\udd5c E\ninst\u271d\u00b9 : OrderedSMul \ud835\udd5c F\ninst\u271d : OrderedSMul E F\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\ng : \ud835\udd5c \u2192 F\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nhf\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 f x \u2264 0\nhg\u2080 : \u2200 \u2983x : \ud835\udd5c\u2984, x \u2208 s \u2192 g x \u2264 0\nhfg : AntivaryOn f g s\n\u22a2 ConcaveOn \ud835\udd5c s (-f \u2022 -g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/SingleObj.lean", "full_name": "MonoidHom.comp_toFunctor", "start": [201, 1], "end": [203, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.biUnion_coe_eq_coe", "start": [300, 1], "end": [302, 44], "traced_tactics": [{"tactic": "induction I <;> simp [WithBot.coe_eq_coe]", "annotated_tactic": ["induction I <;> simp [<a>WithBot.coe_eq_coe</a>]", [{"full_name": "WithBot.coe_eq_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}]], "state_before": "\u03b9 : Type u_1\nI\u271d J : Box \u03b9\nx y : \u03b9 \u2192 \u211d\nI : WithBot (Box \u03b9)\n\u22a2 \u22c3 J, \u22c3 (_ : \u2191J = I), \u2191J = \u2191I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Basic.lean", "full_name": "BoxIntegral.Prepartition.mem_iUnion", "start": [226, 1], "end": [228, 32], "traced_tactics": [{"tactic": "convert Set.mem_iUnion\u2082", "annotated_tactic": ["convert <a>Set.mem_iUnion\u2082</a>", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [67, 9], "def_end_pos": [67, 20]}]], "state_before": "\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 x \u2208 \u03c0.iUnion \u2194 \u2203 J \u2208 \u03c0, x \u2208 J", "state_after": "case h.e'_2.h.e'_2.h.a\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nx\u271d : Box \u03b9\n\u22a2 x\u271d \u2208 \u03c0 \u2227 x \u2208 x\u271d \u2194 \u2203 (_ : x\u271d \u2208 \u03c0), x \u2208 \u2191x\u271d"}, {"tactic": "rw [Box.mem_coe, exists_prop]", "annotated_tactic": ["rw [<a>Box.mem_coe</a>, <a>exists_prop</a>]", [{"full_name": "BoxIntegral.Box.mem_coe", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "case h.e'_2.h.e'_2.h.a\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nx\u271d : Box \u03b9\n\u22a2 x\u271d \u2208 \u03c0 \u2227 x \u2208 x\u271d \u2194 \u2203 (_ : x\u271d \u2208 \u03c0), x \u2208 \u2191x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Submodule.lean", "full_name": "Submodule.comm_trans_rTensorOne", "start": [226, 1], "end": [231, 17], "traced_tactics": [{"tactic": "refine LinearEquiv.toLinearMap_injective <| TensorProduct.ext' fun r m \u21a6 ?_", "annotated_tactic": ["refine <a>LinearEquiv.toLinearMap_injective</a> <| <a>TensorProduct.ext'</a> fun r m \u21a6 ?_", [{"full_name": "LinearEquiv.toLinearMap_injective", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [171, 9], "def_end_pos": [171, 30]}, {"full_name": "TensorProduct.ext'", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 13]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\n\u22a2 TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne = M.lTensorOne", "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)"}, {"tactic": "obtain \u27e8x, h\u27e9 := Algebra.mem_bot.1 r.2", "annotated_tactic": ["obtain \u27e8x, h\u27e9 := <a>Algebra.mem_bot</a>.1 r.2", [{"full_name": "Algebra.mem_bot", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [884, 9], "def_end_pos": [884, 16]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)", "state_after": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R S) x = \u2191r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)"}, {"tactic": "replace h : algebraMap R _ x = r := Subtype.val_injective h", "annotated_tactic": ["replace h : <a>algebraMap</a> R _ x = r := <a>Subtype.val_injective</a> h", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "Subtype.val_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [131, 9], "def_end_pos": [131, 22]}]], "state_before": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R S) x = \u2191r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)", "state_after": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R \u21a5\u22a5) x = r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R \u21a5\u22a5) x = r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) (r \u2297\u209c[R] m) = \u2191M.lTensorOne (r \u2297\u209c[R] m)", "state_after": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R \u21a5\u22a5) x = r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) ((algebraMap R \u21a5\u22a5) x \u2297\u209c[R] m) =\n    \u2191M.lTensorOne ((algebraMap R \u21a5\u22a5) x \u2297\u209c[R] m)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Algebra R S\nM N : Submodule R S\nr : \u21a5\u22a5\nm : \u21a5M\nx : R\nh : (algebraMap R \u21a5\u22a5) x = r\n\u22a2 \u2191(TensorProduct.comm R \u21a5\u22a5 \u21a5M \u226a\u226b\u2097 M.rTensorOne) ((algebraMap R \u21a5\u22a5) x \u2297\u209c[R] m) =\n    \u2191M.lTensorOne ((algebraMap R \u21a5\u22a5) x \u2297\u209c[R] m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "full_name": "MvPFunctor.M.dest_eq_dest'", "start": [201, 1], "end": [204, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "full_name": "CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_\u03c0", "start": [207, 1], "end": [212, 7], "traced_tactics": [{"tactic": "dsimp [limitObjIsoLimitCompEvaluation]", "annotated_tactic": ["dsimp [<a>limitObjIsoLimitCompEvaluation</a>]", [{"full_name": "CategoryTheory.Limits.limitObjIsoLimitCompEvaluation", "def_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "def_pos": [201, 5], "def_end_pos": [201, 35]}]], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (limitObjIsoLimitCompEvaluation F k).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j = (limit.\u03c0 F j).app k", "state_after": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (preservesLimitIso ((evaluation K C).obj k) F).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j = (limit.\u03c0 F j).app k"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (preservesLimitIso ((evaluation K C).obj k) F).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j = (limit.\u03c0 F j).app k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.gcd_mul_left_add_left", "start": [68, 1], "end": [69, 50], "traced_tactics": [{"tactic": "rw [gcd_comm, gcd_mul_left_add_right, gcd_comm]", "annotated_tactic": ["rw [<a>gcd_comm</a>, <a>gcd_mul_left_add_right</a>, <a>gcd_comm</a>]", [{"full_name": "Nat.gcd_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}, {"full_name": "Nat.gcd_mul_left_add_right", "def_path": "Mathlib/Data/Nat/GCD/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 31]}, {"full_name": "Nat.gcd_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}]], "state_before": "m n k : \u2115\n\u22a2 (n * k + m).gcd n = m.gcd n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "full_name": "UniformInducing.of_comp_iff", "start": [76, 1], "end": [80, 34], "traced_tactics": [{"tactic": "refine \u27e8fun h \u21a6 ?_, hg.comp\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 ?_, hg.comp\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformInducing (g \u2218 f) \u2194 UniformInducing f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\nf : \u03b1 \u2192 \u03b2\nh : UniformInducing (g \u2218 f)\n\u22a2 UniformInducing f"}, {"tactic": "rw [uniformInducing_iff, \u2190 hg.comap_uniformity, comap_comap, \u2190 h.comap_uniformity,\n  Function.comp, Function.comp]", "annotated_tactic": ["rw [<a>uniformInducing_iff</a>, \u2190 hg.comap_uniformity, <a>comap_comap</a>, \u2190 h.comap_uniformity,\n    <a>Function.comp</a>, <a>Function.comp</a>]", [{"full_name": "uniformInducing_iff", "def_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "def_pos": [33, 3], "def_end_pos": [33, 9]}, {"full_name": "Filter.comap_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2245, 9], "def_end_pos": [2245, 20]}, {"full_name": "Function.comp", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "Function.comp", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\nf : \u03b1 \u2192 \u03b2\nh : UniformInducing (g \u2218 f)\n\u22a2 UniformInducing f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Bounded.to_sep", "start": [991, 1], "end": [995, 49], "traced_tactics": [{"tactic": "refine h\u2081.mem_lt.imp fun y yx => ?_", "annotated_tactic": ["refine h\u2081.mem_lt.imp fun y yx => ?_", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nt\u2081 t\u2082 : Ordnode \u03b1\no\u2081 : WithBot \u03b1\no\u2082 : WithTop \u03b1\nx : \u03b1\nh\u2081 : t\u2081.Bounded o\u2081 \u2191x\nh\u2082 : t\u2082.Bounded (\u2191x) o\u2082\n\u22a2 All (fun y => All (fun z => y < z) t\u2082) t\u2081", "state_after": "\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nt\u2081 t\u2082 : Ordnode \u03b1\no\u2081 : WithBot \u03b1\no\u2082 : WithTop \u03b1\nx : \u03b1\nh\u2081 : t\u2081.Bounded o\u2081 \u2191x\nh\u2082 : t\u2082.Bounded (\u2191x) o\u2082\ny : \u03b1\nyx : y < x\n\u22a2 All (fun z => y < z) t\u2082"}, {"tactic": "exact h\u2082.mem_gt.imp fun z xz => lt_trans yx xz", "annotated_tactic": ["exact h\u2082.mem_gt.imp fun z xz => <a>lt_trans</a> yx xz", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nt\u2081 t\u2082 : Ordnode \u03b1\no\u2081 : WithBot \u03b1\no\u2082 : WithTop \u03b1\nx : \u03b1\nh\u2081 : t\u2081.Bounded o\u2081 \u2191x\nh\u2082 : t\u2082.Bounded (\u2191x) o\u2082\ny : \u03b1\nyx : y < x\n\u22a2 All (fun z => y < z) t\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "IsBoundedBilinearMap.contDiff", "start": [201, 1], "end": [206, 45], "traced_tactics": [{"tactic": "suffices h : ContDiff \ud835\udd5c \u221e b from h.of_le le_top", "annotated_tactic": ["suffices h : <a>ContDiff</a> \ud835\udd5c \u221e b from h.of_le <a>le_top</a>", [{"full_name": "ContDiff", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1443, 5], "def_end_pos": [1443, 13]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c n b", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 b"}, {"tactic": "rw [contDiff_top_iff_fderiv]", "annotated_tactic": ["rw [<a>contDiff_top_iff_fderiv</a>]", [{"full_name": "contDiff_top_iff_fderiv", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1753, 9], "def_end_pos": [1753, 32]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 b", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 Differentiable \ud835\udd5c b \u2227 ContDiff \ud835\udd5c \u22a4 fun y => fderiv \ud835\udd5c b y"}, {"tactic": "refine \u27e8hb.differentiable, ?_\u27e9", "annotated_tactic": ["refine \u27e8hb.differentiable, ?_\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 Differentiable \ud835\udd5c b \u2227 ContDiff \ud835\udd5c \u22a4 fun y => fderiv \ud835\udd5c b y", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 fun y => fderiv \ud835\udd5c b y"}, {"tactic": "simp only [hb.fderiv]", "annotated_tactic": ["simp only [hb.fderiv]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 fun y => fderiv \ud835\udd5c b y", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 fun y => hb.deriv y"}, {"tactic": "exact hb.isBoundedLinearMap_deriv.contDiff", "annotated_tactic": ["exact hb.isBoundedLinearMap_deriv.contDiff", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhb : IsBoundedBilinearMap \ud835\udd5c b\n\u22a2 ContDiff \ud835\udd5c \u22a4 fun y => hb.deriv y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "ContinuousLinearEquiv.comp_right_differentiableWithinAt_iff", "start": [191, 1], "end": [199, 56], "traced_tactics": [{"tactic": "refine \u27e8fun H => ?_, fun H => H.comp x iso.differentiableWithinAt (mapsTo_preimage _ s)\u27e9", "annotated_tactic": ["refine \u27e8fun H => ?_, fun H => H.comp x iso.differentiableWithinAt (<a>mapsTo_preimage</a> _ s)\u27e9", [{"full_name": "Set.mapsTo_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [273, 9], "def_end_pos": [273, 24]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\n\u22a2 DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x \u2194 DifferentiableWithinAt \ud835\udd5c f s (iso x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (iso x)"}, {"tactic": "have : DifferentiableWithinAt \ud835\udd5c ((f \u2218 iso) \u2218 iso.symm) s (iso x) := by\n  rw [\u2190 iso.symm_apply_apply x] at H\n  apply H.comp (iso x) iso.symm.differentiableWithinAt\n  intro y hy\n  simpa only [mem_preimage, apply_symm_apply] using hy", "annotated_tactic": ["have : <a>DifferentiableWithinAt</a> \ud835\udd5c ((f \u2218 iso) \u2218 iso.symm) s (iso x) := by\n    rw [\u2190 iso.symm_apply_apply x] at H\n    apply H.comp (iso x) iso.symm.differentiableWithinAt\n    intro y hy\n    simpa only [<a>mem_preimage</a>, <a>apply_symm_apply</a>] using hy", [{"full_name": "DifferentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [171, 5], "def_end_pos": [171, 27]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "ContinuousLinearEquiv.apply_symm_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2154, 9], "def_end_pos": [2154, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (iso x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x\nthis : DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u21d1iso) \u2218 \u21d1iso.symm) s (iso x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (iso x)"}, {"tactic": "rwa [Function.comp.assoc, iso.self_comp_symm] at this", "annotated_tactic": ["rwa [<a>Function.comp.assoc</a>, iso.self_comp_symm] at this", [{"full_name": "Function.comp.assoc", "def_path": "Mathlib/Init/Function.lean", "def_pos": [110, 9], "def_end_pos": [110, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x\nthis : DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u21d1iso) \u2218 \u21d1iso.symm) s (iso x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (iso x)", "state_after": "no goals"}, {"tactic": "rw [\u2190 iso.symm_apply_apply x] at H", "annotated_tactic": ["rw [\u2190 iso.symm_apply_apply x] at H", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u21d1iso) \u2218 \u21d1iso.symm) s (iso x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u21d1iso) \u2218 \u21d1iso.symm) s (iso x)"}, {"tactic": "apply H.comp (iso x) iso.symm.differentiableWithinAt", "annotated_tactic": ["apply H.comp (iso x) iso.symm.differentiableWithinAt", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u21d1iso) \u2218 \u21d1iso.symm) s (iso x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\n\u22a2 MapsTo (\u21d1iso.symm) s (\u21d1iso \u207b\u00b9' s)"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\n\u22a2 MapsTo (\u21d1iso.symm) s (\u21d1iso \u207b\u00b9' s)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\ny : F\nhy : y \u2208 s\n\u22a2 iso.symm y \u2208 \u21d1iso \u207b\u00b9' s"}, {"tactic": "simpa only [mem_preimage, apply_symm_apply] using hy", "annotated_tactic": ["simpa only [<a>mem_preimage</a>, <a>apply_symm_apply</a>] using hy", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "ContinuousLinearEquiv.apply_symm_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2154, 9], "def_end_pos": [2154, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u21d1iso) (\u21d1iso \u207b\u00b9' s) (iso.symm (iso x))\ny : F\nhy : y \u2208 s\n\u22a2 iso.symm y \u2208 \u21d1iso \u207b\u00b9' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Adjunction/Unique.lean", "full_name": "CategoryTheory.Adjunction.leftAdjointUniq_trans_app", "start": [170, 1], "end": [175, 6], "traced_tactics": [{"tactic": "rw [\u2190 leftAdjointUniq_trans adj1 adj2 adj3]", "annotated_tactic": ["rw [\u2190 <a>leftAdjointUniq_trans</a> adj1 adj2 adj3]", [{"full_name": "CategoryTheory.Adjunction.leftAdjointUniq_trans", "def_path": "Mathlib/CategoryTheory/Adjunction/Unique.lean", "def_pos": [162, 9], "def_end_pos": [162, 30]}]], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : Category.{u_4, u_2} D\nF F' F'' : C \u2964 D\nG : D \u2964 C\nadj1 : F \u22a3 G\nadj2 : F' \u22a3 G\nadj3 : F'' \u22a3 G\nx : C\n\u22a2 (adj1.leftAdjointUniq adj2).hom.app x \u226b (adj2.leftAdjointUniq adj3).hom.app x = (adj1.leftAdjointUniq adj3).hom.app x", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : Category.{u_4, u_2} D\nF F' F'' : C \u2964 D\nG : D \u2964 C\nadj1 : F \u22a3 G\nadj2 : F' \u22a3 G\nadj3 : F'' \u22a3 G\nx : C\n\u22a2 (adj1.leftAdjointUniq adj2).hom.app x \u226b 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K\nU : Set Z\nhU : IsOpen U\n\u22a2 IsOpen {x | MapsTo (\u21d1(x.comp f)) K U}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Infinite.of_image", "start": [1390, 1], "end": [1391, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Bialgebra/Hom.lean", "full_name": "BialgHom.id_toAlgHom", "start": [239, 1], "end": [240, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "full_name": "CategoryTheory.NatTrans.CommShift.app_shift", "start": [276, 1], "end": [280, 61], "traced_tactics": [{"tactic": "erw [comm_app_assoc, Iso.hom_inv_id_app, Category.comp_id]", 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goals"}, {"tactic": "simp [integral, hG]", "annotated_tactic": ["simp [<a>integral</a>, hG]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [788, 17], "def_end_pos": [788, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhG : \u00acCompleteSpace G\n\u22a2 \u222b (x : \u03b1), 0 \u2202\u03bc = 0", "state_after": "no goals"}]}, 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<a>withDensity_congr_ae</a> (<a>coe_toNNReal_ae_eq</a> hflt),\n    <a>integrable_withDensity_iff_integrable_smul</a>]", [{"full_name": "MeasureTheory.withDensity_congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [83, 9], "def_end_pos": [83, 29]}, {"full_name": "MeasureTheory.coe_toNNReal_ae_eq", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [935, 9], "def_end_pos": [935, 27]}, {"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1001, 9], "def_end_pos": [1001, 51]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Integrable g (\u03bc.withDensity f) \u2194 Integrable (fun x => (f x).toReal \u2022 g x) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Integrable (fun x => (f x).toNNReal \u2022 g x) \u03bc \u2194 Integrable (fun x => (f x).toReal \u2022 g x) \u03bc\n\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace 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\u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Integrable (fun x => (f x).toNNReal \u2022 g x) \u03bc \u2194 Integrable (fun x => (f x).toReal \u2022 g x) \u03bc", "state_after": "no goals"}, {"tactic": "exact hf.ennreal_toNNReal", "annotated_tactic": ["exact hf.ennreal_toNNReal", []], "state_before": "case hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b4\ninst\u271d\u00b3 : NormedAddCommGroup \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup \u03b3\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 E\n\u22a2 Measurable fun x => (f x).toNNReal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.isClique_bot_iff", "start": [107, 1], "end": [108, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.congr_fun", "start": [158, 11], "end": [159, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Squarefree.lean", "full_name": "Nat.squarefree_mul_iff", "start": [383, 1], "end": [386, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.inv_smul_finset_distrib", "start": [2200, 1], "end": [2202, 33], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na\u271d : \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\n\u22a2 (a \u2022 s)\u207b\u00b9 = op a\u207b\u00b9 \u2022 s\u207b\u00b9", "state_after": "case a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na\u271d\u00b9 : \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 (a \u2022 s)\u207b\u00b9 \u2194 a\u271d \u2208 op a\u207b\u00b9 \u2022 s\u207b\u00b9"}, {"tactic": "simp [\u2190 inv_smul_mem_iff]", "annotated_tactic": ["simp [\u2190 <a>inv_smul_mem_iff</a>]", [{"full_name": "Finset.inv_smul_mem_iff", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [2097, 9], "def_end_pos": [2097, 25]}]], "state_before": "case a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b2\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\ns\u271d t : Finset \u03b2\na\u271d\u00b9 : \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 (a \u2022 s)\u207b\u00b9 \u2194 a\u271d \u2208 op a\u207b\u00b9 \u2022 s\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Coloring.lean", "full_name": "SimpleGraph.colorable_of_fintype", "start": [234, 1], "end": [235, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "MeasureTheory.Integrable.compProd_mk_left_ae", "start": [131, 1], "end": [133, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.one_mem_one", "start": [713, 1], "end": [714, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean", "full_name": "CategoryTheory.LocalizerMorphism.isEquivalence_imp", "start": [103, 1], "end": [120, 57], "traced_tactics": [{"tactic": "let E\u2081 := Localization.uniq L\u2081 L\u2081' W\u2081", "annotated_tactic": ["let E\u2081 := <a>Localization.uniq</a> L\u2081 L\u2081' W\u2081", [{"full_name": "CategoryTheory.Localization.uniq", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [455, 5], "def_end_pos": [455, 9]}]], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\n\u22a2 G'.IsEquivalence", "state_after": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\n\u22a2 G'.IsEquivalence"}, {"tactic": "let E\u2082 := Localization.uniq L\u2082 L\u2082' W\u2082", "annotated_tactic": ["let E\u2082 := <a>Localization.uniq</a> L\u2082 L\u2082' W\u2082", [{"full_name": "CategoryTheory.Localization.uniq", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [455, 5], "def_end_pos": [455, 9]}]], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\n\u22a2 G'.IsEquivalence", "state_after": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\n\u22a2 G'.IsEquivalence"}, {"tactic": "let e : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n  calc\n    L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 \u03a6.functor \u22d9 L\u2082 \u22d9 E\u2082.functor :=\n        (Functor.associator _ _ _).symm \u226a\u226b\n          isoWhiskerRight (CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G).symm E\u2082.functor \u226a\u226b\n          Functor.associator _ _ _\n    _ \u2245 \u03a6.functor \u22d9 L\u2082' := isoWhiskerLeft \u03a6.functor (compUniqFunctor L\u2082 L\u2082' W\u2082)\n    _ \u2245 L\u2081' \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081' L\u2082' G'\n    _ \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n          isoWhiskerRight (compUniqFunctor L\u2081 L\u2081' W\u2081).symm G' \u226a\u226b Functor.associator _ _ _", "annotated_tactic": ["let e : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n    calc\n      L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 \u03a6.functor \u22d9 L\u2082 \u22d9 E\u2082.functor :=\n          (<a>Functor.associator</a> _ _ _).<a>symm</a> \u226a\u226b\n            <a>isoWhiskerRight</a> (<a>CatCommSq.iso</a> \u03a6.functor L\u2081 L\u2082 G).<a>symm</a> E\u2082.functor \u226a\u226b\n            <a>Functor.associator</a> _ _ _\n      _ \u2245 \u03a6.functor \u22d9 L\u2082' := <a>isoWhiskerLeft</a> \u03a6.functor (<a>compUniqFunctor</a> L\u2082 L\u2082' W\u2082)\n      _ \u2245 L\u2081' \u22d9 G' := <a>CatCommSq.iso</a> \u03a6.functor L\u2081' L\u2082' G'\n      _ \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n            <a>isoWhiskerRight</a> (<a>compUniqFunctor</a> L\u2081 L\u2081' W\u2081).<a>symm</a> G' \u226a\u226b <a>Functor.associator</a> _ _ _", [{"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}, {"full_name": "CategoryTheory.Iso.symm", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "CategoryTheory.isoWhiskerRight", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [191, 5], "def_end_pos": [191, 20]}, {"full_name": "CategoryTheory.CatCommSq.iso", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}, {"full_name": "CategoryTheory.Iso.symm", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}, {"full_name": "CategoryTheory.isoWhiskerLeft", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [172, 5], "def_end_pos": [172, 19]}, {"full_name": "CategoryTheory.Localization.compUniqFunctor", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [462, 5], "def_end_pos": [462, 20]}, {"full_name": "CategoryTheory.CatCommSq.iso", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}, {"full_name": "CategoryTheory.isoWhiskerRight", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [191, 5], "def_end_pos": [191, 20]}, {"full_name": "CategoryTheory.Localization.compUniqFunctor", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [462, 5], "def_end_pos": [462, 20]}, {"full_name": "CategoryTheory.Iso.symm", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}]], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\n\u22a2 G'.IsEquivalence", "state_after": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\ne : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n  Trans.trans\n    (Trans.trans\n      (Trans.trans\n        ((L\u2081.associator G E\u2082.functor).symm \u226a\u226b\n          isoWhiskerRight (CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G).symm E\u2082.functor \u226a\u226b \u03a6.functor.associator L\u2082 E\u2082.functor)\n        (isoWhiskerLeft \u03a6.functor (compUniqFunctor L\u2082 L\u2082' W\u2082)))\n      (CatCommSq.iso \u03a6.functor L\u2081' L\u2082' G'))\n    (isoWhiskerRight (compUniqFunctor L\u2081 L\u2081' W\u2081).symm G' \u226a\u226b L\u2081.associator (uniq L\u2081 L\u2081' W\u2081).functor G')\n\u22a2 G'.IsEquivalence"}, {"tactic": "have := Functor.isEquivalence_of_iso\n  (liftNatIso L\u2081 W\u2081 _ _ (G \u22d9 E\u2082.functor) (E\u2081.functor \u22d9 G') e)", "annotated_tactic": ["have := <a>Functor.isEquivalence_of_iso</a>\n    (<a>liftNatIso</a> L\u2081 W\u2081 _ _ (G \u22d9 E\u2082.functor) (E\u2081.functor \u22d9 G') e)", [{"full_name": "CategoryTheory.Functor.isEquivalence_of_iso", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [629, 7], "def_end_pos": [629, 27]}, {"full_name": "CategoryTheory.Localization.liftNatIso", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [368, 5], "def_end_pos": [368, 15]}]], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\ne : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n  Trans.trans\n    (Trans.trans\n      (Trans.trans\n        ((L\u2081.associator G E\u2082.functor).symm \u226a\u226b\n          isoWhiskerRight (CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G).symm E\u2082.functor \u226a\u226b \u03a6.functor.associator L\u2082 E\u2082.functor)\n        (isoWhiskerLeft \u03a6.functor (compUniqFunctor L\u2082 L\u2082' W\u2082)))\n      (CatCommSq.iso \u03a6.functor L\u2081' L\u2082' G'))\n    (isoWhiskerRight (compUniqFunctor L\u2081 L\u2081' W\u2081).symm G' \u226a\u226b L\u2081.associator (uniq L\u2081 L\u2081' W\u2081).functor G')\n\u22a2 G'.IsEquivalence", "state_after": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\ne : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n  Trans.trans\n    (Trans.trans\n      (Trans.trans\n        ((L\u2081.associator G E\u2082.functor).symm \u226a\u226b\n          isoWhiskerRight (CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G).symm E\u2082.functor \u226a\u226b \u03a6.functor.associator L\u2082 E\u2082.functor)\n        (isoWhiskerLeft \u03a6.functor (compUniqFunctor L\u2082 L\u2082' W\u2082)))\n      (CatCommSq.iso \u03a6.functor L\u2081' L\u2082' G'))\n    (isoWhiskerRight (compUniqFunctor L\u2081 L\u2081' W\u2081).symm G' \u226a\u226b L\u2081.associator (uniq L\u2081 L\u2081' W\u2081).functor G')\nthis : (E\u2081.functor \u22d9 G').IsEquivalence\n\u22a2 G'.IsEquivalence"}, {"tactic": "exact Functor.isEquivalence_of_comp_left E\u2081.functor G'", "annotated_tactic": ["exact <a>Functor.isEquivalence_of_comp_left</a> E\u2081.functor G'", [{"full_name": "CategoryTheory.Functor.isEquivalence_of_comp_left", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [644, 7], "def_end_pos": [644, 33]}]], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u00b9\u2074 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b9\u00b3 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u00b9\u00b9 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u00b9\u2070 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u2079 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u2078 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u2077 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\ninst\u271d\u2076 : CatCommSq \u03a6.functor L\u2081 L\u2082 G\nD\u2081' : Type u\u2084'\nD\u2082' : Type u\u2085'\ninst\u271d\u2075 : Category.{v\u2084', u\u2084'} D\u2081'\ninst\u271d\u2074 : Category.{v\u2085', u\u2085'} D\u2082'\nL\u2081' : C\u2081 \u2964 D\u2081'\nL\u2082' : C\u2082 \u2964 D\u2082'\ninst\u271d\u00b3 : L\u2081'.IsLocalization W\u2081\ninst\u271d\u00b2 : L\u2082'.IsLocalization W\u2082\nG' : D\u2081' \u2964 D\u2082'\ninst\u271d\u00b9 : CatCommSq \u03a6.functor L\u2081' L\u2082' G'\ninst\u271d : G.IsEquivalence\nE\u2081 : D\u2081 \u224c D\u2081' := uniq L\u2081 L\u2081' W\u2081\nE\u2082 : D\u2082 \u224c D\u2082' := uniq L\u2082 L\u2082' W\u2082\ne : L\u2081 \u22d9 G \u22d9 E\u2082.functor \u2245 L\u2081 \u22d9 E\u2081.functor \u22d9 G' :=\n  Trans.trans\n    (Trans.trans\n      (Trans.trans\n        ((L\u2081.associator G E\u2082.functor).symm \u226a\u226b\n          isoWhiskerRight (CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G).symm E\u2082.functor \u226a\u226b \u03a6.functor.associator L\u2082 E\u2082.functor)\n        (isoWhiskerLeft \u03a6.functor (compUniqFunctor L\u2082 L\u2082' W\u2082)))\n      (CatCommSq.iso \u03a6.functor L\u2081' L\u2082' G'))\n    (isoWhiskerRight (compUniqFunctor L\u2081 L\u2081' W\u2081).symm G' \u226a\u226b L\u2081.associator (uniq L\u2081 L\u2081' W\u2081).functor G')\nthis : (E\u2081.functor \u22d9 G').IsEquivalence\n\u22a2 G'.IsEquivalence", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_pos_iff_support", "start": [1000, 1], "end": [1002, 76], "traced_tactics": [{"tactic": "simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support]", "annotated_tactic": ["simp [<a>pos_iff_ne_zero</a>, hf, <a>Filter.EventuallyEq</a>, <a>ae_iff</a>, <a>Function.support</a>]", [{"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [230, 3], "def_end_pos": [230, 14]}, {"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1470, 5], "def_end_pos": [1470, 17]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [30, 3], "def_end_pos": [30, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 0 < \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2194 0 < \u03bc (support f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/VecNotation.lean", "full_name": "Matrix.cons_add", "start": [455, 1], "end": [458, 55], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03b1 : Type u\nm n o : \u2115\nm' : Type u_1\nn' : Type u_2\no' : Type u_3\ninst\u271d : Add \u03b1\nx : \u03b1\nv : Fin n \u2192 \u03b1\nw : Fin n.succ \u2192 \u03b1\n\u22a2 vecCons x v + w = vecCons (x + vecHead w) (v + vecTail w)", "state_after": "case h\n\u03b1 : Type u\nm n o : \u2115\nm' : Type u_1\nn' : Type u_2\no' : Type u_3\ninst\u271d : Add \u03b1\nx : \u03b1\nv : Fin n \u2192 \u03b1\nw : Fin n.succ \u2192 \u03b1\ni : Fin n.succ\n\u22a2 (vecCons x v + w) i = vecCons (x + vecHead w) (v + vecTail w) i"}, {"tactic": "refine Fin.cases ?_ ?_ i <;> simp [vecHead, vecTail]", "annotated_tactic": ["refine <a>Fin.cases</a> ?_ ?_ i <;> simp [<a>vecHead</a>, <a>vecTail</a>]", [{"full_name": "Fin.cases", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [654, 21], "def_end_pos": [654, 26]}, {"full_name": "Matrix.vecHead", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [93, 5], "def_end_pos": [93, 12]}, {"full_name": "Matrix.vecTail", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [98, 5], "def_end_pos": [98, 12]}]], "state_before": "case h\n\u03b1 : Type u\nm n o : \u2115\nm' : Type u_1\nn' : Type u_2\no' : Type u_3\ninst\u271d : Add \u03b1\nx : \u03b1\nv : Fin n \u2192 \u03b1\nw : Fin n.succ \u2192 \u03b1\ni : Fin n.succ\n\u22a2 (vecCons x v + w) i = vecCons (x + vecHead w) (v + vecTail w) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sublists.lean", "full_name": "List.sublistsLen_zero", "start": [265, 1], "end": [266, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.dropUntil_copy", "start": [1258, 1], "end": [1262, 6], "traced_tactics": [{"tactic": "subst_vars", "annotated_tactic": ["subst_vars", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v\u271d w\u271d x y : V\ninst\u271d : DecidableEq V\nu v w v' w' : V\np : G.Walk v w\nhv : v = v'\nhw : w = w'\nh : u \u2208 (p.copy hv hw).support\n\u22a2 u \u2208 p.support", "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v w x y : V\ninst\u271d : DecidableEq V\nu v' w' : V\np : G.Walk v' w'\nh : u \u2208 (p.copy \u22ef \u22ef).support\n\u22a2 u \u2208 p.support"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v w x y : V\ninst\u271d : DecidableEq V\nu v' w' : V\np : G.Walk v' w'\nh : u \u2208 (p.copy \u22ef \u22ef).support\n\u22a2 u \u2208 p.support", "state_after": "no goals"}, {"tactic": "subst_vars", "annotated_tactic": ["subst_vars", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v\u271d w\u271d x y : V\ninst\u271d : DecidableEq V\nu v w v' w' : V\np : G.Walk v w\nhv : v = v'\nhw : w = w'\nh : u \u2208 (p.copy hv hw).support\n\u22a2 (p.copy hv hw).dropUntil u h = (p.dropUntil u \u22ef).copy \u22ef hw", "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v w x y : V\ninst\u271d : DecidableEq V\nu v' w' : V\np : G.Walk v' w'\nh : u \u2208 (p.copy \u22ef \u22ef).support\n\u22a2 (p.copy \u22ef \u22ef).dropUntil u h = (p.dropUntil u \u22ef).copy \u22ef \u22ef"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu\u271d v w x y : V\ninst\u271d : DecidableEq V\nu v' w' : V\np : G.Walk v' w'\nh : u \u2208 (p.copy \u22ef \u22ef).support\n\u22a2 (p.copy \u22ef \u22ef).dropUntil u h = (p.dropUntil u \u22ef).copy \u22ef \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.eval_prec_succ", "start": [514, 1], "end": [518, 7], "traced_tactics": [{"tactic": "rw [eval, Nat.unpaired, Part.bind_eq_bind, Nat.unpair_pair]", "annotated_tactic": ["rw [<a>eval</a>, <a>Nat.unpaired</a>, <a>Part.bind_eq_bind</a>, <a>Nat.unpair_pair</a>]", [{"full_name": "Nat.Partrec.Code.eval", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [488, 5], "def_end_pos": [488, 9]}, {"full_name": "Nat.unpaired", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [72, 5], "def_end_pos": [72, 13]}, {"full_name": "Part.bind_eq_bind", "def_path": "Mathlib/Data/Part.lean", "def_pos": [613, 9], "def_end_pos": [613, 21]}, {"full_name": "Nat.unpair_pair", "def_path": "Mathlib/Data/Nat/Pairing.lean", "def_pos": [62, 9], "def_end_pos": [62, 20]}]], "state_before": "cf cg : Code\na k : \u2115\n\u22a2 (cf.prec cg).eval (Nat.pair a k.succ) = do\n    let ih \u2190 (cf.prec cg).eval (Nat.pair a k)\n    cg.eval (Nat.pair a (Nat.pair k ih))", "state_after": "cf cg : Code\na k : \u2115\n\u22a2 Nat.rec (cf.eval (a, k.succ).1)\n      (fun y IH => do\n        let i \u2190 IH\n        cg.eval (Nat.pair (a, k.succ).1 (Nat.pair y i)))\n      (a, k.succ).2 =\n    (unpaired\n          (fun a n =>\n            Nat.rec (cf.eval a)\n              (fun y IH => do\n                let i \u2190 IH\n                cg.eval (Nat.pair a (Nat.pair y i)))\n              n)\n          (Nat.pair a k)).bind\n      fun ih => cg.eval (Nat.pair a (Nat.pair k ih))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "cf cg : Code\na k : \u2115\n\u22a2 Nat.rec (cf.eval (a, k.succ).1)\n      (fun y IH => do\n        let i \u2190 IH\n        cg.eval (Nat.pair (a, k.succ).1 (Nat.pair y i)))\n      (a, k.succ).2 =\n    (unpaired\n          (fun a n =>\n            Nat.rec (cf.eval a)\n              (fun y IH => do\n                let i \u2190 IH\n                cg.eval (Nat.pair a (Nat.pair y i)))\n              n)\n          (Nat.pair a k)).bind\n      fun ih => cg.eval (Nat.pair a (Nat.pair k ih))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.sub_apply", "start": [225, 1], "end": [225, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpOrder.lean", "full_name": "MeasureTheory.Lp.coeFn_sup", "start": [92, 1], "end": [93, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Fubini.lean", "full_name": "CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_hom_\u03c0_\u03c0", "start": [526, 1], "end": [538, 7], "traced_tactics": [{"tactic": "dsimp [limitCurrySwapCompLimIsoLimitCurryCompLim]", "annotated_tactic": ["dsimp [<a>limitCurrySwapCompLimIsoLimitCurryCompLim</a>]", [{"full_name": "CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim", "def_path": "Mathlib/CategoryTheory/Limits/Fubini.lean", "def_pos": [515, 19], "def_end_pos": [515, 60]}]], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitCurrySwapCompLimIsoLimitCurryCompLim G).hom \u226b limit.\u03c0 (curry.obj G \u22d9 lim) j \u226b limit.\u03c0 ((curry.obj G).obj j) k =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (((limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n          (HasLimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor \u22d9 G))).hom) \u226b\n        (limitIsoLimitCurryCompLim G).hom) \u226b\n      limit.\u03c0 (curry.obj G \u22d9 lim) j \u226b limit.\u03c0 ((curry.obj G).obj j) k =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j"}, {"tactic": "simp only [Iso.refl_hom, Prod.braiding_counitIso_hom_app, Limits.HasLimit.isoOfEquivalence_hom_\u03c0,\n  Iso.refl_inv, limitIsoLimitCurryCompLim_hom_\u03c0_\u03c0, eqToIso_refl, Category.assoc]", "annotated_tactic": ["simp only [<a>Iso.refl_hom</a>, <a>Prod.braiding_counitIso_hom_app</a>, <a>Limits.HasLimit.isoOfEquivalence_hom_\u03c0</a>,\n    <a>Iso.refl_inv</a>, <a>limitIsoLimitCurryCompLim_hom_\u03c0_\u03c0</a>, <a>eqToIso_refl</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Iso.refl_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [130, 9], "def_end_pos": [130, 14]}, {"full_name": "CategoryTheory.Prod.braiding_counitIso_hom_app", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [165, 3], "def_end_pos": [165, 9]}, {"full_name": "CategoryTheory.Limits.HasLimit.isoOfEquivalence_hom_\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [374, 9], "def_end_pos": [374, 40]}, {"full_name": "CategoryTheory.Iso.refl_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [130, 9], "def_end_pos": [130, 14]}, {"full_name": "CategoryTheory.Limits.limitIsoLimitCurryCompLim_hom_\u03c0_\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Fubini.lean", "def_pos": [454, 9], "def_end_pos": [454, 42]}, {"full_name": "CategoryTheory.eqToIso_refl", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [164, 9], "def_end_pos": [164, 21]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (((limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n          (HasLimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor \u22d9 G))).hom) \u226b\n        (limitIsoLimitCurryCompLim G).hom) \u226b\n      limit.\u03c0 (curry.obj G \u22d9 lim) j \u226b limit.\u03c0 ((curry.obj G).obj j) k =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) ((Prod.braiding K J).inverse.obj (j, k)) \u226b\n        (\ud835\udfd9 ((Prod.braiding K J).functor \u22d9 G)).app ((Prod.braiding K J).inverse.obj (j, k)) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j"}, {"tactic": "erw [NatTrans.id_app]", "annotated_tactic": ["erw [<a>NatTrans.id_app</a>]", [{"full_name": "CategoryTheory.NatTrans.id_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [72, 9], "def_end_pos": [72, 15]}]], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) ((Prod.braiding K J).inverse.obj (j, k)) \u226b\n        (\ud835\udfd9 ((Prod.braiding K J).functor \u22d9 G)).app ((Prod.braiding K J).inverse.obj (j, k)) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) ((Prod.braiding K J).inverse.obj (j, k)) \u226b\n        \ud835\udfd9 ((Prod.swap K J \u22d9 G).obj ((Prod.braiding K J).inverse.obj (j, k))) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) ((Prod.braiding K J).inverse.obj (j, k)) \u226b\n        \ud835\udfd9 ((Prod.swap K J \u22d9 G).obj ((Prod.braiding K J).inverse.obj (j, k))) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) (k, j) \u226b \ud835\udfd9 (G.obj (j, k)) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j"}, {"tactic": "rw [CategoryTheory.Bifunctor.map_id]", "annotated_tactic": ["rw [<a>CategoryTheory.Bifunctor.map_id</a>]", [{"full_name": "CategoryTheory.Bifunctor.map_id", "def_path": "Mathlib/CategoryTheory/Products/Bifunctor.lean", "def_pos": [25, 9], "def_end_pos": [25, 15]}]], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) (k, j) \u226b \ud835\udfd9 (G.obj (j, k)) \u226b G.map (\ud835\udfd9 j, \ud835\udfd9 k) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) (k, j) \u226b \ud835\udfd9 (G.obj (j, k)) \u226b \ud835\udfd9 (G.obj (j, k)) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J K : Type v\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : SmallCategory K\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 K \u2964 C\nG : J \u00d7 K \u2964 C\ninst\u271d : HasLimits C\nj : J\nk : K\n\u22a2 (limitIsoLimitCurryCompLim (Prod.swap K J \u22d9 G)).inv \u226b\n      limit.\u03c0 (Prod.swap K J \u22d9 G) (k, j) \u226b \ud835\udfd9 (G.obj (j, k)) \u226b \ud835\udfd9 (G.obj (j, k)) =\n    limit.\u03c0 (curry.obj (Prod.swap K J \u22d9 G) \u22d9 lim) k \u226b limit.\u03c0 ((curry.obj (Prod.swap K J \u22d9 G)).obj k) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "full_name": "Bool.coe_false", "start": [94, 1], "end": [94, 46], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u22a2 (false = true) = False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize.reduced'", "start": [31, 1], "end": [34, 35], "traced_tactics": [{"tactic": "rw [\u2190 Int.div_eq_ediv_of_dvd (e \u25b8 Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]", "annotated_tactic": ["rw [\u2190 <a>Int.div_eq_ediv_of_dvd</a> (e \u25b8 <a>Int.ofNat_dvd_left</a>.2 (<a>Nat.gcd_dvd_left</a> ..))]", [{"full_name": "Int.div_eq_ediv_of_dvd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 27]}, {"full_name": "Int.ofNat_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [86, 9], "def_end_pos": [86, 21]}]], "state_before": "num : Int\nden g : Nat\nden_nz : den \u2260 0\ne : g = num.natAbs.gcd den\n\u22a2 (num / \u2191g).natAbs.Coprime (den / g)", "state_after": "num : Int\nden g : Nat\nden_nz : den \u2260 0\ne : g = num.natAbs.gcd den\n\u22a2 (num.div \u2191g).natAbs.Coprime (den / g)"}, {"tactic": "exact normalize.reduced den_nz e", "annotated_tactic": ["exact <a>normalize.reduced</a> den_nz e", [{"full_name": "Rat.normalize.reduced", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [60, 9], "def_end_pos": [60, 30]}]], "state_before": "num : Int\nden g : Nat\nden_nz : den \u2260 0\ne : g = num.natAbs.gcd den\n\u22a2 (num.div \u2191g).natAbs.Coprime (den / g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrable_const", "start": [285, 1], "end": [287, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/String/Basic.lean", "full_name": "List.asString_inj", "start": [197, 1], "end": [199, 20], "traced_tactics": [{"tactic": "rw [\u2190 toList_asString l, \u2190 toList_asString l', toList_inj, h]", "annotated_tactic": ["rw [\u2190 <a>toList_asString</a> l, \u2190 <a>toList_asString</a> l', <a>toList_inj</a>, h]", [{"full_name": "List.toList_asString", "def_path": "Mathlib/Data/String/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 24]}, {"full_name": "List.toList_asString", "def_path": "Mathlib/Data/String/Basic.lean", "def_pos": [185, 9], "def_end_pos": [185, 24]}, {"full_name": "String.toList_inj", "def_path": "Mathlib/Data/String/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 19]}]], "state_before": "l l' : List Char\nh : l.asString = l'.asString\n\u22a2 l = l'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Cast/Order.lean", "full_name": "Nat.not_ofNat_le_one", "start": [203, 1], "end": [204, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean", "full_name": "SimpleGraph.IsSRGWith.compl_is_regular", "start": [137, 1], "end": [140, 24], "traced_tactics": [{"tactic": "rw [\u2190 h.card, Nat.sub_sub, add_comm, \u2190 Nat.sub_sub]", "annotated_tactic": ["rw [\u2190 h.card, <a>Nat.sub_sub</a>, <a>add_comm</a>, \u2190 <a>Nat.sub_sub</a>]", [{"full_name": "Nat.sub_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 26]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Nat.sub_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 26]}]], "state_before": "V : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : G.IsSRGWith n k \u2113 \u03bc\n\u22a2 G\u1d9c.IsRegularOfDegree (n - k - 1)", "state_after": "V : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : G.IsSRGWith n k \u2113 \u03bc\n\u22a2 G\u1d9c.IsRegularOfDegree (Fintype.card V - 1 - k)"}, {"tactic": "exact h.regular.compl", "annotated_tactic": ["exact h.regular.compl", []], "state_before": "V : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : G.IsSRGWith n k \u2113 \u03bc\n\u22a2 G\u1d9c.IsRegularOfDegree (Fintype.card V - 1 - k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/LocallyUniformLimit.lean", "full_name": "Complex.differentiableOn_tsum_of_summable_norm", "start": [178, 1], "end": [185, 45], "traced_tactics": [{"tactic": "classical\nhave hc := (tendstoUniformlyOn_tsum hu hF_le).tendstoLocallyUniformlyOn\nrefine hc.differentiableOn (eventually_of_forall fun s => ?_) hU\nexact DifferentiableOn.sum fun i _ => hf i", "annotated_tactic": ["classical\n  have hc := (<a>tendstoUniformlyOn_tsum</a> hu hF_le).<a>tendstoLocallyUniformlyOn</a>\n  refine hc.differentiableOn (<a>eventually_of_forall</a> fun s => ?_) hU\n  exact <a>DifferentiableOn.sum</a> fun i _ => hf i", [{"full_name": "tendstoUniformlyOn_tsum", "def_path": "Mathlib/Analysis/NormedSpace/FunctionSeries.lean", "def_pos": [28, 9], "def_end_pos": [28, 32]}, {"full_name": "TendstoUniformlyOn.tendstoLocallyUniformlyOn", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "def_pos": [640, 19], "def_end_pos": [640, 63]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 29]}, {"full_name": "DifferentiableOn.sum", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "def_pos": [386, 9], "def_end_pos": [386, 29]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\n\u22a2 DifferentiableOn \u2102 (fun w => \u2211' (i : \u03b9), F i w) U", "state_after": "no goals"}, {"tactic": "have hc := (tendstoUniformlyOn_tsum hu hF_le).tendstoLocallyUniformlyOn", "annotated_tactic": ["have hc := (<a>tendstoUniformlyOn_tsum</a> hu hF_le).<a>tendstoLocallyUniformlyOn</a>", [{"full_name": "tendstoUniformlyOn_tsum", "def_path": "Mathlib/Analysis/NormedSpace/FunctionSeries.lean", "def_pos": [28, 9], "def_end_pos": [28, 32]}, {"full_name": "TendstoUniformlyOn.tendstoLocallyUniformlyOn", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "def_pos": [640, 19], "def_end_pos": [640, 63]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\n\u22a2 DifferentiableOn \u2102 (fun w => \u2211' (i : \u03b9), F i w) U", "state_after": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\nhc : TendstoLocallyUniformlyOn (fun t x => \u2211 n \u2208 t, F n x) (fun x => \u2211' (n : \u03b9), F n x) atTop U\n\u22a2 DifferentiableOn \u2102 (fun w => \u2211' (i : \u03b9), F i w) U"}, {"tactic": "refine hc.differentiableOn (eventually_of_forall fun s => ?_) hU", "annotated_tactic": ["refine hc.differentiableOn (<a>eventually_of_forall</a> fun s => ?_) hU", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 29]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\nhc : TendstoLocallyUniformlyOn (fun t x => \u2211 n \u2208 t, F n x) (fun x => \u2211' (n : \u03b9), F n x) atTop U\n\u22a2 DifferentiableOn \u2102 (fun w => \u2211' (i : \u03b9), F i w) U", "state_after": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\nhc : TendstoLocallyUniformlyOn (fun t x => \u2211 n \u2208 t, F n x) (fun x => \u2211' (n : \u03b9), F n x) atTop U\ns : Finset \u03b9\n\u22a2 DifferentiableOn \u2102 (fun x => \u2211 n \u2208 s, F n x) U"}, {"tactic": "exact DifferentiableOn.sum fun i _ => hf i", "annotated_tactic": ["exact <a>DifferentiableOn.sum</a> fun i _ => hf i", [{"full_name": "DifferentiableOn.sum", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "def_pos": [386, 9], "def_end_pos": [386, 29]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nu : \u03b9 \u2192 \u211d\nhu : Summable u\nhf : \u2200 (i : \u03b9), DifferentiableOn \u2102 (F i) U\nhU : IsOpen U\nhF_le : \u2200 (i : \u03b9), \u2200 w \u2208 U, \u2016F i w\u2016 \u2264 u i\nhc : TendstoLocallyUniformlyOn (fun t x => \u2211 n \u2208 t, F n x) (fun x => \u2211' (n : \u03b9), F n x) atTop U\ns : Finset \u03b9\n\u22a2 DifferentiableOn \u2102 (fun x => \u2211 n \u2208 s, F n x) U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "full_name": "Matrix.Represents.add", "start": [131, 1], "end": [133, 66], "traced_tactics": [{"tactic": "delta Matrix.Represents at h h' \u22a2", "annotated_tactic": ["delta <a>Matrix.Represents</a> at h h' \u22a2", [{"full_name": "Matrix.Represents", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "def_pos": [84, 5], "def_end_pos": [84, 22]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : Represents b A f\nh' : Represents b A' f'\n\u22a2 Represents b (A + A') (f + f')", "state_after": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : (PiToModule.fromMatrix R b) A = (PiToModule.fromEnd R b) f\nh' : (PiToModule.fromMatrix R b) A' = (PiToModule.fromEnd R b) f'\n\u22a2 (PiToModule.fromMatrix R b) (A + A') = (PiToModule.fromEnd R b) (f + f')"}, {"tactic": "rw [map_add, map_add, h, h']", "annotated_tactic": ["rw [<a>map_add</a>, <a>map_add</a>, h, h']", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : (PiToModule.fromMatrix R b) A = (PiToModule.fromEnd R b) f\nh' : (PiToModule.fromMatrix R b) A' = (PiToModule.fromEnd R b) f'\n\u22a2 (PiToModule.fromMatrix R b) (A + A') = (PiToModule.fromEnd R b) (f + f')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "WithTop.succ_coe_top", "start": [1094, 1], "end": [1095, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Group/DenselyOrdered.lean", "full_name": "le_of_forall_one_lt_div_le", "start": [32, 1], "end": [34, 80], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv, inv_inv] using h \u03b5\u207b\u00b9 (Left.one_lt_inv_iff.2 \u03b51)", "annotated_tactic": ["simpa only [<a>div_eq_mul_inv</a>, <a>inv_inv</a>] using h \u03b5\u207b\u00b9 (<a>Left.one_lt_inv_iff</a>.2 \u03b51)", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [870, 9], "def_end_pos": [870, 16]}, {"full_name": "Left.one_lt_inv_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [158, 9], "def_end_pos": [158, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Group \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : DenselyOrdered \u03b1\na b c : \u03b1\nh : \u2200 (\u03b5 : \u03b1), 1 < \u03b5 \u2192 a / \u03b5 \u2264 b\n\u03b5 : \u03b1\n\u03b51 : \u03b5 < 1\n\u22a2 a * \u03b5 \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Directed.exists_mem_subset_of_finset_subset_biUnion", "start": [1771, 1], "end": [1780, 36], "traced_tactics": [{"tactic": "induction s using Finset.cons_induction with\n| empty => simp\n| cons b t hbt iht =>\n  simp only [Finset.coe_cons, Set.insert_subset_iff, Set.mem_iUnion] at hs \u22a2\n  rcases hs.imp_right iht with \u27e8\u27e8i, hi\u27e9, j, hj\u27e9\n  rcases h i j with \u27e8k, hik, hjk\u27e9\n  exact \u27e8k, hik hi, hj.trans hjk\u27e9", "annotated_tactic": ["induction s using <a>Finset.cons_induction</a> with\n  | <a>empty</a> => simp\n  | <a>cons</a> b t hbt iht =>\n    simp only [<a>Finset.coe_cons</a>, <a>Set.insert_subset_iff</a>, <a>Set.mem_iUnion</a>] at hs \u22a2\n    rcases hs.imp_right iht with \u27e8\u27e8i, hi\u27e9, j, hj\u27e9\n    rcases h i j with \u27e8k, hik, hjk\u27e9\n    exact \u27e8k, hik hi, hj.trans hjk\u27e9", [{"full_name": "Finset.cons_induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1250, 9], "def_end_pos": [1250, 23]}, {"full_name": "Finset.empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [533, 15], "def_end_pos": [533, 20]}, {"full_name": "Finset.cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [868, 5], "def_end_pos": [868, 9]}, {"full_name": "Finset.coe_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 17]}, {"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 26]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns\u271d : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 i, f i\n\u22a2 \u2203 i, \u2191s \u2286 f i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nhs : \u2191\u2205 \u2286 \u22c3 i, f i\n\u22a2 \u2203 i, \u2191\u2205 \u2286 f i", "state_after": "no goals"}, {"tactic": "simp only [Finset.coe_cons, Set.insert_subset_iff, Set.mem_iUnion] at hs \u22a2", "annotated_tactic": ["simp only [<a>Finset.coe_cons</a>, <a>Set.insert_subset_iff</a>, <a>Set.mem_iUnion</a>] at hs \u22a2", [{"full_name": "Finset.coe_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [922, 9], "def_end_pos": [922, 17]}, {"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 26]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}]], "state_before": "case cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : \u2191(Finset.cons b t hbt) \u2286 \u22c3 i, f i\n\u22a2 \u2203 i, \u2191(Finset.cons b t hbt) \u2286 f i", "state_after": "case cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i"}, {"tactic": "rcases hs.imp_right iht with \u27e8\u27e8i, hi\u27e9, j, hj\u27e9", "annotated_tactic": ["rcases hs.imp_right iht with \u27e8\u27e8i, hi\u27e9, j, hj\u27e9", []], "state_before": "case cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i", "state_after": "case cons.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\ni : \u03b9\nhi : b \u2208 f i\nj : \u03b9\nhj : \u2191t \u2286 f j\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i"}, {"tactic": "rcases h i j with \u27e8k, hik, hjk\u27e9", "annotated_tactic": ["rcases h i j with \u27e8k, hik, hjk\u27e9", []], "state_before": "case cons.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\ni : \u03b9\nhi : b \u2208 f i\nj : \u03b9\nhj : \u2191t \u2286 f j\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i", "state_after": "case cons.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\ni : \u03b9\nhi : b \u2208 f i\nj : \u03b9\nhj : \u2191t \u2286 f j\nk : \u03b9\nhik : f i \u2286 f k\nhjk : f j \u2286 f k\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i"}, {"tactic": "exact \u27e8k, hik hi, hj.trans hjk\u27e9", "annotated_tactic": ["exact \u27e8k, hik hi, hj.trans hjk\u27e9", []], "state_before": "case cons.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9\u271d : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nb : \u03b1\nt : Finset \u03b1\nhbt : b \u2209 t\niht : \u2191t \u2286 \u22c3 i, f i \u2192 \u2203 i, \u2191t \u2286 f i\nhs : (\u2203 i, b \u2208 f i) \u2227 \u2191t \u2286 \u22c3 i, f i\ni : \u03b9\nhi : b \u2208 f i\nj : \u03b9\nhj : \u2191t \u2286 f j\nk : \u03b9\nhik : f i \u2286 f k\nhjk : f j \u2286 f k\n\u22a2 \u2203 i, b \u2208 f i \u2227 \u2191t \u2286 f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.Infinite.ne_real", "start": [624, 1], "end": [625, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/MinMax.lean", "full_name": "max_eq_left_iff", "start": [153, 1], "end": [154, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean", "full_name": "contMDiffAt_extChartAt'", "start": [92, 1], "end": [94, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean", "full_name": "CategoryTheory.Limits.colimitLimitToLimitColimit_injective", "start": [72, 1], "end": [142, 32], "traced_tactics": [{"tactic": "classical\n  cases nonempty_fintype J\n  intro x y h\n  obtain \u27e8kx, x, rfl\u27e9 := jointly_surjective' x\n  obtain \u27e8ky, y, rfl\u27e9 := jointly_surjective' y\n  dsimp at x y\n  replace h := fun j => congr_arg (limit.\u03c0 (curry.obj F \u22d9 colim) j) h\n  simp? [colimit_eq_iff] at h says\n    simp only [Functor.comp_obj, colim_obj, \u03b9_colimitLimitToLimitColimit_\u03c0_apply,\n      colimit_eq_iff, curry_obj_obj_obj, curry_obj_obj_map] at h\n  let k j := (h j).choose\n  let f : \u2200 j, kx \u27f6 k j := fun j => (h j).choose_spec.choose\n  let g : \u2200 j, ky \u27f6 k j := fun j => (h j).choose_spec.choose_spec.choose\n  have w :\n    \u2200 j, F.map ((\ud835\udfd9 j, f j) :\n      (j, kx) \u27f6 (j, k j)) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map ((\ud835\udfd9 j, g j) : (j, ky) \u27f6 (j, k j))\n          (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y) :=\n    fun j => (h j).choose_spec.choose_spec.choose_spec\n  let O : Finset K := Finset.univ.image k \u222a {kx, ky}\n  have kxO : kx \u2208 O := Finset.mem_union.mpr (Or.inr (by simp))\n  have kyO : ky \u2208 O := Finset.mem_union.mpr (Or.inr (by simp))\n  have kjO : \u2200 j, k j \u2208 O := fun j => Finset.mem_union.mpr (Or.inl (by simp))\n  let H : Finset (\u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) :=\n    (Finset.univ.image fun j : J =>\n        \u27e8kx, k j, kxO, Finset.mem_union.mpr (Or.inl (by simp)), f j\u27e9) \u222a\n      Finset.univ.image fun j : J => \u27e8ky, k j, kyO, Finset.mem_union.mpr (Or.inl (by simp)), g j\u27e9\n  obtain \u27e8S, T, W\u27e9 := IsFiltered.sup_exists O H\n  have fH : \u2200 j, (\u27e8kx, k j, kxO, kjO j, f j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n    fun j =>\n    Finset.mem_union.mpr\n      (Or.inl\n        (by\n          simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n            Finset.mem_image, heq_iff_eq]\n          refine \u27e8j, ?_\u27e9\n          simp only [heq_iff_eq] ))\n  have gH :\n    \u2200 j, (\u27e8ky, k j, kyO, kjO j, g j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n    fun j =>\n    Finset.mem_union.mpr\n      (Or.inr\n        (by\n          simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n            Finset.mem_image, heq_iff_eq]\n          refine \u27e8j, ?_\u27e9\n          simp only [heq_iff_eq]))\n  apply colimit_sound' (T kxO) (T kyO)\n  ext j\n  simp only [Functor.comp_map, Limit.map_\u03c0_apply, curry_obj_map_app, swap_map]\n  rw [\u2190 W _ _ (fH j), \u2190 W _ _ (gH j)]\n  simp [Limit.map_\u03c0_apply, w]", "annotated_tactic": ["classical\n    cases <a>nonempty_fintype</a> J\n    -- Suppose we have two terms `x y` in the colimit (over `K`) of the limits (over `J`),\n    -- and that these have the same image under `colimitLimitToLimitColimit F`.\n    intro x y h\n    -- These elements of the colimit have representatives somewhere:\n    obtain \u27e8kx, x, rfl\u27e9 := <a>jointly_surjective'</a> x\n    obtain \u27e8ky, y, rfl\u27e9 := <a>jointly_surjective'</a> y\n    dsimp at x y\n    -- Since the images of `x` and `y` are equal in a limit, they are equal componentwise\n    -- (indexed by `j : J`),\n    replace h := fun j => <a>congr_arg</a> (<a>limit.\u03c0</a> (curry.obj F \u22d9 <a>colim</a>) j) h\n    -- and they are equations in a filtered colimit,\n    -- so for each `j` we have some place `k j` to the right of both `kx` and `ky`\n    simp? [colimit_eq_iff] at h says\n      simp only [<a>Functor.comp_obj</a>, <a>colim_obj</a>, <a>\u03b9_colimitLimitToLimitColimit_\u03c0_apply</a>,\n        <a>colimit_eq_iff</a>, <a>curry_obj_obj_obj</a>, <a>curry_obj_obj_map</a>] at h\n    let k j := (h j).<a>choose</a>\n    let f : \u2200 j, kx \u27f6 k j := fun j => (h j).choose_spec.choose\n    let g : \u2200 j, ky \u27f6 k j := fun j => (h j).choose_spec.choose_spec.choose\n    -- where the images of the components of the representatives become equal:\n    have w :\n      \u2200 j, F.map ((\ud835\udfd9 j, f j) :\n        (j, kx) \u27f6 (j, k j)) (<a>limit.\u03c0</a> ((curry.obj (<a>swap</a> K J \u22d9 F)).<a>obj</a> kx) j x) =\n          F.map ((\ud835\udfd9 j, g j) : (j, ky) \u27f6 (j, k j))\n            (<a>limit.\u03c0</a> ((curry.obj (<a>swap</a> K J \u22d9 F)).<a>obj</a> ky) j y) :=\n      fun j => (h j).choose_spec.choose_spec.choose_spec\n    -- We now use that `K` is filtered, picking some point to the right of all these\n    -- morphisms `f j` and `g j`.\n    let O : <a>Finset</a> K := Finset.univ.image k \u222a {kx, ky}\n    have kxO : kx \u2208 O := Finset.mem_union.mpr (<a>Or.inr</a> (by simp))\n    have kyO : ky \u2208 O := Finset.mem_union.mpr (<a>Or.inr</a> (by simp))\n    have kjO : \u2200 j, k j \u2208 O := fun j => Finset.mem_union.mpr (<a>Or.inl</a> (by simp))\n    let H : <a>Finset</a> (\u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) :=\n      (Finset.univ.image fun j : J =>\n          \u27e8kx, k j, kxO, Finset.mem_union.mpr (<a>Or.inl</a> (by simp)), f j\u27e9) \u222a\n        Finset.univ.image fun j : J => \u27e8ky, k j, kyO, Finset.mem_union.mpr (<a>Or.inl</a> (by simp)), g j\u27e9\n    obtain \u27e8S, T, W\u27e9 := <a>IsFiltered.sup_exists</a> O H\n    have fH : \u2200 j, (\u27e8kx, k j, kxO, kjO j, f j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (<a>Or.inl</a>\n          (by\n            simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]\n            refine \u27e8j, ?_\u27e9\n            simp only [<a>heq_iff_eq</a>] ))\n    have gH :\n      \u2200 j, (\u27e8ky, k j, kyO, kjO j, g j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (<a>Or.inr</a>\n          (by\n            simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]\n            refine \u27e8j, ?_\u27e9\n            simp only [<a>heq_iff_eq</a>]))\n    -- Our goal is now an equation between equivalence classes of representatives of a colimit,\n    -- and so it suffices to show those representative become equal somewhere, in particular at `S`.\n    apply <a>colimit_sound'</a> (T kxO) (T kyO)\n    -- We can check if two elements of a limit (in `Type`)\n    -- are equal by comparing them componentwise.\n    ext j\n    -- Now it's just a calculation using `W` and `w`.\n    simp only [<a>Functor.comp_map</a>, <a>Limit.map_\u03c0_apply</a>, <a>curry_obj_map_app</a>, <a>swap_map</a>]\n    rw [\u2190 W _ _ (fH j), \u2190 W _ _ (gH j)]\n    -- Porting note(#10745): had to add `Limit.map_\u03c0_apply`\n    -- (which was un-tagged simp since \"simp can prove it\")\n    simp [<a>Limit.map_\u03c0_apply</a>, w]", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}, {"full_name": "CategoryTheory.Limits.Types.jointly_surjective'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [616, 9], "def_end_pos": [616, 28]}, {"full_name": "CategoryTheory.Limits.Types.jointly_surjective'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [616, 9], "def_end_pos": [616, 28]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Limits.colim", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1115, 5], "def_end_pos": [1115, 10]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Limits.colim_obj", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1114, 3], "def_end_pos": [1114, 8]}, {"full_name": "CategoryTheory.Limits.\u03b9_colimitLimitToLimitColimit_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/ColimitLimit.lean", "def_pos": [97, 9], "def_end_pos": [97, 45]}, {"full_name": "CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff", "def_path": "Mathlib/CategoryTheory/Limits/TypesFiltered.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "CategoryTheory.curry_obj_obj_obj", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 10], "def_end_pos": [66, 21]}, {"full_name": "CategoryTheory.curry_obj_obj_map", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 22], "def_end_pos": [66, 33]}, {"full_name": "Exists.choose", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [174, 32], "def_end_pos": [174, 45]}, {"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Prod.swap", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [149, 5], "def_end_pos": [149, 9]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Prod.swap", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [149, 5], "def_end_pos": [149, 9]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "CategoryTheory.IsFiltered.sup_exists", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 19]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "CategoryTheory.Limits.Types.colimit_sound'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [585, 9], "def_end_pos": [585, 23]}, {"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.Limits.Types.Limit.map_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}, {"full_name": "CategoryTheory.curry_obj_map_app", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 34], "def_end_pos": [66, 45]}, {"full_name": "CategoryTheory.Prod.swap_map", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 8]}, {"full_name": "CategoryTheory.Limits.Types.Limit.map_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\n\u22a2 Function.Injective (colimitLimitToLimitColimit F)", "state_after": "no goals"}, {"tactic": "cases nonempty_fintype J", "annotated_tactic": ["cases <a>nonempty_fintype</a> J", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\n\u22a2 Function.Injective (colimitLimitToLimitColimit F)", "state_after": "case intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\n\u22a2 Function.Injective (colimitLimitToLimitColimit F)"}, {"tactic": "intro x y h", "annotated_tactic": ["intro x y h", []], "state_before": "case intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\n\u22a2 Function.Injective (colimitLimitToLimitColimit F)", "state_after": "case intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nx y : colimit (curry.obj (swap K J \u22d9 F) \u22d9 lim)\nh : colimitLimitToLimitColimit F x = colimitLimitToLimitColimit F y\n\u22a2 x = y"}, {"tactic": "obtain \u27e8kx, x, rfl\u27e9 := jointly_surjective' x", "annotated_tactic": ["obtain \u27e8kx, x, rfl\u27e9 := <a>jointly_surjective'</a> x", [{"full_name": "CategoryTheory.Limits.Types.jointly_surjective'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [616, 9], "def_end_pos": [616, 28]}]], "state_before": "case intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nx y : colimit (curry.obj (swap K J \u22d9 F) \u22d9 lim)\nh : colimitLimitToLimitColimit F x = colimitLimitToLimitColimit F y\n\u22a2 x = y", "state_after": "case intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\ny : colimit (curry.obj (swap K J \u22d9 F) \u22d9 lim)\nkx : K\nx : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj kx\nh : colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) = colimitLimitToLimitColimit F y\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = y"}, {"tactic": "obtain \u27e8ky, y, rfl\u27e9 := jointly_surjective' y", "annotated_tactic": ["obtain \u27e8ky, y, rfl\u27e9 := <a>jointly_surjective'</a> y", [{"full_name": "CategoryTheory.Limits.Types.jointly_surjective'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [616, 9], "def_end_pos": [616, 28]}]], "state_before": "case intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\ny : colimit (curry.obj (swap K J \u22d9 F) \u22d9 lim)\nkx : K\nx : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj kx\nh : colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) = colimitLimitToLimitColimit F y\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj kx\nky : K\ny : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj ky\nh :\n  colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) =\n    colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "dsimp at x y", "annotated_tactic": ["dsimp at x y", []], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj kx\nky : K\ny : (curry.obj (swap K J \u22d9 F) \u22d9 lim).obj ky\nh :\n  colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) =\n    colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) =\n    colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "replace h := fun j => congr_arg (limit.\u03c0 (curry.obj F \u22d9 colim) j) h", "annotated_tactic": ["replace h := fun j => <a>congr_arg</a> (<a>limit.\u03c0</a> (curry.obj F \u22d9 <a>colim</a>) j) h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Limits.colim", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1115, 5], "def_end_pos": [1115, 10]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x) =\n    colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x)) =\n      limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y))\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "simp? [colimit_eq_iff] at h says\n  simp only [Functor.comp_obj, colim_obj, \u03b9_colimitLimitToLimitColimit_\u03c0_apply,\n    colimit_eq_iff, curry_obj_obj_obj, curry_obj_obj_map] at h", "annotated_tactic": ["simp? [colimit_eq_iff] at h says\n      simp only [<a>Functor.comp_obj</a>, <a>colim_obj</a>, <a>\u03b9_colimitLimitToLimitColimit_\u03c0_apply</a>,\n        <a>colimit_eq_iff</a>, <a>curry_obj_obj_obj</a>, <a>curry_obj_obj_map</a>] at h", [{"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Limits.colim_obj", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1114, 3], "def_end_pos": [1114, 8]}, {"full_name": "CategoryTheory.Limits.\u03b9_colimitLimitToLimitColimit_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/ColimitLimit.lean", "def_pos": [97, 9], "def_end_pos": [97, 45]}, {"full_name": "CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff", "def_path": "Mathlib/CategoryTheory/Limits/TypesFiltered.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "CategoryTheory.curry_obj_obj_obj", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 10], "def_end_pos": [66, 21]}, {"full_name": "CategoryTheory.curry_obj_obj_map", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 22], "def_end_pos": [66, 33]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x)) =\n      limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y))\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "let k j := (h j).choose", "annotated_tactic": ["let k j := (h j).<a>choose</a>", [{"full_name": "Exists.choose", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [174, 32], "def_end_pos": [174, 45]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "let f : \u2200 j, kx \u27f6 k j := fun j => (h j).choose_spec.choose", "annotated_tactic": ["let f : \u2200 j, kx \u27f6 k j := fun j => (h j).choose_spec.choose", []], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "let g : \u2200 j, ky \u27f6 k j := fun j => (h j).choose_spec.choose_spec.choose", "annotated_tactic": ["let g : \u2200 j, ky \u27f6 k j := fun j => (h j).choose_spec.choose_spec.choose", []], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have w :\n  \u2200 j, F.map ((\ud835\udfd9 j, f j) :\n    (j, kx) \u27f6 (j, k j)) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map ((\ud835\udfd9 j, g j) : (j, ky) \u27f6 (j, k j))\n        (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y) :=\n  fun j => (h j).choose_spec.choose_spec.choose_spec", "annotated_tactic": ["have w :\n      \u2200 j, F.map ((\ud835\udfd9 j, f j) :\n        (j, kx) \u27f6 (j, k j)) (<a>limit.\u03c0</a> ((curry.obj (<a>swap</a> K J \u22d9 F)).<a>obj</a> kx) j x) =\n          F.map ((\ud835\udfd9 j, g j) : (j, ky) \u27f6 (j, k j))\n            (<a>limit.\u03c0</a> ((curry.obj (<a>swap</a> K J \u22d9 F)).<a>obj</a> ky) j y) :=\n      fun j => (h j).choose_spec.choose_spec.choose_spec", [{"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Prod.swap", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [149, 5], "def_end_pos": [149, 9]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "CategoryTheory.Limits.limit.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}, {"full_name": "CategoryTheory.Prod.swap", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [149, 5], "def_end_pos": [149, 9]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "let O : Finset K := Finset.univ.image k \u222a {kx, ky}", "annotated_tactic": ["let O : <a>Finset</a> K := Finset.univ.image k \u222a {kx, ky}", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have kxO : kx \u2208 O := Finset.mem_union.mpr (Or.inr (by simp))", "annotated_tactic": ["have kxO : kx \u2208 O := Finset.mem_union.mpr (<a>Or.inr</a> (by simp))", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have kyO : ky \u2208 O := Finset.mem_union.mpr (Or.inr (by simp))", "annotated_tactic": ["have kyO : ky \u2208 O := Finset.mem_union.mpr (<a>Or.inr</a> (by simp))", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have kjO : \u2200 j, k j \u2208 O := fun j => Finset.mem_union.mpr (Or.inl (by simp))", "annotated_tactic": ["have kjO : \u2200 j, k j \u2208 O := fun j => Finset.mem_union.mpr (<a>Or.inl</a> (by simp))", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "let H : Finset (\u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) :=\n  (Finset.univ.image fun j : J =>\n      \u27e8kx, k j, kxO, Finset.mem_union.mpr (Or.inl (by simp)), f j\u27e9) \u222a\n    Finset.univ.image fun j : J => \u27e8ky, k j, kyO, Finset.mem_union.mpr (Or.inl (by simp)), g j\u27e9", "annotated_tactic": ["let H : <a>Finset</a> (\u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) :=\n      (Finset.univ.image fun j : J =>\n          \u27e8kx, k j, kxO, Finset.mem_union.mpr (<a>Or.inl</a> (by simp)), f j\u27e9) \u222a\n        Finset.univ.image fun j : J => \u27e8ky, k j, kyO, Finset.mem_union.mpr (<a>Or.inl</a> (by simp)), g j\u27e9", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "obtain \u27e8S, T, W\u27e9 := IsFiltered.sup_exists O H", "annotated_tactic": ["obtain \u27e8S, T, W\u27e9 := <a>IsFiltered.sup_exists</a> O H", [{"full_name": "CategoryTheory.IsFiltered.sup_exists", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [262, 9], "def_end_pos": [262, 19]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have fH : \u2200 j, (\u27e8kx, k j, kxO, kjO j, f j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n  fun j =>\n  Finset.mem_union.mpr\n    (Or.inl\n      (by\n        simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n          Finset.mem_image, heq_iff_eq]\n        refine \u27e8j, ?_\u27e9\n        simp only [heq_iff_eq] ))", "annotated_tactic": ["have fH : \u2200 j, (\u27e8kx, k j, kxO, kjO j, f j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (<a>Or.inl</a>\n          (by\n            simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]\n            refine \u27e8j, ?_\u27e9\n            simp only [<a>heq_iff_eq</a>] ))", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "have gH :\n  \u2200 j, (\u27e8ky, k j, kyO, kjO j, g j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n  fun j =>\n  Finset.mem_union.mpr\n    (Or.inr\n      (by\n        simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n          Finset.mem_image, heq_iff_eq]\n        refine \u27e8j, ?_\u27e9\n        simp only [heq_iff_eq]))", "annotated_tactic": ["have gH :\n      \u2200 j, (\u27e8ky, k j, kyO, kjO j, g j\u27e9 : \u03a3' (X Y : K) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H :=\n      fun j =>\n      Finset.mem_union.mpr\n        (<a>Or.inr</a>\n          (by\n            simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]\n            refine \u27e8j, ?_\u27e9\n            simp only [<a>heq_iff_eq</a>]))", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "apply colimit_sound' (T kxO) (T kyO)", "annotated_tactic": ["apply <a>colimit_sound'</a> (T kxO) (T kyO)", [{"full_name": "CategoryTheory.Limits.Types.colimit_sound'", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [585, 9], "def_end_pos": [585, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 (curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kxO) x = (curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kyO) y"}, {"tactic": "ext j", "annotated_tactic": ["ext j", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\n\u22a2 (curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kxO) x = (curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kyO) y", "state_after": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j ((curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kxO) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j ((curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kyO) y)"}, {"tactic": "simp only [Functor.comp_map, Limit.map_\u03c0_apply, curry_obj_map_app, swap_map]", "annotated_tactic": ["simp only [<a>Functor.comp_map</a>, <a>Limit.map_\u03c0_apply</a>, <a>curry_obj_map_app</a>, <a>swap_map</a>]", [{"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.Limits.Types.Limit.map_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}, {"full_name": "CategoryTheory.curry_obj_map_app", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 34], "def_end_pos": [66, 45]}, {"full_name": "CategoryTheory.Prod.swap_map", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [148, 3], "def_end_pos": [148, 8]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j ((curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kxO) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j ((curry.obj (swap K J \u22d9 F) \u22d9 lim).map (T kyO) y)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (T kxO)) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (T kyO)) y)"}, {"tactic": "rw [\u2190 W _ _ (fH j), \u2190 W _ _ (gH j)]", "annotated_tactic": ["rw [\u2190 W _ _ (fH j), \u2190 W _ _ (gH j)]", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (T kxO)) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (T kyO)) y)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (f j \u226b T \u22ef)) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (g j \u226b T \u22ef)) y)"}, {"tactic": "simp [Limit.map_\u03c0_apply, w]", "annotated_tactic": ["simp [<a>Limit.map_\u03c0_apply</a>, w]", [{"full_name": "CategoryTheory.Limits.Types.Limit.map_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [306, 9], "def_end_pos": [306, 26]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.w\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\ngH : \u2200 (j : J), \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (f j \u226b T \u22ef)) x) =\n    limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj S) j (lim.map ((curry.obj (swap K J \u22d9 F)).map (g j \u226b T \u22ef)) y)", "state_after": "no goals"}, {"tactic": "simp only [Functor.comp_obj, colim_obj, \u03b9_colimitLimitToLimitColimit_\u03c0_apply,\n  colimit_eq_iff, curry_obj_obj_obj, curry_obj_obj_map] at h", "annotated_tactic": ["simp only [<a>Functor.comp_obj</a>, <a>colim_obj</a>, <a>\u03b9_colimitLimitToLimitColimit_\u03c0_apply</a>,\n        <a>colimit_eq_iff</a>, <a>curry_obj_obj_obj</a>, <a>curry_obj_obj_map</a>] at h", [{"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Limits.colim_obj", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1114, 3], "def_end_pos": [1114, 8]}, {"full_name": "CategoryTheory.Limits.\u03b9_colimitLimitToLimitColimit_\u03c0_apply", "def_path": "Mathlib/CategoryTheory/Limits/ColimitLimit.lean", "def_pos": [97, 9], "def_end_pos": [97, 45]}, {"full_name": "CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff", "def_path": "Mathlib/CategoryTheory/Limits/TypesFiltered.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "CategoryTheory.curry_obj_obj_obj", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 10], "def_end_pos": [66, 21]}, {"full_name": "CategoryTheory.curry_obj_obj_map", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [66, 22], "def_end_pos": [66, 33]}]], "state_before": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x)) =\n      limit.\u03c0 (curry.obj F \u22d9 colim) j (colimitLimitToLimitColimit F (colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y))\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y", "state_after": "case intro.intro.intro.intro.intro\nJ : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\n\u22a2 colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) kx x = colimit.\u03b9 (curry.obj (swap K J \u22d9 F) \u22d9 lim) ky y"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\n\u22a2 kx \u2208 {kx, ky}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\n\u22a2 ky \u2208 {kx, ky}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nj : J\n\u22a2 k j \u2208 Finset.image k Finset.univ", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nj : J\n\u22a2 k j \u2208 Finset.image k Finset.univ", "state_after": "no goals"}, {"tactic": "simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n  Finset.mem_image, heq_iff_eq]", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nj : J\n\u22a2 \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ", "state_after": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nj : J\n\u22a2 \u2203 a, \u27e8kx, \u27e8k a, \u27e8kxO, \u27e8\u22ef, f a\u27e9\u27e9\u27e9\u27e9 = \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9"}, {"tactic": "refine \u27e8j, ?_\u27e9", "annotated_tactic": ["refine \u27e8j, ?_\u27e9", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nj : J\n\u22a2 \u2203 a, \u27e8kx, \u27e8k a, \u27e8kxO, \u27e8\u22ef, f a\u27e9\u27e9\u27e9\u27e9 = \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9", "state_after": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nj : J\n\u22a2 \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 = \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9"}, {"tactic": "simp only [heq_iff_eq]", "annotated_tactic": ["simp only [<a>heq_iff_eq</a>]", [{"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nj : J\n\u22a2 \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 = \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9", "state_after": "no goals"}, {"tactic": "simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,\n  Finset.mem_image, heq_iff_eq]", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>Finset.mem_univ</a>, <a>eq_self_iff_true</a>, <a>exists_prop_of_true</a>,\n              <a>Finset.mem_image</a>, <a>heq_iff_eq</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "exists_prop_of_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [902, 9], "def_end_pos": [902, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 \u2208 Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ", "state_after": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 \u2203 a, \u27e8ky, \u27e8k a, \u27e8kyO, \u27e8\u22ef, g a\u27e9\u27e9\u27e9\u27e9 = \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9"}, {"tactic": "refine \u27e8j, ?_\u27e9", "annotated_tactic": ["refine \u27e8j, ?_\u27e9", []], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 \u2203 a, \u27e8ky, \u27e8k a, \u27e8kyO, \u27e8\u22ef, g a\u27e9\u27e9\u27e9\u27e9 = \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9", "state_after": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 = \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9"}, {"tactic": "simp only [heq_iff_eq]", "annotated_tactic": ["simp only [<a>heq_iff_eq</a>]", [{"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "J : Type u\u2081\nK : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} J\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} K\ninst\u271d\u00b2 : Small.{v, u\u2082} K\nF : J \u00d7 K \u2964 Type v\ninst\u271d\u00b9 : IsFiltered K\ninst\u271d : Finite J\nval\u271d : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J \u22d9 F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J \u22d9 F)).obj ky)\nh :\n  \u2200 (j : J),\n    \u2203 k f g,\n      F.map (\ud835\udfd9 j, f) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n        F.map (\ud835\udfd9 j, g) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nk : J \u2192 K := fun j => \u22ef.choose\nf : (j : J) \u2192 kx \u27f6 k j := fun j => \u22ef.choose\ng : (j : J) \u2192 ky \u27f6 k j := fun j => \u22ef.choose\nw :\n  \u2200 (j : J),\n    F.map (\ud835\udfd9 j, f j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj kx) j x) =\n      F.map (\ud835\udfd9 j, g j) (limit.\u03c0 ((curry.obj (swap K J \u22d9 F)).obj ky) j y)\nO : Finset K := Finset.image k Finset.univ \u222a {kx, ky}\nkxO : kx \u2208 O\nkyO : ky \u2208 O\nkjO : \u2200 (j : J), k j \u2208 O\nH : Finset ((X : K) \u00d7' (Y : K) \u00d7' (_ : X \u2208 O) \u00d7' (_ : Y \u2208 O) \u00d7' (X \u27f6 Y)) :=\n  Finset.image (fun j => \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9) Finset.univ \u222a\n    Finset.image (fun j => \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9) Finset.univ\nS : K\nT : {X : K} \u2192 X \u2208 O \u2192 (X \u27f6 S)\nW : \u2200 {X Y : K} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}, \u27e8X, \u27e8Y, \u27e8mX, \u27e8mY, f\u27e9\u27e9\u27e9\u27e9 \u2208 H \u2192 f \u226b T mY = T mX\nfH : \u2200 (j : J), \u27e8kx, \u27e8k j, \u27e8kxO, \u27e8\u22ef, f j\u27e9\u27e9\u27e9\u27e9 \u2208 H\nj : J\n\u22a2 \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9 = \u27e8ky, \u27e8k j, \u27e8kyO, \u27e8\u22ef, g j\u27e9\u27e9\u27e9\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "full_name": "PowerSeries.order_monomial", "start": [212, 1], "end": [222, 24], "traced_tactics": [{"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\n\u22a2 ((monomial R n) a).order = if a = 0 then \u22a4 else \u2191n", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : a = 0\n\u22a2 ((monomial R n) a).order = \u22a4\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\n\u22a2 ((monomial R n) a).order = \u2191n"}, {"tactic": "rw [h, order_eq_top, LinearMap.map_zero]", "annotated_tactic": ["rw [h, <a>order_eq_top</a>, <a>LinearMap.map_zero</a>]", [{"full_name": "PowerSeries.order_eq_top", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [106, 9], "def_end_pos": [106, 21]}, {"full_name": "LinearMap.map_zero", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [363, 19], "def_end_pos": [363, 27]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : a = 0\n\u22a2 ((monomial R n) a).order = \u22a4", "state_after": "no goals"}, {"tactic": "rw [order_eq]", "annotated_tactic": ["rw [<a>order_eq</a>]", [{"full_name": "PowerSeries.order_eq", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [144, 9], "def_end_pos": [144, 17]}]], "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\n\u22a2 ((monomial R n) a).order = \u2191n", "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\n\u22a2 (\u2200 (i : \u2115), \u2191i = \u2191n \u2192 (coeff R i) ((monomial R n) a) \u2260 0) \u2227 \u2200 (i : \u2115), \u2191i < \u2191n \u2192 (coeff R i) ((monomial R n) a) = 0"}, {"tactic": "constructor <;> intro i hi", "annotated_tactic": ["constructor <;> intro i hi", []], "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\n\u22a2 (\u2200 (i : \u2115), \u2191i = \u2191n \u2192 (coeff R i) ((monomial R n) a) \u2260 0) \u2227 \u2200 (i : \u2115), \u2191i < \u2191n \u2192 (coeff R i) ((monomial R n) a) = 0", "state_after": "case neg.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : \u2191i = \u2191n\n\u22a2 (coeff R i) ((monomial R n) a) \u2260 0\n\ncase neg.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : \u2191i < \u2191n\n\u22a2 (coeff R i) ((monomial R n) a) = 0"}, {"tactic": "rw [PartENat.natCast_inj] at hi", "annotated_tactic": ["rw [<a>PartENat.natCast_inj</a>] at hi", [{"full_name": "PartENat.natCast_inj", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [116, 9], "def_end_pos": [116, 20]}]], "state_before": "case neg.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : \u2191i = \u2191n\n\u22a2 (coeff R i) ((monomial R n) a) \u2260 0", "state_after": "case neg.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i = n\n\u22a2 (coeff R i) ((monomial R n) a) \u2260 0"}, {"tactic": "rwa [hi, coeff_monomial_same]", "annotated_tactic": ["rwa [hi, <a>coeff_monomial_same</a>]", [{"full_name": "PowerSeries.coeff_monomial_same", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}]], "state_before": "case neg.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i = n\n\u22a2 (coeff R i) ((monomial R n) a) \u2260 0", "state_after": "no goals"}, {"tactic": "rw [PartENat.coe_lt_coe] at hi", "annotated_tactic": ["rw [<a>PartENat.coe_lt_coe</a>] at hi", [{"full_name": "PartENat.coe_lt_coe", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [314, 9], "def_end_pos": [314, 19]}]], "state_before": "case neg.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : \u2191i < \u2191n\n\u22a2 (coeff R i) ((monomial R n) a) = 0", "state_after": "case neg.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i < n\n\u22a2 (coeff R i) ((monomial R n) a) = 0"}, {"tactic": "rw [coeff_monomial, if_neg]", "annotated_tactic": ["rw [<a>coeff_monomial</a>, <a>if_neg</a>]", [{"full_name": "PowerSeries.coeff_monomial", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 23]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}]], "state_before": "case neg.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i < n\n\u22a2 (coeff R i) ((monomial R n) a) = 0", "state_after": "case neg.right.hnc\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i < n\n\u22a2 \u00aci = n"}, {"tactic": "exact ne_of_lt hi", "annotated_tactic": ["exact <a>ne_of_lt</a> hi", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case neg.right.hnc\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\n\u03c6 : R\u27e6X\u27e7\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\nh : \u00aca = 0\ni : \u2115\nhi : i < n\n\u22a2 \u00aci = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.arrowCongr_comp", "start": [466, 1], "end": [472, 37], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\n\u22a2 (e\u2081.arrowCongr e\u2083) (g.comp f) = ((e\u2082.arrowCongr e\u2083) g).comp ((e\u2081.arrowCongr e\u2082) f)", "state_after": "case H\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 ((e\u2081.arrowCongr e\u2083) (g.comp f)) x\u271d = (((e\u2082.arrowCongr e\u2083) g).comp ((e\u2081.arrowCongr e\u2082) f)) x\u271d"}, {"tactic": "simp only [arrowCongr, Equiv.coe_fn_mk, AlgHom.comp_apply]", "annotated_tactic": ["simp only [<a>arrowCongr</a>, <a>Equiv.coe_fn_mk</a>, <a>AlgHom.comp_apply</a>]", [{"full_name": "AlgEquiv.arrowCongr", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [455, 5], "def_end_pos": [455, 15]}, {"full_name": "Equiv.coe_fn_mk", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [113, 17], "def_end_pos": [113, 26]}, {"full_name": "AlgHom.comp_apply", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [324, 9], "def_end_pos": [324, 19]}]], "state_before": "case H\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 ((e\u2081.arrowCongr e\u2083) (g.comp f)) x\u271d = (((e\u2082.arrowCongr e\u2083) g).comp ((e\u2081.arrowCongr e\u2082) f)) x\u271d", "state_after": "case H\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 \u2191e\u2083 (g (f (\u2191e\u2081.symm x\u271d))) = \u2191e\u2083 (g (\u2191e\u2082.symm (\u2191e\u2082 (f (\u2191e\u2081.symm x\u271d)))))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case H\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 \u2191e\u2083 (g (f (\u2191e\u2081.symm x\u271d))) = \u2191e\u2083 (g (\u2191e\u2082.symm (\u2191e\u2082 (f (\u2191e\u2081.symm x\u271d)))))", "state_after": "case H.h.e_6.h.h.e_6.h\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 f (\u2191e\u2081.symm x\u271d) = \u2191e\u2082.symm (\u2191e\u2082 (f (\u2191e\u2081.symm x\u271d)))"}, {"tactic": "exact (e\u2082.symm_apply_apply _).symm", "annotated_tactic": ["exact (e\u2082.symm_apply_apply _).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case H.h.e_6.h.h.e_6.h\nR : Type uR\nA\u2081 : Type uA\u2081\nA\u2082 : Type uA\u2082\nA\u2083 : Type uA\u2083\nA\u2081' : Type uA\u2081'\nA\u2082' : Type uA\u2082'\nA\u2083' : Type uA\u2083'\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : Semiring A\u2081\ninst\u271d\u00b9\u2070 : Semiring A\u2082\ninst\u271d\u2079 : Semiring A\u2083\ninst\u271d\u2078 : Semiring A\u2081'\ninst\u271d\u2077 : Semiring A\u2082'\ninst\u271d\u2076 : Semiring A\u2083'\ninst\u271d\u2075 : Algebra R A\u2081\ninst\u271d\u2074 : Algebra R A\u2082\ninst\u271d\u00b3 : Algebra R A\u2083\ninst\u271d\u00b2 : Algebra R A\u2081'\ninst\u271d\u00b9 : Algebra R A\u2082'\ninst\u271d : Algebra R A\u2083'\ne : A\u2081 \u2243\u2090[R] A\u2082\ne\u2081 : A\u2081 \u2243\u2090[R] A\u2081'\ne\u2082 : A\u2082 \u2243\u2090[R] A\u2082'\ne\u2083 : A\u2083 \u2243\u2090[R] A\u2083'\nf : A\u2081 \u2192\u2090[R] A\u2082\ng : A\u2082 \u2192\u2090[R] A\u2083\nx\u271d : A\u2081'\n\u22a2 f (\u2191e\u2081.symm x\u271d) = \u2191e\u2082.symm (\u2191e\u2082 (f (\u2191e\u2081.symm x\u271d)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.lsum_single", "start": [491, 1], "end": [493, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Counit.lean", "full_name": "MvPolynomial.ACounit_surjective", "start": [60, 1], "end": [60, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Height.lean", "full_name": "Set.le_chainHeight_iff", "start": [119, 1], "end": [120, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Pi/Lemmas.lean", "full_name": "Sigma.uncurry_inv", "start": [543, 1], "end": [545, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.measure_ge_le_exp_mul_mgf", "start": [326, 1], "end": [345, 66], "traced_tactics": [{"tactic": "rcases ht.eq_or_lt with ht_zero_eq | ht_pos", "annotated_tactic": ["rcases ht.eq_or_lt with ht_zero_eq | ht_pos", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-t * \u03b5) * mgf X \u03bc t\n\ncase inr\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-t * \u03b5) * mgf X \u03bc t"}, {"tactic": "calc\n  (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal = (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal := by\n    congr with \u03c9\n    simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt]\n    exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n      fun h => le_of_mul_le_mul_left h ht_pos\u27e9\n  _ \u2264 (exp (t * \u03b5))\u207b\u00b9 * \u03bc[fun \u03c9 => exp (t * X \u03c9)] := by\n    have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n        \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n      mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _\n    rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)]\n  _ = exp (-t * \u03b5) * mgf X \u03bc t := by rw [neg_mul, exp_neg]; rfl", "annotated_tactic": ["calc\n    (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).<a>toReal</a> = (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> := by\n      congr with \u03c9\n      simp only [<a>Set.mem_setOf_eq</a>, <a>exp_le_exp</a>, <a>gt_iff_lt</a>]\n      exact \u27e8fun h => <a>mul_le_mul_of_nonneg_left</a> h ht_pos.le,\n        fun h => <a>le_of_mul_le_mul_left</a> h ht_pos\u27e9\n    _ \u2264 (<a>exp</a> (t * \u03b5))\u207b\u00b9 * \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] := by\n      have : <a>exp</a> (t * \u03b5) * (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> \u2264\n          \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] :=\n        <a>mul_meas_ge_le_integral_of_nonneg</a> (<a>ae_of_all</a> _ fun x => (<a>exp_pos</a> _).<a>le</a>) h_int _\n      rwa [<a>mul_comm</a> (<a>exp</a> (t * \u03b5))\u207b\u00b9, \u2190 <a>div_eq_mul_inv</a>, <a>le_div_iff'</a> (<a>exp_pos</a> _)]\n    _ = <a>exp</a> (-t * \u03b5) * <a>mgf</a> X \u03bc t := by rw [<a>neg_mul</a>, <a>exp_neg</a>]; rfl", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Real.exp_le_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1237, 9], "def_end_pos": [1237, 19]}, {"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [205, 9], "def_end_pos": [205, 34]}, {"full_name": "le_of_mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [232, 9], "def_end_pos": [232, 30]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "MeasureTheory.mul_meas_ge_le_integral_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1797, 9], "def_end_pos": [1797, 42]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "ProbabilityTheory.mgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [103, 5], "def_end_pos": [103, 8]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Real.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [862, 16], "def_end_pos": [862, 23]}]], "state_before": "case inr\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "no goals"}, {"tactic": "rw [ht_zero_eq.symm]", "annotated_tactic": ["rw [ht_zero_eq.symm]", []], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-0 * \u03b5) * mgf X \u03bc 0"}, {"tactic": "simp only [neg_zero, zero_mul, exp_zero, mgf_zero', one_mul]", "annotated_tactic": ["simp only [<a>neg_zero</a>, <a>zero_mul</a>, <a>exp_zero</a>, <a>mgf_zero'</a>, <a>one_mul</a>]", [{"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "Real.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [825, 9], "def_end_pos": [825, 17]}, {"full_name": "ProbabilityTheory.mgf_zero'", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 rexp (-0 * \u03b5) * mgf X \u03bc 0", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 (\u03bc Set.univ).toReal"}, {"tactic": "rw [ENNReal.toReal_le_toReal (measure_ne_top \u03bc _) (measure_ne_top \u03bc _)]", "annotated_tactic": ["rw [<a>ENNReal.toReal_le_toReal</a> (<a>measure_ne_top</a> \u03bc _) (<a>measure_ne_top</a> \u03bc _)]", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [76, 9], "def_end_pos": [76, 25]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 (\u03bc Set.univ).toReal", "state_after": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 \u03bc {\u03c9 | \u03b5 \u2264 X \u03c9} \u2264 \u03bc Set.univ"}, {"tactic": "exact measure_mono (Set.subset_univ _)", "annotated_tactic": ["exact <a>measure_mono</a> (<a>Set.subset_univ</a> _)", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [657, 9], "def_end_pos": [657, 20]}]], "state_before": "case inl\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_zero_eq : 0 = t\n\u22a2 \u03bc {\u03c9 | \u03b5 \u2264 X \u03c9} \u2264 \u03bc Set.univ", "state_after": "no goals"}, {"tactic": "congr with \u03c9", "annotated_tactic": ["congr with \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal = (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal", "state_after": "case e_a.h.e_6.h.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03b5 \u2264 X \u03c9} \u2194 \u03c9 \u2208 {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}"}, {"tactic": "simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>exp_le_exp</a>, <a>gt_iff_lt</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Real.exp_le_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1237, 9], "def_end_pos": [1237, 19]}, {"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}]], "state_before": "case e_a.h.e_6.h.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | \u03b5 \u2264 X \u03c9} \u2194 \u03c9 \u2208 {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}", "state_after": "case e_a.h.e_6.h.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03b5 \u2264 X \u03c9 \u2194 t * \u03b5 \u2264 t * X \u03c9"}, {"tactic": "exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n  fun h => le_of_mul_le_mul_left h ht_pos\u27e9", "annotated_tactic": ["exact \u27e8fun h => <a>mul_le_mul_of_nonneg_left</a> h ht_pos.le,\n        fun h => <a>le_of_mul_le_mul_left</a> h ht_pos\u27e9", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [205, 9], "def_end_pos": [205, 34]}, {"full_name": "le_of_mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [232, 9], "def_end_pos": [232, 30]}]], "state_before": "case e_a.h.e_6.h.h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u03c9 : \u03a9\n\u22a2 \u03b5 \u2264 X \u03c9 \u2194 t * \u03b5 \u2264 t * X \u03c9", "state_after": "no goals"}, {"tactic": "have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n    \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n  mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _", "annotated_tactic": ["have : <a>exp</a> (t * \u03b5) * (\u03bc {\u03c9 | <a>exp</a> (t * \u03b5) \u2264 <a>exp</a> (t * X \u03c9)}).<a>toReal</a> \u2264\n          \u03bc[fun \u03c9 => <a>exp</a> (t * X \u03c9)] :=\n        <a>mul_meas_ge_le_integral_of_nonneg</a> (<a>ae_of_all</a> _ fun x => (<a>exp_pos</a> _).<a>le</a>) h_int _", [{"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "MeasureTheory.mul_meas_ge_le_integral_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1797, 9], "def_end_pos": [1797, 42]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal \u2264 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\nthis : rexp (t * \u03b5) * (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal \u2264 \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc\n\u22a2 (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal \u2264 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc"}, {"tactic": "rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)]", "annotated_tactic": ["rwa [<a>mul_comm</a> (<a>exp</a> (t * \u03b5))\u207b\u00b9, \u2190 <a>div_eq_mul_inv</a>, <a>le_div_iff'</a> (<a>exp_pos</a> _)]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\nthis : rexp (t * \u03b5) * (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal \u2264 \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc\n\u22a2 (\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}).toReal \u2264 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [neg_mul, exp_neg]", "annotated_tactic": ["rw [<a>neg_mul</a>, <a>exp_neg</a>]", [{"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Real.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [862, 16], "def_end_pos": [862, 23]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = rexp (-t * \u03b5) * mgf X \u03bc t", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = (rexp (t * \u03b5))\u207b\u00b9 * mgf X \u03bc t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\n\u03b5 : \u211d\nht : 0 \u2264 t\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\nht_pos : 0 < t\n\u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = (rexp (t * \u03b5))\u207b\u00b9 * mgf X \u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Order.lean", "full_name": "FirstOrder.Language.realize_noTopOrder_iff", "start": [231, 1], "end": [237, 24], "traced_tactics": [{"tactic": "simp only [noTopOrderSentence, Sentence.Realize, Formula.Realize, BoundedFormula.realize_all,\n  BoundedFormula.realize_ex, BoundedFormula.realize_not, Term.realize, Term.realize_le,\n  Sum.elim_inr]", "annotated_tactic": ["simp only [<a>noTopOrderSentence</a>, <a>Sentence.Realize</a>, <a>Formula.Realize</a>, <a>BoundedFormula.realize_all</a>,\n    <a>BoundedFormula.realize_ex</a>, <a>BoundedFormula.realize_not</a>, <a>Term.realize</a>, <a>Term.realize_le</a>,\n    <a>Sum.elim_inr</a>]", [{"full_name": "FirstOrder.Language.noTopOrderSentence", "def_path": "Mathlib/ModelTheory/Order.lean", "def_pos": [136, 5], "def_end_pos": [136, 23]}, {"full_name": "FirstOrder.Language.Sentence.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [716, 12], "def_end_pos": [716, 28]}, {"full_name": "FirstOrder.Language.Formula.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [606, 12], "def_end_pos": [606, 19]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_all", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [338, 9], "def_end_pos": [338, 20]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_ex", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [343, 9], "def_end_pos": [343, 19]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_not", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [266, 9], "def_end_pos": [266, 20]}, {"full_name": "FirstOrder.Language.Term.realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [71, 5], "def_end_pos": [71, 12]}, {"full_name": "FirstOrder.Language.Term.realize_le", "def_path": "Mathlib/ModelTheory/Order.lean", "def_pos": [212, 9], "def_end_pos": [212, 24]}, {"full_name": "Sum.elim_inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [88, 17], "def_end_pos": [88, 25]}]], "state_before": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\n\u22a2 M \u22a8 Language.order.noTopOrderSentence \u2194 NoTopOrder M", "state_after": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\n\u22a2 (\u2200 (a : M), \u2203 a_1, \u00acFin.snoc (Fin.snoc default a) a_1 1 \u2264 Fin.snoc (Fin.snoc default a) a_1 0) \u2194 NoTopOrder M"}, {"tactic": "refine \u27e8fun h => \u27e8fun a => h a\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => \u27e8fun a => h a\u27e9, ?_\u27e9", []], "state_before": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\n\u22a2 (\u2200 (a : M), \u2203 a_1, \u00acFin.snoc (Fin.snoc default a) a_1 1 \u2264 Fin.snoc (Fin.snoc default a) a_1 0) \u2194 NoTopOrder M", "state_after": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\n\u22a2 NoTopOrder M \u2192 \u2200 (a : M), \u2203 a_1, \u00acFin.snoc (Fin.snoc default a) a_1 1 \u2264 Fin.snoc (Fin.snoc default a) a_1 0"}, {"tactic": "intro h a", "annotated_tactic": ["intro h a", []], "state_before": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\n\u22a2 NoTopOrder M \u2192 \u2200 (a : M), \u2203 a_1, \u00acFin.snoc (Fin.snoc default a) a_1 1 \u2264 Fin.snoc (Fin.snoc default a) a_1 0", "state_after": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\nh : NoTopOrder M\na : M\n\u22a2 \u2203 a_1, \u00acFin.snoc (Fin.snoc default a) a_1 1 \u2264 Fin.snoc (Fin.snoc default a) a_1 0"}, {"tactic": "exact exists_not_le a", "annotated_tactic": ["exact <a>exists_not_le</a> a", [{"full_name": "NoTopOrder.exists_not_le", "def_path": "Mathlib/Order/Max.lean", "def_pos": [49, 3], "def_end_pos": [49, 16]}]], "state_before": "L : Language\n\u03b1 : Type w\nM : Type w'\nn : \u2115\ninst\u271d : LE M\nh : NoTopOrder M\na : M\n\u22a2 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\u2964 A\n\u22a2 J.W (adj.unit.app P)", "state_after": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst\u271d : Category.{u_4, u_2} A\nG : (C\u1d52\u1d56 \u2964 A) \u2964 Sheaf J A\nadj : G \u22a3 sheafToPresheaf J A\nP : C\u1d52\u1d56 \u2964 A\n\u22a2 LeftBousfield.W (fun x => x \u2208 Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)"}, {"tactic": "exact LeftBousfield.W_adj_unit_app adj P", "annotated_tactic": ["exact <a>LeftBousfield.W_adj_unit_app</a> adj P", [{"full_name": "CategoryTheory.Localization.LeftBousfield.W_adj_unit_app", "def_path": "Mathlib/CategoryTheory/Localization/Bousfield.lean", "def_pos": [112, 7], "def_end_pos": [112, 21]}]], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst\u271d : Category.{u_4, u_2} A\nG : (C\u1d52\u1d56 \u2964 A) \u2964 Sheaf J A\nadj : G \u22a3 sheafToPresheaf J A\nP : C\u1d52\u1d56 \u2964 A\n\u22a2 LeftBousfield.W (fun x => x \u2208 Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/List.lean", "full_name": "Vector.tendsto_cons", "start": [200, 1], "end": [203, 77], "traced_tactics": [{"tactic": "rw [tendsto_subtype_rng, cons_val]", "annotated_tactic": ["rw [<a>tendsto_subtype_rng</a>, <a>cons_val</a>]", [{"full_name": "tendsto_subtype_rng", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 28]}, {"full_name": "Vector.cons_val", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun p => p.1 ::\u1d65 p.2) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a ::\u1d65 l))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun x => \u2191(x.1 ::\u1d65 x.2)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a :: \u2191l))"}, {"tactic": "exact tendsto_fst.cons (Tendsto.comp continuousAt_subtype_val tendsto_snd)", "annotated_tactic": ["exact tendsto_fst.cons (<a>Tendsto.comp</a> <a>continuousAt_subtype_val</a> <a>tendsto_snd</a>)", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3098, 9], "def_end_pos": [3098, 21]}, {"full_name": "continuousAt_subtype_val", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1136, 9], "def_end_pos": [1136, 33]}, {"full_name": "Filter.tendsto_snd", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [142, 9], "def_end_pos": [142, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun x => \u2191(x.1 ::\u1d65 x.2)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a :: \u2191l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean", "full_name": "Ideal.map_includeRight_eq", "start": [580, 1], "end": [646, 40], "traced_tactics": [{"tactic": "rw [\u2190 Submodule.carrier_inj]", "annotated_tactic": ["rw [\u2190 <a>Submodule.carrier_inj</a>]", [{"full_name": "Submodule.carrier_inj", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 Submodule.restrictScalars R (map includeRight I) =\n    LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (Submodule.restrictScalars R (map includeRight I)).carrier =\n    (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier"}, {"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (Submodule.restrictScalars R (map includeRight I)).carrier =\n    (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (Submodule.restrictScalars R (map includeRight I)).carrier \u2264\n    (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier\n\ncase a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier \u2264\n    (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (Submodule.restrictScalars R (map includeRight I)).carrier \u2264\n    (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\n\u22a2 x \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier \u2192\n    x \u2208 (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier"}, {"tactic": "simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,\n  Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem, LinearMap.mem_range]", "annotated_tactic": ["simp only [<a>AddSubsemigroup.mem_carrier</a>, <a>AddSubmonoid.mem_toSubsemigroup</a>,\n      <a>Submodule.mem_toAddSubmonoid</a>, <a>Submodule.restrictScalars_mem</a>, <a>LinearMap.mem_range</a>]", [{"full_name": "AddSubsemigroup.mem_carrier", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [111, 3], "def_end_pos": [111, 14]}, {"full_name": "AddSubmonoid.mem_toSubsemigroup", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [161, 3], "def_end_pos": [161, 14]}, {"full_name": "Submodule.mem_toAddSubmonoid", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 27]}, {"full_name": "Submodule.restrictScalars_mem", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [49, 9], "def_end_pos": [49, 28]}, {"full_name": "LinearMap.mem_range", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\n\u22a2 x \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier \u2192\n    x \u2208 (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\n\u22a2 x \u2208 map includeRight I \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x"}, {"tactic": "intro hx", "annotated_tactic": ["intro hx", []], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\n\u22a2 x \u2208 map includeRight I \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 map includeRight I\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x"}, {"tactic": "rw [Ideal.map, \u2190 submodule_span_eq] at hx", "annotated_tactic": ["rw [<a>Ideal.map</a>, \u2190 <a>submodule_span_eq</a>] at hx", [{"full_name": "Ideal.map", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [36, 5], "def_end_pos": [36, 8]}, {"full_name": "Ideal.submodule_span_eq", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 26]}]], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 map includeRight I\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x"}, {"tactic": "refine Submodule.span_induction hx ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>Submodule.span_induction</a> hx ?_ ?_ ?_ ?_", [{"full_name": "Submodule.span_induction", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [182, 9], "def_end_pos": [182, 23]}]], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 x \u2208 \u21d1includeRight '' \u2191I, \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x\n\ncase a.refine_2\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = 0\n\ncase a.refine_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x) \u2192\n      (\u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = y) \u2192\n        \u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = x + y\n\ncase a.refine_4\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 (a x : A \u2297[R] B),\n    (\u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x) \u2192\n      \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = a \u2022 x"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 x \u2208 \u21d1includeRight '' \u2191I, \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] B\n\u22a2 x \u2208 \u21d1includeRight '' \u2191I \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x"}, {"tactic": "simp only [includeRight_apply, Set.mem_image, SetLike.mem_coe]", "annotated_tactic": ["simp only [<a>includeRight_apply</a>, <a>Set.mem_image</a>, <a>SetLike.mem_coe</a>]", [{"full_name": "Algebra.TensorProduct.includeRight_apply", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [418, 9], "def_end_pos": [418, 27]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}]], "state_before": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] B\n\u22a2 x \u2208 \u21d1includeRight '' \u2191I \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] B\n\u22a2 (\u2203 x_1 \u2208 I, 1 \u2297\u209c[R] x_1 = x) \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x"}, {"tactic": "rintro \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, hy, rfl\u27e9", []], "state_before": "case a.refine_1\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] B\n\u22a2 (\u2203 x_1 \u2208 I, 1 \u2297\u209c[R] x_1 = x) \u2192 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x", "state_after": "case a.refine_1.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\ny : B\nhy : y \u2208 I\n\u22a2 \u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = 1 \u2297\u209c[R] y"}, {"tactic": "use 1 \u2297\u209c[R] \u27e8y, hy\u27e9", "annotated_tactic": ["use 1 \u2297\u209c[R] \u27e8y, hy\u27e9", []], "state_before": "case a.refine_1.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\ny : B\nhy : y \u2208 I\n\u22a2 \u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = 1 \u2297\u209c[R] y", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\ny : B\nhy : y \u2208 I\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (1 \u2297\u209c[R] \u27e8y, hy\u27e9) = 1 \u2297\u209c[R] y"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\ny : B\nhy : y \u2208 I\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (1 \u2297\u209c[R] \u27e8y, hy\u27e9) = 1 \u2297\u209c[R] y", "state_after": "no goals"}, {"tactic": "use 0", "annotated_tactic": ["use 0", []], "state_before": "case a.refine_2\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = 0", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 = 0"}, {"tactic": "simp only [map_zero]", "annotated_tactic": ["simp only [<a>map_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 = 0", "state_after": "no goals"}, {"tactic": "rintro x y \u27e8x, hx, rfl\u27e9 \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro x y \u27e8x, hx, rfl\u27e9 \u27e8y, hy, rfl\u27e9", []], "state_before": "case a.refine_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x) \u2192\n      (\u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = y) \u2192\n        \u2203 y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = x + y", "state_after": "case a.refine_3.intro.refl.intro.refl\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n        (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y"}, {"tactic": "use x + y", "annotated_tactic": ["use x + y", []], "state_before": "case a.refine_3.intro.refl.intro.refl\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n        (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y) =\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y"}, {"tactic": "simp only [map_add]", "annotated_tactic": ["simp only [<a>map_add</a>]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y) =\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y", "state_after": "no goals"}, {"tactic": "rintro a x \u27e8x, hx, rfl\u27e9", "annotated_tactic": ["rintro a x \u27e8x, hx, rfl\u27e9", []], "state_before": "case a.refine_4\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\n\u22a2 \u2200 (a x : A \u2297[R] B),\n    (\u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x) \u2192\n      \u2203 y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = a \u2022 x", "state_after": "case a.refine_4.intro.refl\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A \u2297[R] B\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x"}, {"tactic": "induction a with\n| zero =>\n  use 0\n  simp only [map_zero, smul_eq_mul, zero_mul]\n| tmul a b =>\n  induction x with\n  | zero =>\n    use 0\n    simp only [map_zero, smul_eq_mul, mul_zero]\n  | tmul x y =>\n    use (a * x) \u2297\u209c[R] (b \u2022y)\n    simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype, smul_eq_mul, tmul_mul_tmul]\n    rfl\n  | add x y hx hy =>\n    obtain \u27e8x', hx'\u27e9 := hx\n    obtain \u27e8y', hy'\u27e9 := hy\n    use x' + y'\n    simp only [map_add, hx', smul_add, hy']\n| add a b ha hb =>\n  obtain \u27e8x', ha'\u27e9 := ha\n  obtain \u27e8y', hb'\u27e9 := hb\n  use x' + y'\n  simp only [map_add, ha', add_smul, hb']", "annotated_tactic": ["induction a with\n      | zero =>\n        use 0\n        simp only [<a>map_zero</a>, <a>smul_eq_mul</a>, <a>zero_mul</a>]\n      | tmul a b =>\n        induction x with\n        | zero =>\n          use 0\n          simp only [<a>map_zero</a>, <a>smul_eq_mul</a>, <a>mul_zero</a>]\n        | tmul x y =>\n          use (a * x) \u2297\u209c[R] (b \u2022y)\n          simp only [<a>LinearMap.lTensor_tmul</a>, <a>Submodule.coeSubtype</a>, <a>smul_eq_mul</a>, <a>tmul_mul_tmul</a>]\n          rfl\n        | add x y hx hy =>\n          obtain \u27e8x', hx'\u27e9 := hx\n          obtain \u27e8y', hy'\u27e9 := hy\n          use x' + y'\n          simp only [<a>map_add</a>, hx', <a>smul_add</a>, hy']\n      | add a b ha hb =>\n        obtain \u27e8x', ha'\u27e9 := ha\n        obtain \u27e8y', hb'\u27e9 := hb\n        use x' + y'\n        simp only [<a>map_add</a>, ha', <a>add_smul</a>, hb']", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "add_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case a.refine_4.intro.refl\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A \u2297[R] B\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "no goals"}, {"tactic": "use 0", "annotated_tactic": ["use 0", []], "state_before": "case a.refine_4.intro.refl.zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      0 \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =\n    0 \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x"}, {"tactic": "simp only [map_zero, smul_eq_mul, zero_mul]", "annotated_tactic": ["simp only [<a>map_zero</a>, <a>smul_eq_mul</a>, <a>zero_mul</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =\n    0 \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "no goals"}, {"tactic": "induction x with\n| zero =>\n  use 0\n  simp only [map_zero, smul_eq_mul, mul_zero]\n| tmul x y =>\n  use (a * x) \u2297\u209c[R] (b \u2022y)\n  simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype, smul_eq_mul, tmul_mul_tmul]\n  rfl\n| add x y hx hy =>\n  obtain \u27e8x', hx'\u27e9 := hx\n  obtain \u27e8y', hy'\u27e9 := hy\n  use x' + y'\n  simp only [map_add, hx', smul_add, hy']", "annotated_tactic": ["induction x with\n        | zero =>\n          use 0\n          simp only [<a>map_zero</a>, <a>smul_eq_mul</a>, <a>mul_zero</a>]\n        | tmul x y =>\n          use (a * x) \u2297\u209c[R] (b \u2022y)\n          simp only [<a>LinearMap.lTensor_tmul</a>, <a>Submodule.coeSubtype</a>, <a>smul_eq_mul</a>, <a>tmul_mul_tmul</a>]\n          rfl\n        | add x y hx hy =>\n          obtain \u27e8x', hx'\u27e9 := hx\n          obtain \u27e8y', hy'\u27e9 := hy\n          use x' + y'\n          simp only [<a>map_add</a>, hx', <a>smul_add</a>, hy']", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}]], "state_before": "case a.refine_4.intro.refl.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na : A\nb : B\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "no goals"}, {"tactic": "use 0", "annotated_tactic": ["use 0", []], "state_before": "case a.refine_4.intro.refl.tmul.zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0"}, {"tactic": "simp only [map_zero, smul_eq_mul, mul_zero]", "annotated_tactic": ["simp only [<a>map_zero</a>, <a>smul_eq_mul</a>, <a>mul_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx : A \u2297[R] B\nhx : x \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0", "state_after": "no goals"}, {"tactic": "use (a * x) \u2297\u209c[R] (b \u2022y)", "annotated_tactic": ["use (a * x) \u2297\u209c[R] (b \u2022y)", []], "state_before": "case a.refine_4.intro.refl.tmul.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx : A\ny : \u21a5(Submodule.restrictScalars R I)\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x \u2297\u209c[R] y)", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx : A\ny : \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) ((a * x) \u2297\u209c[R] (b \u2022 y)) =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x \u2297\u209c[R] y)"}, {"tactic": "simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype, smul_eq_mul, tmul_mul_tmul]", "annotated_tactic": ["simp only [<a>LinearMap.lTensor_tmul</a>, <a>Submodule.coeSubtype</a>, <a>smul_eq_mul</a>, <a>tmul_mul_tmul</a>]", [{"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx : A\ny : \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) ((a * x) \u2297\u209c[R] (b \u2022 y)) =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x \u2297\u209c[R] y)", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx : A\ny : \u21a5(Submodule.restrictScalars R I)\n\u22a2 (a * x) \u2297\u209c[R] \u2191(b \u2022 y) = (a * x) \u2297\u209c[R] (b * \u2191y)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx : A\ny : \u21a5(Submodule.restrictScalars R I)\n\u22a2 (a * x) \u2297\u209c[R] \u2191(b \u2022 y) = (a * x) \u2297\u209c[R] (b * \u2191y)", "state_after": "no goals"}, {"tactic": "obtain \u27e8x', hx'\u27e9 := hx", "annotated_tactic": ["obtain \u27e8x', hx'\u27e9 := hx", []], "state_before": "case a.refine_4.intro.refl.tmul.add\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx\u271d : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx :\n  \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\nhy :\n  \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)", "state_after": "case a.refine_4.intro.refl.tmul.add.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy :\n  \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)"}, {"tactic": "obtain \u27e8y', hy'\u27e9 := hy", "annotated_tactic": ["obtain \u27e8y', hy'\u27e9 := hy", []], "state_before": "case a.refine_4.intro.refl.tmul.add.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy :\n  \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)", "state_after": "case a.refine_4.intro.refl.tmul.add.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y x' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)"}, {"tactic": "use x' + y'", "annotated_tactic": ["use x' + y'", []], "state_before": "case a.refine_4.intro.refl.tmul.add.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y x' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\n\u22a2 \u2203 y_1,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =\n      a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y x' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x' + y') =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)"}, {"tactic": "simp only [map_add, hx', smul_add, hy']", "annotated_tactic": ["simp only [<a>map_add</a>, hx', <a>smul_add</a>, hy']", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\na : A\nb : B\nx y x' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhy' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x' + y') =\n    a \u2297\u209c[R] b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)", "state_after": "no goals"}, {"tactic": "obtain \u27e8x', ha'\u27e9 := ha", "annotated_tactic": ["obtain \u27e8x', ha'\u27e9 := ha", []], "state_before": "case a.refine_4.intro.refl.add\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nha :\n  \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\nhb :\n  \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "case a.refine_4.intro.refl.add.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nhb :\n  \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x"}, {"tactic": "obtain \u27e8y', hb'\u27e9 := hb", "annotated_tactic": ["obtain \u27e8y', hb'\u27e9 := hb", []], "state_before": "case a.refine_4.intro.refl.add.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nhb :\n  \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "case a.refine_4.intro.refl.add.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhb' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x"}, {"tactic": "use x' + y'", "annotated_tactic": ["use x' + y'", []], "state_before": "case a.refine_4.intro.refl.add.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhb' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 \u2203 y,\n    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =\n      (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhb' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x' + y') =\n    (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x"}, {"tactic": "simp only [map_add, ha', add_smul, hb']", "annotated_tactic": ["simp only [<a>map_add</a>, ha', <a>add_smul</a>, hb']", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "add_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx\u271d : A \u2297[R] B\nhx : x\u271d \u2208 Submodule.span (A \u2297[R] B) (\u21d1includeRight '' \u2191I)\nx : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\na b : A \u2297[R] B\nx' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nha' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =\n    a \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\ny' : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhb' :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y' =\n    b \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x' + y') =\n    (a + b) \u2022 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x", "state_after": "no goals"}, {"tactic": "rintro x \u27e8y, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y, rfl\u27e9", []], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier \u2264\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "case a.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\ny : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "induction y with\n| zero =>\n    rw [map_zero]\n    apply zero_mem\n| tmul a b =>\n    simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype]\n    suffices a \u2297\u209c[R] (b : B) = (a \u2297\u209c[R] (1 : B)) * ((1 : A) \u2297\u209c[R] (b : B)) by\n      rw [this]\n      simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,\n        Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem]\n      apply Ideal.mul_mem_left\n      apply Ideal.mem_map_of_mem includeRight\n      exact Submodule.coe_mem b\n    simp only [Submodule.coe_restrictScalars, Algebra.TensorProduct.tmul_mul_tmul,\n      mul_one, one_mul]\n| add x y hx hy =>\n    rw [map_add]\n    apply Submodule.add_mem _ hx hy", "annotated_tactic": ["induction y with\n    | zero =>\n        rw [<a>map_zero</a>]\n        apply <a>zero_mem</a>\n    | tmul a b =>\n        simp only [<a>LinearMap.lTensor_tmul</a>, <a>Submodule.coeSubtype</a>]\n        suffices a \u2297\u209c[R] (b : B) = (a \u2297\u209c[R] (1 : B)) * ((1 : A) \u2297\u209c[R] (b : B)) by\n          rw [this]\n          simp only [<a>AddSubsemigroup.mem_carrier</a>, <a>AddSubmonoid.mem_toSubsemigroup</a>,\n            <a>Submodule.mem_toAddSubmonoid</a>, <a>Submodule.restrictScalars_mem</a>]\n          apply <a>Ideal.mul_mem_left</a>\n          -- Note: adding `includeRight` as a hint fixes a timeout #8386\n          apply <a>Ideal.mem_map_of_mem</a> <a>includeRight</a>\n          exact <a>Submodule.coe_mem</a> b\n        simp only [<a>Submodule.coe_restrictScalars</a>, <a>Algebra.TensorProduct.tmul_mul_tmul</a>,\n          <a>mul_one</a>, <a>one_mul</a>]\n    | add x y hx hy =>\n        rw [<a>map_add</a>]\n        apply <a>Submodule.add_mem</a> _ hx hy", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}, {"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "AddSubsemigroup.mem_carrier", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [111, 3], "def_end_pos": [111, 14]}, {"full_name": "AddSubmonoid.mem_toSubsemigroup", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [161, 3], "def_end_pos": [161, 14]}, {"full_name": "Submodule.mem_toAddSubmonoid", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 27]}, {"full_name": "Submodule.restrictScalars_mem", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [49, 9], "def_end_pos": [49, 28]}, {"full_name": "Ideal.mul_mem_left", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}, {"full_name": "Ideal.mem_map_of_mem", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "Algebra.TensorProduct.includeRight", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 17]}, {"full_name": "Submodule.coe_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 16]}, {"full_name": "Submodule.coe_restrictScalars", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [39, 9], "def_end_pos": [39, 28]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [220, 19], "def_end_pos": [220, 26]}]], "state_before": "case a.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\ny : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "no goals"}, {"tactic": "rw [map_zero]", "annotated_tactic": ["rw [<a>map_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "case a.intro.zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "case a.intro.zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 0 \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "apply zero_mem", "annotated_tactic": ["apply <a>zero_mem</a>", [{"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}]], "state_before": "case a.intro.zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\n\u22a2 0 \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "no goals"}, {"tactic": "simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype]", "annotated_tactic": ["simp only [<a>LinearMap.lTensor_tmul</a>, <a>Submodule.coeSubtype</a>]", [{"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}]], "state_before": "case a.intro.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (a \u2297\u209c[R] b) \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "case a.intro.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\n\u22a2 a \u2297\u209c[R] \u2191b \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "suffices a \u2297\u209c[R] (b : B) = (a \u2297\u209c[R] (1 : B)) * ((1 : A) \u2297\u209c[R] (b : B)) by\n  rw [this]\n  simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,\n    Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem]\n  apply Ideal.mul_mem_left\n  apply Ideal.mem_map_of_mem includeRight\n  exact Submodule.coe_mem b", "annotated_tactic": ["suffices a \u2297\u209c[R] (b : B) = (a \u2297\u209c[R] (1 : B)) * ((1 : A) \u2297\u209c[R] (b : B)) by\n          rw [this]\n          simp only [<a>AddSubsemigroup.mem_carrier</a>, <a>AddSubmonoid.mem_toSubsemigroup</a>,\n            <a>Submodule.mem_toAddSubmonoid</a>, <a>Submodule.restrictScalars_mem</a>]\n          apply <a>Ideal.mul_mem_left</a>\n          -- Note: adding `includeRight` as a hint fixes a timeout #8386\n          apply <a>Ideal.mem_map_of_mem</a> <a>includeRight</a>\n          exact <a>Submodule.coe_mem</a> b", [{"full_name": "AddSubsemigroup.mem_carrier", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [111, 3], "def_end_pos": [111, 14]}, {"full_name": "AddSubmonoid.mem_toSubsemigroup", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [161, 3], "def_end_pos": [161, 14]}, {"full_name": "Submodule.mem_toAddSubmonoid", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 27]}, {"full_name": "Submodule.restrictScalars_mem", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [49, 9], "def_end_pos": [49, 28]}, {"full_name": "Ideal.mul_mem_left", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}, {"full_name": "Ideal.mem_map_of_mem", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "Algebra.TensorProduct.includeRight", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 17]}, {"full_name": "Submodule.coe_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 16]}]], "state_before": "case a.intro.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\n\u22a2 a \u2297\u209c[R] \u2191b \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "case a.intro.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\n\u22a2 a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b"}, {"tactic": "simp only [Submodule.coe_restrictScalars, Algebra.TensorProduct.tmul_mul_tmul,\n  mul_one, one_mul]", "annotated_tactic": ["simp only [<a>Submodule.coe_restrictScalars</a>, <a>Algebra.TensorProduct.tmul_mul_tmul</a>,\n          <a>mul_one</a>, <a>one_mul</a>]", [{"full_name": "Submodule.coe_restrictScalars", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [39, 9], "def_end_pos": [39, 28]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case a.intro.tmul\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\n\u22a2 a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 a \u2297\u209c[R] \u2191b \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,\n  Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem]", "annotated_tactic": ["simp only [<a>AddSubsemigroup.mem_carrier</a>, <a>AddSubmonoid.mem_toSubsemigroup</a>,\n            <a>Submodule.mem_toAddSubmonoid</a>, <a>Submodule.restrictScalars_mem</a>]", [{"full_name": "AddSubsemigroup.mem_carrier", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [111, 3], "def_end_pos": [111, 14]}, {"full_name": "AddSubmonoid.mem_toSubsemigroup", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [161, 3], "def_end_pos": [161, 14]}, {"full_name": "Submodule.mem_toAddSubmonoid", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 27]}, {"full_name": "Submodule.restrictScalars_mem", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [49, 9], "def_end_pos": [49, 28]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b \u2208 (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b \u2208 map includeRight I"}, {"tactic": "apply Ideal.mul_mem_left", "annotated_tactic": ["apply <a>Ideal.mul_mem_left</a>", [{"full_name": "Ideal.mul_mem_left", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b \u2208 map includeRight I", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 1 \u2297\u209c[R] \u2191b \u2208 map includeRight I"}, {"tactic": "apply Ideal.mem_map_of_mem includeRight", "annotated_tactic": ["apply <a>Ideal.mem_map_of_mem</a> <a>includeRight</a>", [{"full_name": "Ideal.mem_map_of_mem", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "Algebra.TensorProduct.includeRight", "def_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "def_pos": [408, 5], "def_end_pos": [408, 17]}]], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 1 \u2297\u209c[R] \u2191b \u2208 map includeRight I", "state_after": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 \u2191b \u2208 I"}, {"tactic": "exact Submodule.coe_mem b", "annotated_tactic": ["exact <a>Submodule.coe_mem</a> b", [{"full_name": "Submodule.coe_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 16]}]], "state_before": "case a\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\na : A\nb : \u21a5(Submodule.restrictScalars R I)\nthis : a \u2297\u209c[R] \u2191b = a \u2297\u209c[R] 1 * 1 \u2297\u209c[R] \u2191b\n\u22a2 \u2191b \u2208 I", "state_after": "no goals"}, {"tactic": "rw [map_add]", "annotated_tactic": ["rw [<a>map_add</a>]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "case a.intro.add\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\nhy :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y) \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "case a.intro.add\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\nhy :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier"}, {"tactic": "apply Submodule.add_mem _ hx hy", "annotated_tactic": ["apply <a>Submodule.add_mem</a> _ hx hy", [{"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [220, 19], "def_end_pos": [220, 26]}]], "state_before": "case a.intro.add\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nI : Ideal B\nx y : A \u2297[R] \u21a5(Submodule.restrictScalars R I)\nhx :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\nhy :\n  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier\n\u22a2 (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +\n      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y \u2208\n    (Submodule.restrictScalars R (map includeRight I)).carrier", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.einfsep_lt_top_iff", "start": [268, 1], "end": [269, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.pair_preimage", "start": [415, 1], "end": [417, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EpiMono.lean", "full_name": "CategoryTheory.isSplitMono_of_mono", "start": [238, 1], "end": [239, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "fderivWithin_cos", "start": [960, 1], "end": [962, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "full_name": "MvPFunctor.M.bisim'", "start": [302, 1], "end": [314, 20], "traced_tactics": [{"tactic": "have := M.bisim\u2080 P (EqvGen R) ?_ ?_", "annotated_tactic": ["have := <a>M.bisim\u2080</a> P (<a>EqvGen</a> R) ?_ ?_", [{"full_name": "MvPFunctor.M.bisim\u2080", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [270, 9], "def_end_pos": [270, 17]}, {"full_name": "EqvGen", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [21, 11], "def_end_pos": [21, 17]}]], "state_before": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\n\u22a2 x = y", "state_after": "case refine_3\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\nthis : \u2200 (x y : P.M \u03b1), EqvGen R x y \u2192 x = y\n\u22a2 x = y\n\ncase refine_1\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\n\u22a2 Equivalence (EqvGen R)\n\ncase refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\n\u22a2 \u2200 (x y : P.M \u03b1),\n    EqvGen R x y \u2192 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y"}, {"tactic": "solve_by_elim [EqvGen.rel]", "annotated_tactic": ["solve_by_elim [<a>EqvGen.rel</a>]", [{"full_name": "EqvGen.rel", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [22, 5], "def_end_pos": [22, 8]}]], "state_before": "case refine_3\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\nthis : \u2200 (x y : P.M \u03b1), EqvGen R x y \u2192 x = y\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "apply EqvGen.is_equivalence", "annotated_tactic": ["apply <a>EqvGen.is_equivalence</a>", [{"full_name": "EqvGen.is_equivalence", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [28, 9], "def_end_pos": [28, 30]}]], "state_before": "case refine_1\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\n\u22a2 Equivalence (EqvGen R)", "state_after": "no goals"}, {"tactic": "clear r x y", "annotated_tactic": ["clear r x y", []], "state_before": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nr : R x y\n\u22a2 \u2200 (x y : P.M \u03b1),\n    EqvGen R x y \u2192 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y", "state_after": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\n\u22a2 \u2200 (x y : P.M \u03b1),\n    EqvGen R x y \u2192 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y"}, {"tactic": "introv Hr", "annotated_tactic": ["introv Hr", []], "state_before": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\n\u22a2 \u2200 (x y : P.M \u03b1),\n    EqvGen R x y \u2192 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y", "state_after": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nHr : EqvGen R x y\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y"}, {"tactic": "have : \u2200 x y, R x y \u2192 EqvGen R x y := @EqvGen.rel _ R", "annotated_tactic": ["have : \u2200 x y, R x y \u2192 <a>EqvGen</a> R x y := @<a>EqvGen.rel</a> _ R", [{"full_name": "EqvGen", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [21, 11], "def_end_pos": [21, 17]}, {"full_name": "EqvGen.rel", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [22, 5], "def_end_pos": [22, 8]}]], "state_before": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nHr : EqvGen R x y\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y", "state_after": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nHr : EqvGen R x y\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y"}, {"tactic": "induction Hr", "annotated_tactic": ["induction Hr", []], "state_before": "case refine_2\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nHr : EqvGen R x y\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y", "state_after": "case refine_2.rel\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : R x\u271d y\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\n\ncase refine_2.refl\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d : P.M \u03b1\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d\n\ncase refine_2.symm\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : EqvGen R x\u271d y\u271d\na_ih\u271d : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d\n\ncase refine_2.trans\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d z\u271d : P.M \u03b1\na\u271d\u00b9 : EqvGen R x\u271d y\u271d\na\u271d : EqvGen R y\u271d z\u271d\na_ih\u271d\u00b9 : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\na_ih\u271d : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d"}, {"tactic": "all_goals aesop", "annotated_tactic": ["all_goals aesop", []], "state_before": "case refine_2.refl\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d : P.M \u03b1\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d\n\ncase refine_2.symm\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : EqvGen R x\u271d y\u271d\na_ih\u271d : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d\n\ncase refine_2.trans\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d z\u271d : P.M \u03b1\na\u271d\u00b9 : EqvGen R x\u271d y\u271d\na\u271d : EqvGen R y\u271d z\u271d\na_ih\u271d\u00b9 : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\na_ih\u271d : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d", "state_after": "no goals"}, {"tactic": "rw [\u2190 Quot.factor_mk_eq R (EqvGen R) this]", "annotated_tactic": ["rw [\u2190 <a>Quot.factor_mk_eq</a> R (<a>EqvGen</a> R) this]", [{"full_name": "Quot.factor_mk_eq", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [90, 9], "def_end_pos": [90, 21]}, {"full_name": "EqvGen", "def_path": "Mathlib/Init/Data/Quot.lean", "def_pos": [21, 11], "def_end_pos": [21, 17]}]], "state_before": "case refine_2.rel\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : R x\u271d y\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d", "state_after": "case refine_2.rel\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : R x\u271d y\u271d\n\u22a2 (TypeVec.id ::: Quot.factor R (EqvGen R) this \u2218 Quot.mk R) <$$> dest P x\u271d =\n    (TypeVec.id ::: Quot.factor R (EqvGen R) this \u2218 Quot.mk R) <$$> dest P y\u271d"}, {"tactic": "rwa [appendFun_comp_id, \u2190 MvFunctor.map_map, \u2190 MvFunctor.map_map, h]", "annotated_tactic": ["rwa [<a>appendFun_comp_id</a>, \u2190 <a>MvFunctor.map_map</a>, \u2190 <a>MvFunctor.map_map</a>, h]", [{"full_name": "TypeVec.appendFun_comp_id", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [270, 9], "def_end_pos": [270, 26]}, {"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [114, 9], "def_end_pos": [114, 16]}, {"full_name": "MvFunctor.map_map", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [114, 9], "def_end_pos": [114, 16]}]], "state_before": "case refine_2.rel\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d : P.M \u03b1\na\u271d : R x\u271d y\u271d\n\u22a2 (TypeVec.id ::: Quot.factor R (EqvGen R) this \u2218 Quot.mk R) <$$> dest P x\u271d =\n    (TypeVec.id ::: Quot.factor R (EqvGen R) this \u2218 Quot.mk R) <$$> dest P y\u271d", "state_after": "no goals"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "case refine_2.trans\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\nR : P.M \u03b1 \u2192 P.M \u03b1 \u2192 Prop\nh : \u2200 (x y : P.M \u03b1), R x y \u2192 (TypeVec.id ::: Quot.mk R) <$$> dest P x = (TypeVec.id ::: Quot.mk R) <$$> dest P y\nx y : P.M \u03b1\nthis : \u2200 (x y : P.M \u03b1), R x y \u2192 EqvGen R x y\nx\u271d y\u271d z\u271d : P.M \u03b1\na\u271d\u00b9 : EqvGen R x\u271d y\u271d\na\u271d : EqvGen R y\u271d z\u271d\na_ih\u271d\u00b9 : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d\na_ih\u271d : (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P y\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d\n\u22a2 (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P x\u271d = (TypeVec.id ::: Quot.mk (EqvGen R)) <$$> dest P z\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Parity.lean", "full_name": "Nat.even_or_odd'", "start": [281, 1], "end": [282, 67], "traced_tactics": [{"tactic": "simpa only [\u2190 two_mul, exists_or, Odd, Even] using even_or_odd n", "annotated_tactic": ["simpa only [\u2190 <a>two_mul</a>, <a>exists_or</a>, <a>Odd</a>, <a>Even</a>] using <a>even_or_odd</a> n", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [179, 9], "def_end_pos": [179, 16]}, {"full_name": "exists_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}, {"full_name": "Odd", "def_path": "Mathlib/Algebra/Ring/Parity.lean", "def_pos": [111, 5], "def_end_pos": [111, 8]}, {"full_name": "Even", "def_path": "Mathlib/Algebra/Group/Even.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Nat.even_or_odd", "def_path": "Mathlib/Algebra/Ring/Parity.lean", "def_pos": [278, 7], "def_end_pos": [278, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n\u271d n : \u2115\n\u22a2 \u2203 k, n = 2 * k \u2228 n = 2 * k + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "full_name": "AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_add_smoothingFn", "start": [900, 1], "end": [906, 58], "traced_tactics": [{"tactic": "refine IsTheta.norm_left ?_", "annotated_tactic": ["refine <a>IsTheta.norm_left</a> ?_", [{"full_name": "Asymptotics.IsTheta.norm_left", "def_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "def_pos": [158, 30], "def_end_pos": [158, 47]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\np : \u211d\nhp : p \u2260 0\n\u22a2 (fun x => \u2016deriv (fun z => z ^ p * (1 + \u03b5 z)) x\u2016) =\u0398[atTop] fun z => z ^ (p - 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\np : \u211d\nhp : p \u2260 0\n\u22a2 (deriv fun z => z ^ p * (1 + \u03b5 z)) =\u0398[atTop] fun z => z ^ (p - 1)"}, {"tactic": "calc (fun x => deriv (fun z => z ^ p * (1 + \u03b5 z)) x) =\u0398[atTop] fun z => p * z ^ (p-1) :=\n          (isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn hp).isTheta\n  _ =\u0398[atTop] fun z => z ^ (p-1) :=\n          IsTheta.const_mul_left hp <| isTheta_refl _ _", "annotated_tactic": ["calc (fun x => <a>deriv</a> (fun z => z ^ p * (1 + \u03b5 z)) x) =\u0398[<a>atTop</a>] fun z => p * z ^ (p-1) :=\n            (<a>isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn</a> hp).<a>isTheta</a>\n    _ =\u0398[<a>atTop</a>] fun z => z ^ (p-1) :=\n            <a>IsTheta.const_mul_left</a> hp <| <a>isTheta_refl</a> _ _", [{"full_name": "deriv", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [150, 5], "def_end_pos": [150, 10]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn", "def_path": "Mathlib/Computability/AkraBazzi/AkraBazzi.lean", "def_pos": [868, 7], "def_end_pos": [868, 56]}, {"full_name": "Asymptotics.IsEquivalent.isTheta", "def_path": "Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean", "def_pos": [95, 9], "def_end_pos": [95, 29]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Asymptotics.IsTheta.const_mul_left", "def_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "def_pos": [312, 35], "def_end_pos": [312, 57]}, {"full_name": "Asymptotics.isTheta_refl", "def_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "def_pos": [60, 9], "def_end_pos": [60, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Nonempty \u03b1\nT : \u2115 \u2192 \u211d\ng : \u211d \u2192 \u211d\na b : \u03b1 \u2192 \u211d\nr : \u03b1 \u2192 \u2115 \u2192 \u2115\nR : AkraBazziRecurrence T g a b r\np : \u211d\nhp : p \u2260 0\n\u22a2 (deriv fun z => z ^ p * (1 + \u03b5 z)) =\u0398[atTop] fun z => z ^ (p - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "isOpenMap_mul_right", "start": [122, 1], "end": [123, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "Antitone.tendsto_mulIndicator", "start": [82, 1], "end": [85, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_comm", "start": [83, 1], "end": [85, 33], "traced_tactics": [{"tactic": "unfold angle", "annotated_tactic": ["unfold <a>angle</a>", [{"full_name": "InnerProductGeometry.angle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 angle x y = angle y x", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (\u27eax, y\u27eb_\u211d / (\u2016x\u2016 * \u2016y\u2016)) = arccos (\u27eay, x\u27eb_\u211d / (\u2016y\u2016 * \u2016x\u2016))"}, {"tactic": "rw [real_inner_comm, mul_comm]", "annotated_tactic": ["rw [<a>real_inner_comm</a>, <a>mul_comm</a>]", [{"full_name": "real_inner_comm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 24]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (\u27eax, y\u27eb_\u211d / (\u2016x\u2016 * \u2016y\u2016)) = arccos (\u27eay, x\u27eb_\u211d / (\u2016y\u2016 * \u2016x\u2016))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/ToDFinsupp.lean", "full_name": "Finsupp.toDFinsupp_neg", "start": [162, 1], "end": [163, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sort.lean", "full_name": "List.Sorted.rel_get_of_lt", "start": [134, 1], "end": [136, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/ZPowers.lean", "full_name": "Subgroup.mem_zpowers_iff", "start": [56, 1], "end": [57, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Membership.lean", "full_name": "Subsemigroup.mem_sSup_of_directed_on", "start": [67, 1], "end": [70, 17], "traced_tactics": [{"tactic": "simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,\n  exists_prop]", "annotated_tactic": ["simp only [<a>sSup_eq_iSup'</a>, <a>mem_iSup_of_directed</a> hS.directed_val, <a>SetCoe.exists</a>, <a>Subtype.coe_mk</a>,\n    <a>exists_prop</a>]", [{"full_name": "sSup_eq_iSup'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [565, 9], "def_end_pos": [565, 22]}, {"full_name": "Subsemigroup.mem_iSup_of_directed", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Membership.lean", "def_pos": [47, 9], "def_end_pos": [47, 29]}, {"full_name": "SetCoe.exists", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [167, 9], "def_end_pos": [167, 22]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03b9 : Sort u_1\nM : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d : Mul M\nS : Set (Subsemigroup M)\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 sSup S \u2194 \u2203 s \u2208 S, x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "full_name": "ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus", "start": [161, 1], "end": [163, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.card_join", "start": [77, 1], "end": [78, 46], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nS : Multiset (Multiset \u03b1)\n\u22a2 card (join 0) = (map (\u21d1card) 0).sum", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nS : Multiset (Multiset \u03b1)\n\u22a2 \u2200 (a : Multiset \u03b1) (s : Multiset (Multiset \u03b1)),\n    card s.join = (map (\u21d1card) s).sum \u2192 card (a ::\u2098 s).join = (map (\u21d1card) (a ::\u2098 s)).sum", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "add_le_mul", "start": [925, 1], "end": [927, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Join.lean", "full_name": "convexHull_union", "start": [197, 1], "end": [200, 101], "traced_tactics": [{"tactic": "rw [\u2190 convexHull_convexHull_union_left, \u2190 convexHull_convexHull_union_right]", "annotated_tactic": ["rw [\u2190 <a>convexHull_convexHull_union_left</a>, \u2190 <a>convexHull_convexHull_union_right</a>]", [{"full_name": "convexHull_convexHull_union_left", "def_path": "Mathlib/Analysis/Convex/Hull.lean", "def_pos": [134, 9], "def_end_pos": [134, 41]}, {"full_name": "convexHull_convexHull_union_right", "def_path": "Mathlib/Analysis/Convex/Hull.lean", "def_pos": [139, 9], "def_end_pos": [139, 42]}]], "state_before": "\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : s.Nonempty\nht : t.Nonempty\n\u22a2 (convexHull \ud835\udd5c) (s \u222a t) = convexJoin \ud835\udd5c ((convexHull \ud835\udd5c) s) ((convexHull \ud835\udd5c) t)", "state_after": "\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : s.Nonempty\nht : t.Nonempty\n\u22a2 (convexHull \ud835\udd5c) ((convexHull \ud835\udd5c) s \u222a (convexHull \ud835\udd5c) t) = convexJoin \ud835\udd5c ((convexHull \ud835\udd5c) s) ((convexHull \ud835\udd5c) t)"}, {"tactic": "exact (convex_convexHull \ud835\udd5c s).convexHull_union (convex_convexHull \ud835\udd5c t) hs.convexHull ht.convexHull", "annotated_tactic": ["exact (<a>convex_convexHull</a> \ud835\udd5c s).<a>convexHull_union</a> (<a>convex_convexHull</a> \ud835\udd5c t) hs.convexHull ht.convexHull", [{"full_name": "convex_convexHull", "def_path": "Mathlib/Analysis/Convex/Hull.lean", "def_pos": [53, 9], "def_end_pos": [53, 26]}, {"full_name": "Convex.convexHull_union", "def_path": "Mathlib/Analysis/Convex/Join.lean", "def_pos": [190, 19], "def_end_pos": [190, 42]}, {"full_name": "convex_convexHull", "def_path": "Mathlib/Analysis/Convex/Hull.lean", "def_pos": [53, 9], "def_end_pos": [53, 26]}]], "state_before": "\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : s.Nonempty\nht : t.Nonempty\n\u22a2 (convexHull \ud835\udd5c) ((convexHull \ud835\udd5c) s \u222a (convexHull \ud835\udd5c) t) = convexJoin \ud835\udd5c ((convexHull \ud835\udd5c) s) ((convexHull \ud835\udd5c) t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Multiset.lean", "full_name": "DFinsupp.toMultiset_injective", "start": [132, 1], "end": [133, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "t2_iff_ultrafilter", "start": [1492, 1], "end": [1494, 96], "traced_tactics": [{"tactic": "simp only [\u2190 exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]", "annotated_tactic": ["simp only [\u2190 <a>exists_ultrafilter_iff</a>, <a>and_imp</a>, <a>le_inf_iff</a>, <a>exists_imp</a>]", [{"full_name": "Filter.exists_ultrafilter_iff", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [470, 9], "def_end_pos": [470, 31]}, {"full_name": "and_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [115, 17], "def_end_pos": [115, 24]}, {"full_name": "le_inf_iff", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [394, 9], "def_end_pos": [394, 19]}, {"full_name": "exists_imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [200, 9], "def_end_pos": [200, 19]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\n\u22a2 (\u2200 {x y : X}, (\ud835\udcdd x \u2293 \ud835\udcdd y).NeBot \u2192 x = y) \u2194 \u2200 {x y : X} (f : Ultrafilter X), \u2191f \u2264 \ud835\udcdd x \u2192 \u2191f \u2264 \ud835\udcdd y \u2192 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.inter_get_eq", "start": [835, 1], "end": [837, 26], "traced_tactics": [{"tactic": "simp [inter_def]", "annotated_tactic": ["simp [<a>inter_def</a>]", [{"full_name": "Part.inter_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [706, 9], "def_end_pos": [706, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 (a \u2229 b).get hab = a.get \u22ef \u2229 b.get \u22ef", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 (a.bind fun y => map (fun x => y \u2229 x) b).get \u22ef = a.get \u22ef \u2229 b.get \u22ef"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : Part \u03b1\nhab : (a \u2229 b).Dom\n\u22a2 (a.bind fun y => map (fun x => y \u2229 x) b).get \u22ef = a.get \u22ef \u2229 b.get \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pi.lean", "full_name": "Finset.pi_subset", "start": [131, 1], "end": [132, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.biprod.conePointUniqueUpToIso_inv", "start": [1848, 1], "end": [1856, 17], "traced_tactics": [{"tactic": "refine biprod.hom_ext' _ _ (hb.isLimit.hom_ext fun j => ?_) (hb.isLimit.hom_ext fun j => ?_)", "annotated_tactic": ["refine <a>biprod.hom_ext'</a> _ _ (hb.isLimit.hom_ext fun j => ?_) (hb.isLimit.hom_ext fun j => ?_)", [{"full_name": "CategoryTheory.Limits.biprod.hom_ext'", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [1725, 9], "def_end_pos": [1725, 24]}]], "state_before": "J : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv = desc b.inl b.inr", "state_after": "case refine_1\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 (inl \u226b (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv) \u226b b.toCone.\u03c0.app j =\n    (inl \u226b desc b.inl b.inr) \u226b b.toCone.\u03c0.app j\n\ncase refine_2\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 (inr \u226b (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv) \u226b b.toCone.\u03c0.app j =\n    (inr \u226b desc b.inl b.inr) \u226b b.toCone.\u03c0.app j"}, {"tactic": "all_goals\n  simp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]\n  rcases j with \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["all_goals\n    simp only [<a>Category.assoc</a>, <a>IsLimit.conePointUniqueUpToIso_inv_comp</a>]\n    rcases j with \u27e8\u27e8\u27e9\u27e9", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_inv_comp", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [159, 9], "def_end_pos": [159, 40]}]], "state_before": "case refine_1\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 (inl \u226b (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv) \u226b b.toCone.\u03c0.app j =\n    (inl \u226b desc b.inl b.inr) \u226b b.toCone.\u03c0.app j\n\ncase refine_2\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 (inr \u226b (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv) \u226b b.toCone.\u03c0.app j =\n    (inr \u226b desc b.inl b.inr) \u226b b.toCone.\u03c0.app j", "state_after": "case refine_1.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inl \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.left } =\n    inl \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.left }\n\ncase refine_1.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inl \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inl \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }\n\ncase refine_2.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.left } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.left }\n\ncase refine_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }"}, {"tactic": "all_goals simp", "annotated_tactic": ["all_goals simp", []], "state_before": "case refine_1.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inl \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.left } =\n    inl \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.left }\n\ncase refine_1.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inl \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inl \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }\n\ncase refine_2.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.left } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.left }\n\ncase refine_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }", "state_after": "no goals"}, {"tactic": "simp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]", "annotated_tactic": ["simp only [<a>Category.assoc</a>, <a>IsLimit.conePointUniqueUpToIso_inv_comp</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_inv_comp", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [159, 9], "def_end_pos": [159, 40]}]], "state_before": "case refine_2\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 (inr \u226b (hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit X Y)).inv) \u226b b.toCone.\u03c0.app j =\n    (inr \u226b desc b.inl b.inr) \u226b b.toCone.\u03c0.app j", "state_after": "case refine_2\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app j = inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app j"}, {"tactic": "rcases j with \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases j with \u27e8\u27e8\u27e9\u27e9", []], "state_before": "case refine_2\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\nj : Discrete WalkingPair\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app j = inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app j", "state_after": "case refine_2.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.left } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.left }\n\ncase refine_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : b.IsBilimit\n\u22a2 inr \u226b (BinaryBiproduct.bicone X Y).toCone.\u03c0.app { as := WalkingPair.right } =\n    inr \u226b desc b.inl b.inr \u226b b.toCone.\u03c0.app { as := WalkingPair.right }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Holder.lean", "full_name": "HolderOnWith.ediam_image_le_of_subset", "start": [164, 1], "end": [166, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.toMeasure_ofFinset_apply", "start": [191, 1], "end": [193, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Basic.lean", "full_name": "Polynomial.natCast_mul", "start": [963, 1], "end": [964, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Cast/Order.lean", "full_name": "Nat.abs_ofNat", "start": [232, 1], "end": [234, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.sup", "start": [762, 1], "end": [765, 18], "traced_tactics": [{"tactic": "rw [\u2190 mem\u2112p_one_iff_integrable] at hf hg \u22a2", "annotated_tactic": ["rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>] at hf hg \u22a2", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 9], "def_end_pos": [442, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\u271d\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\n\u03b2 : Type u_5\ninst\u271d : NormedLatticeAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\n\u22a2 Integrable (f \u2294 g) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\u271d\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\n\u03b2 : Type u_5\ninst\u271d : NormedLatticeAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1 \u03bc\nhg : Mem\u2112p g 1 \u03bc\n\u22a2 Mem\u2112p (f \u2294 g) 1 \u03bc"}, {"tactic": "exact hf.sup hg", "annotated_tactic": ["exact hf.sup hg", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : NormedAddCommGroup \u03b2\u271d\ninst\u271d\u00b9 : NormedAddCommGroup \u03b3\n\u03b2 : Type u_5\ninst\u271d : NormedLatticeAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f 1 \u03bc\nhg : Mem\u2112p g 1 \u03bc\n\u22a2 Mem\u2112p (f \u2294 g) 1 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_const", "start": [990, 1], "end": [990, 94], "traced_tactics": [{"tactic": "rw [iSup, range_const, sSup_singleton]", "annotated_tactic": ["rw [<a>iSup</a>, <a>range_const</a>, <a>sSup_singleton</a>]", [{"full_name": "iSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [60, 5], "def_end_pos": [60, 9]}, {"full_name": "Set.range_const", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1006, 9], "def_end_pos": [1006, 20]}, {"full_name": "sSup_singleton", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b9\n\u22a2 \u2a06 x, a = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/FinsetOps.lean", "full_name": "Multiset.ndinter_subset_right", "start": [262, 1], "end": [263, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "full_name": "Pell.y_dvd_iff", "start": [436, 1], "end": [452, 46], "traced_tactics": [{"tactic": "have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=\n  Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))", "annotated_tactic": ["have co : <a>Nat.Coprime</a> (<a>yn</a> a1 m) (<a>xn</a> a1 (m * (n / m))) :=\n          <a>Nat.Coprime.symm</a> <| (<a>xy_coprime</a> a1 _).<a>coprime_dvd_right</a> (<a>y_mul_dvd</a> a1 m (n / m))", [{"full_name": "Nat.Coprime", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [20, 18], "def_end_pos": [20, 25]}, {"full_name": "Pell.yn", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [117, 5], "def_end_pos": [117, 7]}, {"full_name": "Pell.xn", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [112, 5], "def_end_pos": [112, 7]}, {"full_name": "Nat.Coprime.symm", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [28, 9], "def_end_pos": [28, 21]}, {"full_name": "Pell.xy_coprime", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [395, 9], "def_end_pos": [395, 19]}, {"full_name": "Nat.Coprime.coprime_dvd_right", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [93, 9], "def_end_pos": [93, 34]}, {"full_name": "Pell.y_mul_dvd", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [430, 9], "def_end_pos": [430, 18]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\n\u22a2 False", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\n\u22a2 False"}, {"tactic": "have m0 : 0 < m :=\n  m.eq_zero_or_pos.resolve_left fun e => by\n    rw [e, Nat.mod_zero] at hp;rw [e] at h\n    exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm", "annotated_tactic": ["have m0 : 0 < m :=\n          m.eq_zero_or_pos.resolve_left fun e => by\n            rw [e, <a>Nat.mod_zero</a>] at hp;rw [e] at h\n            exact <a>_root_.ne_of_lt</a> (<a>strictMono_y</a> a1 hp) (<a>eq_zero_of_zero_dvd</a> h).<a>symm</a>", [{"full_name": "Nat.mod_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [125, 17], "def_end_pos": [125, 25]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Pell.strictMono_y", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [402, 9], "def_end_pos": [402, 21]}, {"full_name": "eq_zero_of_zero_dvd", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [27, 9], "def_end_pos": [27, 28]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\n\u22a2 False", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False"}, {"tactic": "rw [\u2190 Nat.mod_add_div n m, yn_add] at h", "annotated_tactic": ["rw [\u2190 <a>Nat.mod_add_div</a> n m, <a>yn_add</a>] at h", [{"full_name": "Nat.mod_add_div", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [201, 9], "def_end_pos": [201, 20]}, {"full_name": "Pell.yn_add", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [371, 9], "def_end_pos": [371, 15]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 xn a1 (n % m) * yn a1 (m * (n / m)) + yn a1 (n % m) * xn a1 (m * (n / m))\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False"}, {"tactic": "exact\n  not_le_of_gt (strictMono_y _ <| Nat.mod_lt n m0)\n    (Nat.le_of_dvd (strictMono_y _ hp) <|\n      co.dvd_of_dvd_mul_right <|\n        (Nat.dvd_add_iff_right <| (y_mul_dvd _ _ _).mul_left _).2 h)", "annotated_tactic": ["exact\n          <a>not_le_of_gt</a> (<a>strictMono_y</a> _ <| <a>Nat.mod_lt</a> n m0)\n            (<a>Nat.le_of_dvd</a> (<a>strictMono_y</a> _ hp) <|\n              co.dvd_of_dvd_mul_right <|\n                (<a>Nat.dvd_add_iff_right</a> <| (<a>y_mul_dvd</a> _ _ _).<a>mul_left</a> _).2 h)", [{"full_name": "not_le_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [148, 9], "def_end_pos": [148, 21]}, {"full_name": "Pell.strictMono_y", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [402, 9], "def_end_pos": [402, 21]}, {"full_name": "Nat.mod_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [142, 9], "def_end_pos": [142, 15]}, {"full_name": "Nat.le_of_dvd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [46, 9], "def_end_pos": [46, 18]}, {"full_name": "Pell.strictMono_y", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [402, 9], "def_end_pos": [402, 21]}, {"full_name": "Nat.dvd_add_iff_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [33, 19], "def_end_pos": [33, 36]}, {"full_name": "Pell.y_mul_dvd", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [430, 9], "def_end_pos": [430, 18]}, {"full_name": "Dvd.dvd.mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [213, 7], "def_end_pos": [213, 23]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 xn a1 (n % m) * yn a1 (m * (n / m)) + yn a1 (n % m) * xn a1 (m * (n / m))\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [e, Nat.mod_zero] at hp", "annotated_tactic": ["rw [e, <a>Nat.mod_zero</a>] at hp", [{"full_name": "Nat.mod_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [125, 17], "def_end_pos": [125, 25]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False"}, {"tactic": "rw [e] at h", "annotated_tactic": ["rw [e] at h", []], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 0 \u2223 yn a1 n\nhp : n > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False"}, {"tactic": "exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm", "annotated_tactic": ["exact <a>_root_.ne_of_lt</a> (<a>strictMono_y</a> a1 hp) (<a>eq_zero_of_zero_dvd</a> h).<a>symm</a>", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Pell.strictMono_y", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [402, 9], "def_end_pos": [402, 21]}, {"full_name": "eq_zero_of_zero_dvd", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [27, 9], "def_end_pos": [27, 28]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 0 \u2223 yn a1 n\nhp : n > 0\nco : (yn a1 m).Coprime (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [e]", "annotated_tactic": ["rw [e]", []], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nx\u271d : m \u2223 n\nk : \u2115\ne : n = m * k\n\u22a2 yn a1 m \u2223 yn a1 n", "state_after": "a : \u2115\na1 : 1 < a\nm n : \u2115\nx\u271d : m \u2223 n\nk : \u2115\ne : n = m * k\n\u22a2 yn a1 m \u2223 yn a1 (m * k)"}, {"tactic": "apply y_mul_dvd", "annotated_tactic": ["apply <a>y_mul_dvd</a>", [{"full_name": "Pell.y_mul_dvd", "def_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "def_pos": [430, 9], "def_end_pos": [430, 18]}]], "state_before": "a : \u2115\na1 : 1 < a\nm n : \u2115\nx\u271d : m \u2223 n\nk : \u2115\ne : n = m * k\n\u22a2 yn a1 m \u2223 yn a1 (m * k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Cast/Order.lean", "full_name": "Rat.preimage_cast_uIcc", "start": [125, 1], "end": [126, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorization/Root.lean", "full_name": "Nat.floorRoot_def", "start": [52, 1], "end": [56, 99], "traced_tactics": [{"tactic": "unfold floorRoot", "annotated_tactic": ["unfold <a>floorRoot</a>", [{"full_name": "Nat.floorRoot", "def_path": "Mathlib/Data/Nat/Factorization/Root.lean", "def_pos": [49, 5], "def_end_pos": [49, 14]}]], "state_before": "a b n : \u2115\n\u22a2 n.floorRoot a = if n = 0 \u2228 a = 0 then 0 else (a.factorization \u230a/\u230b n).prod fun x x_1 => x ^ x_1", "state_after": "a b n : \u2115\n\u22a2 (if n = 0 \u2228 a = 0 then 0 else a.factorization.prod fun p k => p ^ (k / n)) =\n    if n = 0 \u2228 a = 0 then 0 else (a.factorization \u230a/\u230b n).prod fun x x_1 => x ^ x_1"}, {"tactic": "split_ifs with h <;> simp [Finsupp.floorDiv_def, prod_mapRange_index pow_zero]", "annotated_tactic": ["split_ifs with h <;> simp [<a>Finsupp.floorDiv_def</a>, <a>prod_mapRange_index</a> <a>pow_zero</a>]", [{"full_name": "Finsupp.floorDiv_def", "def_path": "Mathlib/Algebra/Order/Floor/Div.lean", "def_pos": [240, 7], "def_end_pos": [240, 19]}, {"full_name": "Finsupp.prod_mapRange_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [80, 9], "def_end_pos": [80, 28]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}]], "state_before": "a b n : \u2115\n\u22a2 (if n = 0 \u2228 a = 0 then 0 else a.factorization.prod fun p k => p ^ (k / n)) =\n    if n = 0 \u2228 a = 0 then 0 else (a.factorization \u230a/\u230b n).prod fun x x_1 => x ^ x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_def", "start": [175, 1], "end": [176, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "StrictMono.mul_const'", "start": [1424, 1], "end": [1425, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Comp.lean", "full_name": "HasDerivWithinAt.comp_hasFDerivWithinAt_of_eq", "start": [217, 1], "end": [221, 58], "traced_tactics": [{"tactic": "rw [hy] at hh", "annotated_tactic": ["rw [hy] at hh", []], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns\u271d t\u271d : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\ny : \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\ns : Set E\nt : Set \ud835\udd5c'\nx : E\nhh : HasDerivWithinAt h\u2082 h\u2082' t y\nhf : HasFDerivWithinAt f f' s x\nhst : MapsTo f s t\nhy : y = f x\n\u22a2 HasFDerivWithinAt (h\u2082 \u2218 f) (h\u2082' \u2022 f') s x", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns\u271d t\u271d : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\ny : \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\ns : Set E\nt : Set \ud835\udd5c'\nx : E\nhh : HasDerivWithinAt h\u2082 h\u2082' t (f x)\nhf : HasFDerivWithinAt f f' s x\nhst : MapsTo f s t\nhy : y = f x\n\u22a2 HasFDerivWithinAt (h\u2082 \u2218 f) (h\u2082' \u2022 f') s x"}, {"tactic": "exact hh.comp_hasFDerivWithinAt x hf hst", "annotated_tactic": ["exact hh.comp_hasFDerivWithinAt x hf hst", []], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns\u271d t\u271d : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\ny : \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\ns : Set E\nt : Set \ud835\udd5c'\nx : E\nhh : HasDerivWithinAt h\u2082 h\u2082' t (f x)\nhf : HasFDerivWithinAt f f' s x\nhst : MapsTo f s t\nhy : y = f x\n\u22a2 HasFDerivWithinAt (h\u2082 \u2218 f) (h\u2082' \u2022 f') s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/MellinTransform.lean", "full_name": "hasMellin_sub", "start": [180, 1], "end": [183, 60], "traced_tactics": [{"tactic": "simpa only [MellinConvergent, smul_sub] using hf.sub hg", "annotated_tactic": ["simpa only [<a>MellinConvergent</a>, <a>smul_sub</a>] using hf.sub hg", [{"full_name": "MellinConvergent", "def_path": "Mathlib/Analysis/MellinTransform.lean", "def_pos": [43, 5], "def_end_pos": [43, 21]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : \u211d \u2192 E\ns : \u2102\nhf : MellinConvergent f s\nhg : MellinConvergent g s\n\u22a2 MellinConvergent (fun t => f t - g t) s", "state_after": "no goals"}, {"tactic": "simpa only [mellin, smul_sub] using integral_sub hf hg", "annotated_tactic": ["simpa only [<a>mellin</a>, <a>smul_sub</a>] using <a>integral_sub</a> hf hg", [{"full_name": "mellin", "def_path": "Mathlib/Analysis/MellinTransform.lean", "def_pos": [96, 5], "def_end_pos": [96, 11]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}, {"full_name": "MeasureTheory.integral_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [901, 9], "def_end_pos": [901, 21]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : \u211d \u2192 E\ns : \u2102\nhf : MellinConvergent f s\nhg : MellinConvergent g s\n\u22a2 mellin (fun t => f t - g t) s = mellin f s - mellin g s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Separation.lean", "full_name": "inseparable_iff_ker_uniformity", "start": [134, 1], "end": [135, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "pi_Ioi_mem_nhds", "start": [759, 1], "end": [760, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/PrimeIdeal.lean", "full_name": "Order.Ideal.PrimePair.I_isPrime", "start": [110, 1], "end": [114, 29], "traced_tactics": [{"tactic": "rw [IF.compl_I_eq_F]", "annotated_tactic": ["rw [IF.compl_I_eq_F]", []], "state_before": "P : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\n\u22a2 IsPFilter (\u2191IF.I)\u1d9c", "state_after": "P : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\n\u22a2 IsPFilter \u2191IF.F"}, {"tactic": "exact IF.F.isPFilter", "annotated_tactic": ["exact IF.F.isPFilter", []], "state_before": "P : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\n\u22a2 IsPFilter \u2191IF.F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.countP_zero", "start": [2284, 1], "end": [2285, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.comap_iSup", "start": [2439, 1], "end": [2440, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Int.addRight_one_isCycle", "start": [728, 1], "end": [729, 44], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2124\nx\u271d : (Equiv.addRight 1) n \u2260 n\n\u22a2 (Equiv.addRight 1 ^ n) 0 = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Minimal.lean", "full_name": "IsAntichain.maximals_lowerClosure", "start": [312, 1], "end": [314, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/ZPow.lean", "full_name": "deriv_zpow'", "start": [96, 1], "end": [97, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "InfHom.id_comp", "start": [629, 9], "end": [629, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometryEquiv.vadd_vsub", "start": [762, 1], "end": [765, 18], "traced_tactics": [{"tactic": "convert (vaddConst \ud835\udd5c (f p)).symm.isometry.comp (hf.comp (vaddConst \ud835\udd5c p).isometry)", "annotated_tactic": ["convert (<a>vaddConst</a> \ud835\udd5c (f p)).symm.isometry.comp (hf.comp (<a>vaddConst</a> \ud835\udd5c p).<a>isometry</a>)", [{"full_name": "AffineIsometryEquiv.vaddConst", "def_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "def_pos": [692, 5], "def_end_pos": [692, 14]}, {"full_name": "AffineIsometryEquiv.vaddConst", "def_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "def_pos": [692, 5], "def_end_pos": [692, 14]}, {"full_name": "AffineIsometryEquiv.isometry", "def_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "def_pos": [438, 19], "def_end_pos": [438, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\nV\u2081 : Type u_3\nV\u2081' : Type u_4\nV\u2082 : Type u_5\nV\u2083 : Type u_6\nV\u2084 : Type u_7\nP\u2081 : Type u_8\nP\u2081' : Type u_9\nP : Type u_10\nP\u2082 : Type u_11\nP\u2083 : Type u_12\nP\u2084 : Type u_13\ninst\u271d\u00b2\u2075 : NormedField \ud835\udd5c\ninst\u271d\u00b2\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b2\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b2\u00b9 : NormedAddTorsor V P\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c V\u2081\ninst\u271d\u00b9\u2078 : PseudoMetricSpace P\u2081\ninst\u271d\u00b9\u2077 : NormedAddTorsor V\u2081 P\u2081\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup V\u2081'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c V\u2081'\ninst\u271d\u00b9\u2074 : MetricSpace P\u2081'\ninst\u271d\u00b9\u00b3 : NormedAddTorsor V\u2081' P\u2081'\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c V\u2082\ninst\u271d\u00b9\u2070 : PseudoMetricSpace P\u2082\ninst\u271d\u2079 : NormedAddTorsor V\u2082 P\u2082\ninst\u271d\u2078 : SeminormedAddCommGroup V\u2083\ninst\u271d\u2077 : NormedSpace \ud835\udd5c V\u2083\ninst\u271d\u2076 : PseudoMetricSpace P\u2083\ninst\u271d\u2075 : NormedAddTorsor V\u2083 P\u2083\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2084\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c V\u2084\ninst\u271d\u00b2 : PseudoMetricSpace P\u2084\ninst\u271d\u00b9 : NormedAddTorsor V\u2084 P\u2084\ne : P \u2243\u1d43\u2071[\ud835\udd5c] P\u2082\n\u03b1 : Type u_14\ninst\u271d : TopologicalSpace \u03b1\nf : P \u2192 P\u2082\nhf : Isometry f\np : P\ng : V \u2192 V\u2082\nhg : \u2200 (v : V), g v = f (v +\u1d65 p) -\u1d65 f p\n\u22a2 Isometry g", "state_after": "case h.e'_5\n\ud835\udd5c : Type u_1\nV : Type u_2\nV\u2081 : Type u_3\nV\u2081' : Type u_4\nV\u2082 : Type u_5\nV\u2083 : Type u_6\nV\u2084 : Type u_7\nP\u2081 : Type u_8\nP\u2081' : Type u_9\nP : Type u_10\nP\u2082 : Type u_11\nP\u2083 : Type u_12\nP\u2084 : Type u_13\ninst\u271d\u00b2\u2075 : NormedField \ud835\udd5c\ninst\u271d\u00b2\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b2\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b2\u00b9 : NormedAddTorsor V P\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c V\u2081\ninst\u271d\u00b9\u2078 : PseudoMetricSpace P\u2081\ninst\u271d\u00b9\u2077 : NormedAddTorsor V\u2081 P\u2081\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup V\u2081'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c V\u2081'\ninst\u271d\u00b9\u2074 : MetricSpace P\u2081'\ninst\u271d\u00b9\u00b3 : NormedAddTorsor V\u2081' P\u2081'\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c V\u2082\ninst\u271d\u00b9\u2070 : PseudoMetricSpace P\u2082\ninst\u271d\u2079 : NormedAddTorsor V\u2082 P\u2082\ninst\u271d\u2078 : SeminormedAddCommGroup V\u2083\ninst\u271d\u2077 : NormedSpace \ud835\udd5c V\u2083\ninst\u271d\u2076 : PseudoMetricSpace P\u2083\ninst\u271d\u2075 : NormedAddTorsor V\u2083 P\u2083\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2084\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c V\u2084\ninst\u271d\u00b2 : PseudoMetricSpace P\u2084\ninst\u271d\u00b9 : NormedAddTorsor V\u2084 P\u2084\ne : P \u2243\u1d43\u2071[\ud835\udd5c] P\u2082\n\u03b1 : Type u_14\ninst\u271d : TopologicalSpace \u03b1\nf : P \u2192 P\u2082\nhf : Isometry f\np : P\ng : V \u2192 V\u2082\nhg : \u2200 (v : V), g v = f (v +\u1d65 p) -\u1d65 f p\n\u22a2 g = \u21d1(vaddConst \ud835\udd5c (f p)).symm \u2218 f \u2218 \u21d1(vaddConst \ud835\udd5c p)"}, {"tactic": "exact funext hg", "annotated_tactic": ["exact <a>funext</a> hg", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "case h.e'_5\n\ud835\udd5c : Type u_1\nV : Type u_2\nV\u2081 : Type u_3\nV\u2081' : Type u_4\nV\u2082 : Type u_5\nV\u2083 : Type u_6\nV\u2084 : Type u_7\nP\u2081 : Type u_8\nP\u2081' : Type u_9\nP : Type u_10\nP\u2082 : Type u_11\nP\u2083 : Type u_12\nP\u2084 : Type u_13\ninst\u271d\u00b2\u2075 : NormedField \ud835\udd5c\ninst\u271d\u00b2\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b2\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b2\u00b9 : NormedAddTorsor V P\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c V\u2081\ninst\u271d\u00b9\u2078 : PseudoMetricSpace P\u2081\ninst\u271d\u00b9\u2077 : NormedAddTorsor V\u2081 P\u2081\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup V\u2081'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c V\u2081'\ninst\u271d\u00b9\u2074 : MetricSpace P\u2081'\ninst\u271d\u00b9\u00b3 : NormedAddTorsor V\u2081' P\u2081'\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c V\u2082\ninst\u271d\u00b9\u2070 : PseudoMetricSpace P\u2082\ninst\u271d\u2079 : NormedAddTorsor V\u2082 P\u2082\ninst\u271d\u2078 : SeminormedAddCommGroup V\u2083\ninst\u271d\u2077 : NormedSpace \ud835\udd5c V\u2083\ninst\u271d\u2076 : PseudoMetricSpace P\u2083\ninst\u271d\u2075 : NormedAddTorsor V\u2083 P\u2083\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2084\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c V\u2084\ninst\u271d\u00b2 : PseudoMetricSpace P\u2084\ninst\u271d\u00b9 : NormedAddTorsor V\u2084 P\u2084\ne : P \u2243\u1d43\u2071[\ud835\udd5c] P\u2082\n\u03b1 : Type u_14\ninst\u271d : TopologicalSpace \u03b1\nf : P \u2192 P\u2082\nhf : Isometry f\np : P\ng : V \u2192 V\u2082\nhg : \u2200 (v : V), g v = f (v +\u1d65 p) -\u1d65 f p\n\u22a2 g = \u21d1(vaddConst \ud835\udd5c (f p)).symm \u2218 f \u2218 \u21d1(vaddConst \ud835\udd5c p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.compactSpace_iff_isBounded_univ", "start": [328, 1], "end": [330, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Div.lean", "full_name": "Polynomial.degree_modByMonic_lt", "start": [136, 1], "end": [163, 24], "traced_tactics": [{"tactic": "have _wf := div_wf_lemma \u27e8h.1, h.2\u27e9 hq", "annotated_tactic": ["have _wf := <a>div_wf_lemma</a> \u27e8h.1, h.2\u27e9 hq", [{"full_name": "Polynomial.div_wf_lemma", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [91, 9], "def_end_pos": [91, 21]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n\u22a2 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\n\u22a2 (p %\u2098 q).degree < q.degree"}, {"tactic": "have :=\n  degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq", "annotated_tactic": ["have :=\n        degree_modByMonic_lt (p - q * (<a>C</a> (<a>leadingCoeff</a> p) * <a>X</a> ^ (<a>natDegree</a> p - <a>natDegree</a> q))) hq", [{"full_name": "Polynomial.C", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [501, 5], "def_end_pos": [501, 6]}, {"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [72, 5], "def_end_pos": [72, 17]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 6]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\n\u22a2 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))) %\u2098 q).degree < q.degree\n\u22a2 (p %\u2098 q).degree < q.degree"}, {"tactic": "unfold modByMonic at this \u22a2", "annotated_tactic": ["unfold <a>modByMonic</a> at this \u22a2", [{"full_name": "Polynomial.modByMonic", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))) %\u2098 q).degree < q.degree\n\u22a2 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if hq : q.Monic then (p.divModByMonicAux hq).2 else p).degree < q.degree"}, {"tactic": "unfold divModByMonicAux", "annotated_tactic": ["unfold <a>divModByMonicAux</a>", [{"full_name": "Polynomial.divModByMonicAux", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [106, 19], "def_end_pos": [106, 35]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if hq : q.Monic then (p.divModByMonicAux hq).2 else p).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if h : q.Monic then\n        (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n            let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n            let_fun _wf := \u22ef;\n            let dm := (p - q * z).divModByMonicAux \u22ef;\n            (z + dm.1, dm.2)\n          else (0, p)).2\n      else p).degree <\n    q.degree"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if h : q.Monic then\n        (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n            let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n            let_fun _wf := \u22ef;\n            let dm := (p - q * z).divModByMonicAux \u22ef;\n            (z + dm.1, dm.2)\n          else (0, p)).2\n      else p).degree <\n    q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if h : q.Monic then\n        (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n            (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n          else (0, p)).2\n      else p).degree <\n    q.degree"}, {"tactic": "rw [dif_pos hq] at this \u22a2", "annotated_tactic": ["rw [<a>dif_pos</a> hq] at this \u22a2", [{"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis :\n  (if hq : q.Monic then ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2\n      else p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree <\n    q.degree\n\u22a2 (if h : q.Monic then\n        (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n            (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n          else (0, p)).2\n      else p).degree <\n    q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2.degree < q.degree\n\u22a2 (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n          (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n            ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n        else (0, p)).2.degree <\n    q.degree"}, {"tactic": "rw [if_pos h]", "annotated_tactic": ["rw [<a>if_pos</a> h]", [{"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2.degree < q.degree\n\u22a2 (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n          (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n            ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n        else (0, p)).2.degree <\n    q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2.degree < q.degree\n\u22a2 (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n            ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n          ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2).2.degree <\n    q.degree"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis\u271d : DecidableEq R := Classical.decEq R\nh : q.degree \u2264 p.degree \u2227 p \u2260 0\n_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree\nthis : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2.degree < q.degree\n\u22a2 (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n            ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n          ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2).2.degree <\n    q.degree", "state_after": "no goals"}, {"tactic": "unfold modByMonic divModByMonicAux", "annotated_tactic": ["unfold <a>modByMonic</a> <a>divModByMonicAux</a>", [{"full_name": "Polynomial.modByMonic", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Polynomial.divModByMonicAux", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [106, 19], "def_end_pos": [106, 35]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192\n    (if h : q.Monic then\n          (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n              let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n              let_fun _wf := \u22ef;\n              let dm := (p - q * z).divModByMonicAux \u22ef;\n              (z + dm.1, dm.2)\n            else (0, p)).2\n        else p).degree <\n      q.degree"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192\n    (if h : q.Monic then\n          (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n              let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n              let_fun _wf := \u22ef;\n              let dm := (p - q * z).divModByMonicAux \u22ef;\n              (z + dm.1, dm.2)\n            else (0, p)).2\n        else p).degree <\n      q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192\n    (if h : q.Monic then\n          (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n              (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                  ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n            else (0, p)).2\n        else p).degree <\n      q.degree"}, {"tactic": "rw [dif_pos hq, if_neg h]", "annotated_tactic": ["rw [<a>dif_pos</a> hq, <a>if_neg</a> h]", [{"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192\n    (if h : q.Monic then\n          (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n              (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                  ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n            else (0, p)).2\n        else p).degree <\n      q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192 (0, p).2.degree < q.degree"}, {"tactic": "exact lt_of_not_ge", "annotated_tactic": ["exact <a>lt_of_not_ge</a>", [{"full_name": "lt_of_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [334, 9], "def_end_pos": [334, 21]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acq.degree \u2264 p.degree \u2192 (0, p).2.degree < q.degree", "state_after": "no goals"}, {"tactic": "intro hp", "annotated_tactic": ["intro hp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\n\u22a2 \u00acp \u2260 0 \u2192 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (p %\u2098 q).degree < q.degree"}, {"tactic": "unfold modByMonic divModByMonicAux", "annotated_tactic": ["unfold <a>modByMonic</a> <a>divModByMonicAux</a>", [{"full_name": "Polynomial.modByMonic", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Polynomial.divModByMonicAux", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [106, 19], "def_end_pos": [106, 35]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (p %\u2098 q).degree < q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (if h : q.Monic then\n        (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n            let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n            let_fun _wf := \u22ef;\n            let dm := (p - q * z).divModByMonicAux \u22ef;\n            (z + dm.1, dm.2)\n          else (0, p)).2\n      else p).degree <\n    q.degree"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (if h : q.Monic then\n        (if h_1 : q.degree \u2264 p.degree \u2227 p \u2260 0 then\n            let z := C p.leadingCoeff * X ^ (p.natDegree - q.natDegree);\n            let_fun _wf := \u22ef;\n            let dm := (p - q * z).divModByMonicAux \u22ef;\n            (z + dm.1, dm.2)\n          else (0, p)).2\n      else p).degree <\n    q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (if h : q.Monic then\n        (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n            (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n          else (0, p)).2\n      else p).degree <\n    q.degree"}, {"tactic": "rw [dif_pos hq, if_neg h, Classical.not_not.1 hp]", "annotated_tactic": ["rw [<a>dif_pos</a> hq, <a>if_neg</a> h, <a>Classical.not_not</a>.1 hp]", [{"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}, {"full_name": "Classical.not_not", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [135, 17], "def_end_pos": [135, 24]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (if h : q.Monic then\n        (if q.degree \u2264 p.degree \u2227 \u00acp = 0 then\n            (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) +\n                ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).1,\n              ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux \u22ef).2)\n          else (0, p)).2\n      else p).degree <\n    q.degree", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (0, 0).2.degree < q.degree"}, {"tactic": "exact lt_of_le_of_ne bot_le (Ne.symm (mt degree_eq_bot.1 hq.ne_zero))", "annotated_tactic": ["exact <a>lt_of_le_of_ne</a> <a>bot_le</a> (<a>Ne.symm</a> (<a>mt</a> <a>degree_eq_bot</a>.1 hq.ne_zero))", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [198, 9], "def_end_pos": [198, 23]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Ring R\np\u271d q\u271d : R[X]\ninst\u271d : Nontrivial R\np q : R[X]\nhq : q.Monic\nthis : DecidableEq R := Classical.decEq R\nh : \u00ac(q.degree \u2264 p.degree \u2227 p \u2260 0)\nhp : \u00acp \u2260 0\n\u22a2 (0, 0).2.degree < q.degree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "Metric.measure_closedBall_pos", "start": [229, 1], "end": [230, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Rank.lean", "full_name": "LieModule.rank_le_natTrailingDegree_charpoly_ad", "start": [78, 1], "end": [80, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/BaseChange.lean", "full_name": "LieSubmodule.mem_baseChange_iff", "start": [201, 1], "end": [204, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/LeftRightLim.lean", "full_name": "Antitone.rightLim_le_leftLim", "start": [310, 1], "end": [311, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Derivation/Basic.lean", "full_name": "LieDerivation.toFun_eq_coe", "start": [64, 1], "end": [64, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Basis.dualBasis_apply", "start": [455, 1], "end": [456, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EffectiveEpi/Basic.lean", "full_name": "CategoryTheory.effectiveEpi_iff_effectiveEpiFamily", "start": [206, 1], "end": [208, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean", "full_name": "OrthonormalBasis.measurePreserving_repr", "start": [91, 1], "end": [92, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf", "start": [762, 1], "end": [765, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PNat/Prime.lean", "full_name": "Nat.Primes.coe_pnat_injective", "start": [35, 1], "end": [36, 31], "traced_tactics": [{"tactic": "injection h", "annotated_tactic": ["injection h", []], "state_before": "p q : Primes\nh : \u2191p = \u2191q\n\u22a2 \u2191p = \u2191q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.fst_map_id_prod", "start": [952, 1], "end": [955, 56], "traced_tactics": [{"tactic": "rw [fst_map_prod _ measurable_id' hf, kernel.map_id']", "annotated_tactic": ["rw [<a>fst_map_prod</a> _ <a>measurable_id'</a> hf, <a>kernel.map_id'</a>]", [{"full_name": "ProbabilityTheory.kernel.fst_map_prod", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [944, 7], "def_end_pos": [944, 19]}, {"full_name": "measurable_id'", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [570, 9], "def_end_pos": [570, 23]}, {"full_name": "ProbabilityTheory.kernel.map_id'", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [648, 7], "def_end_pos": [648, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b4 : Type u_4\nm\u03b4 : MeasurableSpace \u03b4\n\u03b3\u271d : Type u_5\nm\u03b3\u271d : MeasurableSpace \u03b3\u271d\nf\u271d : \u03b2 \u2192 \u03b3\u271d\ng : \u03b3\u271d \u2192 \u03b1\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\n\u03b3 : Type u_6\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b2 \u2192 \u03b3\nhf : Measurable f\n\u22a2 fst (map \u03ba (fun a => (a, f a)) \u22ef) = \u03ba", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Iio_insert", "start": [932, 1], "end": [933, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "volume_regionBetween_eq_lintegral", "start": [533, 1], "end": [557, 87], "traced_tactics": [{"tactic": "have h\u2081 :\n  (fun y => ENNReal.ofReal ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n    ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n  (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal", "annotated_tactic": ["have h\u2081 :\n    (fun y => <a>ENNReal.ofReal</a> ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n      <a>ENNReal.ofReal</a> ((<a>AEMeasurable.mk</a> g hg - <a>AEMeasurable.mk</a> f hf) y) :=\n    (hg.ae_eq_mk.sub hf.ae_eq_mk).<a>fun_comp</a> <a>ENNReal.ofReal</a>", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1554, 9], "def_end_pos": [1554, 30]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "have h\u2082 :\n  (\u03bc.restrict s).prod volume (regionBetween f g s) =\n    (\u03bc.restrict s).prod volume\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) := by\n  apply measure_congr\n  apply EventuallyEq.rfl.inter\n  exact\n    ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n      (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["have h\u2082 :\n    (\u03bc.restrict s).<a>prod</a> <a>volume</a> (<a>regionBetween</a> f g s) =\n      (\u03bc.restrict s).<a>prod</a> <a>volume</a>\n        (<a>regionBetween</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s) := by\n    apply <a>measure_congr</a>\n    apply EventuallyEq.rfl.inter\n    exact\n      ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n        (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [326, 27], "def_end_pos": [326, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [367, 3], "def_end_pos": [367, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [445, 5], "def_end_pos": [445, 18]}, {"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [326, 27], "def_end_pos": [326, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [367, 3], "def_end_pos": [367, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [445, 5], "def_end_pos": [445, 18]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [269, 9], "def_end_pos": [269, 22]}, {"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 19], "def_end_pos": [1525, 35]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1616, 9], "def_end_pos": [1616, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae h\u2081, \u2190\n  volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> h\u2081, \u2190\n    <a>volume_regionBetween_eq_lintegral'</a> hf.measurable_mk hg.measurable_mk hs]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [328, 9], "def_end_pos": [328, 27]}, {"full_name": "volume_regionBetween_eq_lintegral'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [511, 9], "def_end_pos": [511, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) =\n    (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "convert h\u2082 using 1", "annotated_tactic": ["convert h\u2082 using 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) =\n    (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = ((\u03bc.restrict s).prod volume) (regionBetween f g s)\n\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "apply measure_congr", "annotated_tactic": ["apply <a>measure_congr</a>", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [269, 9], "def_end_pos": [269, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae ((\u03bc.restrict s).prod volume)] regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s"}, {"tactic": "apply EventuallyEq.rfl.inter", "annotated_tactic": ["apply EventuallyEq.rfl.inter", []], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae ((\u03bc.restrict s).prod volume)] regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae ((\u03bc.restrict s).prod volume)] fun p =>\n    Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2"}, {"tactic": "exact\n  ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n    (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["exact\n      ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n        (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 19], "def_end_pos": [1525, 35]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1616, 9], "def_end_pos": [1616, 27]}]], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae ((\u03bc.restrict s).prod volume)] fun p =>\n    Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 35]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = ((\u03bc.restrict s).prod volume) (regionBetween f g s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = ((\u03bc.prod volume).restrict (s \u00d7\u02e2 univ)) (regionBetween f g s)"}, {"tactic": "exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm", "annotated_tactic": ["exact (<a>Measure.restrict_eq_self</a> _ (<a>regionBetween_subset</a> f g s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [124, 9], "def_end_pos": [124, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 29]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween f g s) = ((\u03bc.prod volume).restrict (s \u00d7\u02e2 univ)) (regionBetween f g s)", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [761, 9], "def_end_pos": [761, 35]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    ((\u03bc.prod volume).restrict (s \u00d7\u02e2 univ)) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "exact\n  (Measure.restrict_eq_self _\n      (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm", "annotated_tactic": ["exact\n      (<a>Measure.restrict_eq_self</a> _\n          (<a>regionBetween_subset</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [124, 9], "def_end_pos": [124, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 29]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [432, 5], "def_end_pos": [432, 7]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f (\u03bc.restrict s)\nhg : AEMeasurable g (\u03bc.restrict s)\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (\u03bc.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  ((\u03bc.restrict s).prod volume) (regionBetween f g s) =\n    ((\u03bc.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 (\u03bc.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    ((\u03bc.prod volume).restrict (s \u00d7\u02e2 univ)) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/Verschiebung.lean", "full_name": "WittVector.ghostComponent_zero_verschiebungFun", "start": [58, 1], "end": [61, 70], "traced_tactics": [{"tactic": "rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,\n  verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul]", "annotated_tactic": ["rw [<a>ghostComponent_apply</a>, <a>aeval_wittPolynomial</a>, <a>Finset.range_one</a>, <a>Finset.sum_singleton</a>,\n    <a>verschiebungFun_coeff_zero</a>, <a>pow_zero</a>, <a>pow_zero</a>, <a>pow_one</a>, <a>one_mul</a>]", [{"full_name": "WittVector.ghostComponent_apply", "def_path": "Mathlib/RingTheory/WittVector/Basic.lean", "def_pos": [311, 9], "def_end_pos": [311, 29]}, {"full_name": "aeval_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [146, 9], "def_end_pos": [146, 29]}, {"full_name": "Finset.range_one", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2950, 9], "def_end_pos": [2950, 18]}, {"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [381, 3], "def_end_pos": [381, 14]}, {"full_name": "WittVector.verschiebungFun_coeff_zero", "def_path": "Mathlib/RingTheory/WittVector/Verschiebung.lean", "def_pos": [47, 9], "def_end_pos": [47, 35]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "p : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nx : \ud835\udd4e R\n\u22a2 (ghostComponent 0) x.verschiebungFun = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": 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NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nx\u271d : \u2191m \u2264 n\n\u22a2 Differentiable \ud835\udd5c 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": 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\u2227 a \u2208 t) id"}, {"tactic": "rw [Sigma.nhds_eq]", "annotated_tactic": ["rw [<a>Sigma.nhds_eq</a>]", [{"full_name": "Sigma.nhds_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1647, 9], "def_end_pos": [1647, 22]}]], "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\na : (i : \u03b9) \u00d7 E i\n\u22a2 (\ud835\udcdd a).HasBasis (fun t => t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 t) id", "state_after": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\na : (i : \u03b9) \u00d7 E i\n\u22a2 (map (Sigma.mk a.fst) (\ud835\udcdd a.snd)).HasBasis (fun t => t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 t) id"}, {"tactic": "convert (((hs a.1).nhds_hasBasis).map _).to_image_id", "annotated_tactic": ["convert (((hs a.1).<a>nhds_hasBasis</a>).<a>map</a> _).<a>to_image_id</a>", [{"full_name": "TopologicalSpace.IsTopologicalBasis.nhds_hasBasis", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [160, 9], "def_end_pos": [160, 41]}, {"full_name": "Filter.HasBasis.map", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [798, 9], "def_end_pos": [798, 21]}, {"full_name": "Filter.HasBasis.to_image_id", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [432, 9], "def_end_pos": [432, 29]}]], "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\na : (i : \u03b9) \u00d7 E i\n\u22a2 (map (Sigma.mk a.fst) (\ud835\udcdd a.snd)).HasBasis (fun t => t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 t) id", "state_after": "case h.e'_4.h.a\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\na : (i : \u03b9) \u00d7 E i\nx\u271d : Set ((i : \u03b9) \u00d7 E i)\n\u22a2 x\u271d \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 x\u271d \u2194\n    x\u271d \u2208 (fun i => Sigma.mk a.fst '' i) '' {i | i \u2208 s a.fst \u2227 a.snd \u2208 i}"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "case h.e'_4.h.a\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\na : (i : \u03b9) \u00d7 E i\nx\u271d : Set ((i : \u03b9) \u00d7 E i)\n\u22a2 x\u271d \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 x\u271d \u2194\n    x\u271d \u2208 (fun i => Sigma.mk a.fst '' i) '' {i | i \u2208 s a.fst \u2227 a.snd \u2208 i}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.ext_iff", "start": [173, 1], "end": [174, 39], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u21a5(Lp E p \u03bc)\nh : f = g\n\u22a2 \u2191\u2191f =\u1da0[ae \u03bc] \u2191\u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/ReImTopology.lean", "full_name": "Complex.interior_setOf_le_im", "start": [109, 1], "end": [110, 63], "traced_tactics": [{"tactic": "simpa only [interior_Ici] using interior_preimage_im (Ici a)", "annotated_tactic": ["simpa only [<a>interior_Ici</a>] using <a>interior_preimage_im</a> (<a>Ici</a> a)", [{"full_name": "interior_Ici", "def_path": "Mathlib/Topology/Order/DenselyOrdered.lean", "def_pos": [87, 9], "def_end_pos": [87, 21]}, {"full_name": "Complex.interior_preimage_im", "def_path": "Mathlib/Analysis/Complex/ReImTopology.lean", "def_pos": [73, 9], "def_end_pos": [73, 29]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "a : \u211d\n\u22a2 interior {z | a \u2264 z.im} = {z | a < z.im}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/WF.lean", "full_name": "Batteries.RBNode.Balanced.ins", "start": [234, 11], "end": [255, 37], "traced_tactics": [{"tactic": "exact .balanced (.red .nil .nil)", "annotated_tactic": ["exact .balanced (.red .nil .nil)", []], "state_before": "case nil\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\n\u22a2 RedRed (nil.isRed = red) (ins cmp v nil) 0", "state_after": "no goals"}, {"tactic": "unfold ins", "annotated_tactic": ["unfold <a>ins</a>", [{"full_name": "Batteries.RBNode.ins", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [306, 19], "def_end_pos": [306, 22]}]], "state_before": "case red\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node red a x b).isRed = red) (ins cmp v (node red a x b)) n", "state_after": "case red\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node red a x b).isRed = red)\n    (match cmp v x with\n    | Ordering.lt => node red (ins cmp v a) x b\n    | Ordering.gt => node red a x (ins cmp v b)\n    | Ordering.eq => node red a v b)\n    n"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case red\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node red a x b).isRed = red)\n    (match cmp v x with\n    | Ordering.lt => node red (ins cmp v a) x b\n    | Ordering.gt => node red a x (ins cmp v b)\n    | Ordering.eq => node red a v b)\n    n", "state_after": "case red.h_1\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\n\u22a2 RedRed ((node red a x b).isRed = red) (node red (ins cmp v a) x b) n\n\ncase red.h_2\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x (ins cmp v b)) n\n\ncase red.h_3\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.eq\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a v b) n"}, {"tactic": "match ins cmp v a, ihl with\n| _, .balanced .nil => exact .balanced (.red .nil hr)\n| _, .balanced (.red ha hb) => exact .redred rfl (.red ha hb) hr\n| _, .balanced (.black ha hb) => exact .balanced (.red (.black ha hb) hr)\n| _, .redred h .. => cases hl <;> cases h", "annotated_tactic": ["match <a>ins</a> cmp v a, ihl with\n      | _, .balanced .nil => exact .balanced (.red .nil hr)\n      | _, .balanced (.red ha hb) => exact .redred <a>rfl</a> (.red ha hb) hr\n      | _, .balanced (.black ha hb) => exact .balanced (.red (.black ha hb) hr)\n      | _, .redred h .. => cases hl <;> cases h", [{"full_name": "Batteries.RBNode.ins", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [306, 19], "def_end_pos": [306, 22]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case red.h_1\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\n\u22a2 RedRed ((node red a x b).isRed = red) (node red (ins cmp v a) x b) n", "state_after": "no goals"}, {"tactic": "exact .balanced (.red .nil hr)", "annotated_tactic": ["exact .balanced (.red .nil hr)", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\nhl : a.Balanced black 0\nhr : b.Balanced black 0\nihr : RedRed (b.isRed = red) (ins cmp v b) 0\n\u22a2 RedRed ((node red a x b).isRed = red) (node red nil x b) 0", "state_after": "no goals"}, {"tactic": "exact .redred rfl (.red ha hb) hr", "annotated_tactic": ["exact .redred <a>rfl</a> (.red ha hb) hr", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn\u271d : Nat\nb : RBNode \u03b1\nx : \u03b1\nihl : RedRed (a.isRed = red) (ins cmp v a) n\u271d\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\nn : Nat\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nha : l\u271d.Balanced black n\nhb : r\u271d.Balanced black n\nhl : a.Balanced black n\nhr : b.Balanced black n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node red a x b).isRed = red) (node red (node red l\u271d v\u271d r\u271d) x b) n", "state_after": "no goals"}, {"tactic": "exact .balanced (.red (.black ha hb) hr)", "annotated_tactic": ["exact .balanced (.red (.black ha hb) hr)", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nn\u271d : Nat\nc\u2081\u271d c\u2082\u271d : RBColor\nha : l\u271d.Balanced c\u2081\u271d n\u271d\nhb : r\u271d.Balanced c\u2082\u271d n\u271d\nhl : a.Balanced black (n\u271d + 1)\nhr : b.Balanced black (n\u271d + 1)\nihr : RedRed (b.isRed = red) (ins cmp v b) (n\u271d + 1)\n\u22a2 RedRed ((node red a x b).isRed = red) (node red (node black l\u271d v\u271d r\u271d) x b) (n\u271d + 1)", "state_after": "no goals"}, {"tactic": "cases hl <;> cases h", "annotated_tactic": ["cases hl <;> cases h", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn\u271d : Nat\nb : RBNode \u03b1\nx : \u03b1\nihl : RedRed (a.isRed = red) (ins cmp v a) n\u271d\nx\u271d\u00b9 : Ordering\nheq\u271d : cmp v x = Ordering.lt\nn : Nat\na\u271d\u00b2 : RBNode \u03b1\nc\u2081\u271d : RBColor\nb\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nx\u271d : \u03b1\nh : a.isRed = red\na\u271d\u00b9 : a\u271d\u00b2.Balanced c\u2081\u271d n\na\u271d : b\u271d.Balanced c\u2082\u271d n\nhl : a.Balanced black n\nhr : b.Balanced black n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node red a x b).isRed = red) (node red (node red a\u271d\u00b2 x\u271d b\u271d) x b) n", "state_after": "no goals"}, {"tactic": "match ins cmp v b, ihr with\n| _, .balanced .nil => exact .balanced (.red hl .nil)\n| _, .balanced (.red ha hb) => exact .redred rfl hl (.red ha hb)\n| _, .balanced (.black ha hb) => exact .balanced (.red hl (.black ha hb))\n| _, .redred h .. => cases hr <;> cases h", "annotated_tactic": ["match <a>ins</a> cmp v b, ihr with\n      | _, .balanced .nil => exact .balanced (.red hl .nil)\n      | _, .balanced (.red ha hb) => exact .redred <a>rfl</a> hl (.red ha hb)\n      | _, .balanced (.black ha hb) => exact .balanced (.red hl (.black ha hb))\n      | _, .redred h .. => cases hr <;> cases h", [{"full_name": "Batteries.RBNode.ins", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [306, 19], "def_end_pos": [306, 22]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case red.h_2\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x (ins cmp v b)) n", "state_after": "no goals"}, {"tactic": "exact .balanced (.red hl .nil)", "annotated_tactic": ["exact .balanced (.red hl .nil)", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\nhl : a.Balanced black 0\nhr : b.Balanced black 0\nihl : RedRed (a.isRed = red) (ins cmp v a) 0\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x nil) 0", "state_after": "no goals"}, {"tactic": "exact .redred rfl hl (.red ha hb)", "annotated_tactic": ["exact .redred <a>rfl</a> hl (.red ha hb)", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn\u271d : Nat\nb : RBNode \u03b1\nx : \u03b1\nihr : RedRed (b.isRed = red) (ins cmp v b) n\u271d\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\nn : Nat\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nha : l\u271d.Balanced black n\nhb : r\u271d.Balanced black n\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x (node red l\u271d v\u271d r\u271d)) n", "state_after": "no goals"}, {"tactic": "exact .balanced (.red hl (.black ha hb))", "annotated_tactic": ["exact .balanced (.red hl (.black ha hb))", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nn\u271d : Nat\nc\u2081\u271d c\u2082\u271d : RBColor\nha : l\u271d.Balanced c\u2081\u271d n\u271d\nhb : r\u271d.Balanced c\u2082\u271d n\u271d\nhl : a.Balanced black (n\u271d + 1)\nhr : b.Balanced black (n\u271d + 1)\nihl : RedRed (a.isRed = red) (ins cmp v a) (n\u271d + 1)\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x (node black l\u271d v\u271d r\u271d)) (n\u271d + 1)", "state_after": "no goals"}, {"tactic": "cases hr <;> cases h", "annotated_tactic": ["cases hr <;> cases h", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn\u271d\u00b9 : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn\u271d : Nat\nb : RBNode \u03b1\nx : \u03b1\nihr : RedRed (b.isRed = red) (ins cmp v b) n\u271d\nx\u271d\u00b9 : Ordering\nheq\u271d : cmp v x = Ordering.gt\nn : Nat\na\u271d\u00b2 : RBNode \u03b1\nc\u2081\u271d : RBColor\nb\u271d : RBNode \u03b1\nc\u2082\u271d : RBColor\nx\u271d : \u03b1\nh : b.isRed = red\na\u271d\u00b9 : a\u271d\u00b2.Balanced c\u2081\u271d n\na\u271d : b\u271d.Balanced c\u2082\u271d n\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a x (node red a\u271d\u00b2 x\u271d b\u271d)) n", "state_after": "no goals"}, {"tactic": "exact .balanced (.red hl hr)", "annotated_tactic": ["exact .balanced (.red hl hr)", []], "state_before": "case red.h_3\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nn : Nat\nb : RBNode \u03b1\nx : \u03b1\nhl : a.Balanced black n\nhr : b.Balanced black n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.eq\n\u22a2 RedRed ((node red a x b).isRed = red) (node red a v b) n", "state_after": "no goals"}, {"tactic": "unfold ins", "annotated_tactic": ["unfold <a>ins</a>", [{"full_name": "Batteries.RBNode.ins", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [306, 19], "def_end_pos": [306, 22]}]], "state_before": "case black\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node black a x b).isRed = red) (ins cmp v (node black a x b)) (n + 1)", "state_after": "case black\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node black a x b).isRed = red)\n    (match cmp v x with\n    | Ordering.lt => (ins cmp v a).balance1 x b\n    | Ordering.gt => a.balance2 x (ins cmp v b)\n    | Ordering.eq => node black a v b)\n    (n + 1)"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case black\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\n\u22a2 RedRed ((node black a x b).isRed = red)\n    (match cmp v x with\n    | Ordering.lt => (ins cmp v a).balance1 x b\n    | Ordering.gt => a.balance2 x (ins cmp v b)\n    | Ordering.eq => node black a v b)\n    (n + 1)", "state_after": "case black.h_1\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\n\u22a2 RedRed ((node black a x b).isRed = red) ((ins cmp v a).balance1 x b) (n + 1)\n\ncase black.h_2\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\n\u22a2 RedRed ((node black a x b).isRed = red) (a.balance2 x (ins cmp v b)) (n + 1)\n\ncase black.h_3\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.eq\n\u22a2 RedRed ((node black a x b).isRed = red) (node black a v b) (n + 1)"}, {"tactic": "exact have \u27e8c, h\u27e9 := ihl.balance1 hr; .balanced h", "annotated_tactic": ["exact have \u27e8c, h\u27e9 := ihl.balance1 hr; .balanced h", []], "state_before": "case black.h_1\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.lt\n\u22a2 RedRed ((node black a x b).isRed = red) ((ins cmp v a).balance1 x b) (n + 1)", "state_after": "no goals"}, {"tactic": "exact have \u27e8c, h\u27e9 := ihr.balance2 hl; .balanced h", "annotated_tactic": ["exact have \u27e8c, h\u27e9 := ihr.balance2 hl; .balanced h", []], "state_before": "case black.h_2\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.gt\n\u22a2 RedRed ((node black a x b).isRed = red) (a.balance2 x (ins cmp v b)) (n + 1)", "state_after": "no goals"}, {"tactic": "exact .balanced (.black hl hr)", "annotated_tactic": ["exact .balanced (.black hl hr)", []], "state_before": "case black.h_3\n\u03b1 : Type u_1\nc : RBColor\nn\u271d : Nat\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt a : RBNode \u03b1\nca : RBColor\nn : Nat\nb : RBNode \u03b1\ncb : RBColor\nx : \u03b1\nhl : a.Balanced ca n\nhr : b.Balanced cb n\nihl : RedRed (a.isRed = red) (ins cmp v a) n\nihr : RedRed (b.isRed = red) (ins cmp v b) n\nx\u271d : Ordering\nheq\u271d : cmp v x = Ordering.eq\n\u22a2 RedRed ((node black a x b).isRed = red) (node black a v b) (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.sdiff_lt_left", "start": [1693, 1], "end": [1694, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "ConcaveOn.inf", "start": [614, 1], "end": [615, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalSubalgebra.gc_map_comap", "start": [362, 1], "end": [364, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.exists_eventually_atBot", "start": [359, 1], "end": [363, 22], "traced_tactics": [{"tactic": "simp_rw [eventually_atBot, \u2190 exists_swap (\u03b1 := \u03b1)]", "annotated_tactic": ["simp_rw [<a>eventually_atBot</a>, \u2190 <a>exists_swap</a> (\u03b1 := \u03b1)]", [{"full_name": "Filter.eventually_atBot", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [188, 9], "def_end_pos": [188, 25]}, {"full_name": "exists_swap", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [670, 9], "def_end_pos": [670, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 b, \u2200\u1da0 (a : \u03b1) in atBot, r a b) \u2194 \u2200\u1da0 (a\u2080 : \u03b1) in atBot, \u2203 b, \u2200 a \u2264 a\u2080, r a b", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 x y, \u2200 b \u2264 x, r b y) \u2194 \u2203 a, \u2200 b \u2264 a, \u2203 b_1, \u2200 a \u2264 b, r a b_1"}, {"tactic": "exact exists_congr fun a \u21a6 .symm <| forall_le_iff <| Antitone.exists fun _ _ _ hb H n hn \u21a6\n  H n (hn.trans hb)", "annotated_tactic": ["exact <a>exists_congr</a> fun a \u21a6 .symm <| <a>forall_le_iff</a> <| <a>Antitone.exists</a> fun _ _ _ hb H n hn \u21a6\n    H n (hn.trans hb)", [{"full_name": "exists_congr", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [210, 9], "def_end_pos": [210, 21]}, {"full_name": "forall_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [560, 9], "def_end_pos": [560, 22]}, {"full_name": "Antitone.exists", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [552, 9], "def_end_pos": [552, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : Nonempty \u03b1\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\n\u22a2 (\u2203 x y, \u2200 b \u2264 x, r b y) \u2194 \u2203 a, \u2200 b \u2264 a, \u2203 b_1, \u2200 a \u2264 b, r a b_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.coe_smul", "start": [253, 1], "end": [254, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Dual.lean", "full_name": "Matroid.Base.inter_basis_iff_compl_inter_basis_dual", "start": [203, 1], "end": [207, 58], "traced_tactics": [{"tactic": "refine \u27e8hB.compl_inter_basis_of_inter_basis, fun h \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8hB.compl_inter_basis_of_inter_basis, fun h \u21a6 ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nI B X : Set \u03b1\nhB : M.Base B\nhX : autoParam (X \u2286 M.E) _auto\u271d\n\u22a2 M.Basis (B \u2229 X) X \u2194 M\u2736.Basis (M.E \\ B \u2229 (M.E \\ X)) (M.E \\ X)", "state_after": "\u03b1 : Type u_1\nM : Matroid \u03b1\nI B X : Set \u03b1\nhB : M.Base B\nhX : autoParam (X \u2286 M.E) _auto\u271d\nh : M\u2736.Basis (M.E \\ B \u2229 (M.E \\ X)) (M.E \\ X)\n\u22a2 M.Basis (B \u2229 X) X"}, {"tactic": "simpa [inter_eq_self_of_subset_right hX, inter_eq_self_of_subset_right hB.subset_ground] using\n  hB.compl_base_dual.compl_inter_basis_of_inter_basis h", "annotated_tactic": ["simpa [<a>inter_eq_self_of_subset_right</a> hX, <a>inter_eq_self_of_subset_right</a> hB.subset_ground] using\n    hB.compl_base_dual.compl_inter_basis_of_inter_basis h", [{"full_name": "Set.inter_eq_self_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [960, 9], "def_end_pos": [960, 38]}, {"full_name": "Set.inter_eq_self_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [960, 9], "def_end_pos": [960, 38]}]], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nI B X : Set \u03b1\nhB : M.Base B\nhX : autoParam (X \u2286 M.E) _auto\u271d\nh : M\u2736.Basis (M.E \\ B \u2229 (M.E \\ X)) (M.E \\ X)\n\u22a2 M.Basis (B \u2229 X) X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/UInt.lean", "full_name": "UInt8.ext", "start": [10, 8], "end": [11, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/RatLemmas.lean", "full_name": "Rat.not_countably_generated_nhds_infty_opc", "start": [65, 1], "end": [69, 47], "traced_tactics": [{"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "p q : \u211a\ns t : Set \u211a\n\u22a2 \u00ac(\ud835\udcdd \u221e).IsCountablyGenerated", "state_after": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\n\u22a2 False"}, {"tactic": "have : IsCountablyGenerated (comap (OnePoint.some : \u211a \u2192 \u211a\u221e) (\ud835\udcdd \u221e)) := by infer_instance", "annotated_tactic": ["have : <a>IsCountablyGenerated</a> (<a>comap</a> (<a>OnePoint.some</a> : \u211a \u2192 \u211a\u221e) (\ud835\udcdd \u221e)) := by infer_instance", [{"full_name": "Filter.IsCountablyGenerated", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [996, 7], "def_end_pos": [996, 27]}, {"full_name": "Filter.comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1997, 5], "def_end_pos": [1997, 10]}, {"full_name": "OnePoint.some", "def_path": "Mathlib/Topology/Compactification/OnePoint.lean", "def_pos": [73, 27], "def_end_pos": [73, 31]}]], "state_before": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\n\u22a2 False", "state_after": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\nthis : (comap OnePoint.some (\ud835\udcdd \u221e)).IsCountablyGenerated\n\u22a2 False"}, {"tactic": "rw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this", "annotated_tactic": ["rw [<a>OnePoint.comap_coe_nhds_infty</a>, <a>coclosedCompact_eq_cocompact</a>] at this", [{"full_name": "OnePoint.comap_coe_nhds_infty", "def_path": "Mathlib/Topology/Compactification/OnePoint.lean", "def_pos": [350, 9], "def_end_pos": [350, 29]}, {"full_name": "Filter.coclosedCompact_eq_cocompact", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1337, 9], "def_end_pos": [1337, 44]}]], "state_before": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\nthis : (comap OnePoint.some (\ud835\udcdd \u221e)).IsCountablyGenerated\n\u22a2 False", "state_after": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\nthis : (cocompact \u211a).IsCountablyGenerated\n\u22a2 False"}, {"tactic": "exact not_countably_generated_cocompact this", "annotated_tactic": ["exact <a>not_countably_generated_cocompact</a> this", [{"full_name": "Rat.not_countably_generated_cocompact", "def_path": "Mathlib/Topology/Instances/RatLemmas.lean", "def_pos": [56, 9], "def_end_pos": [56, 42]}]], "state_before": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\nthis : (cocompact \u211a).IsCountablyGenerated\n\u22a2 False", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "p q : \u211a\ns t : Set \u211a\na\u271d : (\ud835\udcdd \u221e).IsCountablyGenerated\n\u22a2 (comap OnePoint.some (\ud835\udcdd \u221e)).IsCountablyGenerated", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.takeWhile_cons_of_neg", "start": [3010, 1], "end": [3012, 27], "traced_tactics": [{"tactic": "simp [takeWhile_cons, h]", "annotated_tactic": ["simp [<a>takeWhile_cons</a>, h]", [{"full_name": "List.takeWhile_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1718, 9], "def_end_pos": [1718, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nx : \u03b1\nh : \u00acp x = true\n\u22a2 takeWhile p (x :: l) = []", "state_after": "no goals"}]}, {"url": 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"Mathlib/Data/PFunctor/Multivariate/M.lean", "full_name": "MvPFunctor.M.bisim_lemma", "start": [223, 1], "end": [233, 44], "traced_tactics": [{"tactic": "generalize ef : @splitFun n _ (append1 \u03b1 (M P \u03b1)) f' f\u2081' = ff at e\u2081", "annotated_tactic": ["generalize ef : @<a>splitFun</a> n _ (<a>append1</a> \u03b1 (<a>M</a> P \u03b1)) f' f\u2081' = ff at e\u2081", [{"full_name": "TypeVec.splitFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [144, 5], "def_end_pos": [144, 13]}, {"full_name": "TypeVec.append1", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [94, 5], "def_end_pos": [94, 12]}, {"full_name": "MvPFunctor.M", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [101, 5], "def_end_pos": [101, 6]}]], "state_before": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', splitFun f' f\u2081'\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9"}, {"tactic": "let he\u2081' := PFunctor.M.dest a\u2081", "annotated_tactic": ["let he\u2081' := <a>PFunctor.M.dest</a> a\u2081", [{"full_name": "PFunctor.M.dest", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [287, 5], "def_end_pos": [287, 9]}]], "state_before": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9"}, {"tactic": "rcases e\u2081' : he\u2081' with \u27e8a\u2081', g\u2081'\u27e9", "annotated_tactic": ["rcases e\u2081' : he\u2081' with \u27e8a\u2081', g\u2081'\u27e9", []], "state_before": "n : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "case mk\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\na\u2081' : P.last.A\ng\u2081' : P.last.B a\u2081' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a\u2081', g\u2081'\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9"}, {"tactic": "rw [M.dest_eq_dest' _ e\u2081'] at e\u2081", "annotated_tactic": ["rw [<a>M.dest_eq_dest'</a> _ e\u2081'] at e\u2081", [{"full_name": "MvPFunctor.M.dest_eq_dest'", "def_path": "Mathlib/Data/PFunctor/Multivariate/M.lean", "def_pos": [201, 9], "def_end_pos": [201, 24]}]], "state_before": "case mk\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\ne\u2081 : dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', ff\u27e9\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\na\u2081' : P.last.A\ng\u2081' : P.last.B a\u2081' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a\u2081', g\u2081'\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "case mk\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\na\u2081' : P.last.A\ng\u2081' : P.last.B a\u2081' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a\u2081', g\u2081'\u27e9\ne\u2081 : dest' P e\u2081' f\u2081 = \u27e8a', ff\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9"}, {"tactic": "cases e\u2081", "annotated_tactic": ["cases e\u2081", []], "state_before": "case mk\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nff : P.B a' \u27f9 \u03b1 ::: P.M \u03b1\nef : splitFun f' f\u2081' = ff\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\na\u2081' : P.last.A\ng\u2081' : P.last.B a\u2081' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a\u2081', g\u2081'\u27e9\ne\u2081 : dest' P e\u2081' f\u2081 = \u27e8a', ff\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "case mk.refl\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\ng\u2081' : P.last.B a' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a', g\u2081'\u27e9\nef : splitFun f' f\u2081' = splitFun (pathDestLeft P e\u2081' f\u2081) fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9"}, {"tactic": "exact \u27e8_, e\u2081', splitFun_inj ef\u27e9", "annotated_tactic": ["exact \u27e8_, e\u2081', <a>splitFun_inj</a> ef\u27e9", [{"full_name": "TypeVec.splitFun_inj", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [229, 9], "def_end_pos": [229, 21]}]], "state_before": "case mk.refl\nn : \u2115\nP : MvPFunctor.{u} (n + 1)\n\u03b1 : TypeVec.{u} n\na\u2081 : P.mp.A\nf\u2081 : P.mp.B a\u2081 \u27f9 \u03b1\na' : P.A\nf' : (P.B a').drop \u27f9 \u03b1\nf\u2081' : (P.B a').last \u2192 P.M \u03b1\nhe\u2081' : \u2191P.last P.last.M := PFunctor.M.dest a\u2081\ng\u2081' : P.last.B a' \u2192 P.last.M\ne\u2081' : he\u2081' = \u27e8a', g\u2081'\u27e9\nef : splitFun f' f\u2081' = splitFun (pathDestLeft P e\u2081' f\u2081) fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9\n\u22a2 \u2203 g\u2081',\n    \u2203 (e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = pathDestLeft P e\u2081' f\u2081 \u2227 f\u2081' = fun x => \u27e8g\u2081' x, pathDestRight P e\u2081' f\u2081 x\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Dart.lean", "full_name": "SimpleGraph.dart_edge_eq_mk'_iff'", "start": [118, 1], "end": [123, 7], "traced_tactics": [{"tactic": "rintro \u27e8\u27e8a, b\u27e9, h\u27e9 u v", "annotated_tactic": ["rintro \u27e8\u27e8a, b\u27e9, h\u27e9 u v", []], "state_before": "V : Type u_1\nG : SimpleGraph V\n\u22a2 \u2200 {d : G.Dart} {u v : V}, d.edge = s(u, v) \u2194 d.toProd.1 = u \u2227 d.toProd.2 = v \u2228 d.toProd.1 = v \u2227 d.toProd.2 = u", "state_after": "case mk.mk\nV : Type u_1\nG : SimpleGraph V\na b : V\nh : G.Adj (a, b).1 (a, b).2\nu v : V\n\u22a2 { toProd := (a, b), adj := h }.edge = s(u, v) \u2194\n    { toProd := (a, b), adj := h }.toProd.1 = u \u2227 { toProd := (a, b), adj := h }.toProd.2 = v \u2228\n      { toProd := (a, b), adj := h }.toProd.1 = v \u2227 { toProd := (a, b), adj := h }.toProd.2 = u"}, {"tactic": "rw [dart_edge_eq_mk'_iff]", "annotated_tactic": ["rw [<a>dart_edge_eq_mk'_iff</a>]", [{"full_name": "SimpleGraph.dart_edge_eq_mk'_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Dart.lean", "def_pos": [112, 9], "def_end_pos": [112, 29]}]], "state_before": "case mk.mk\nV : Type u_1\nG : SimpleGraph V\na b : V\nh : G.Adj (a, b).1 (a, b).2\nu v : V\n\u22a2 { toProd := (a, b), adj := h }.edge = s(u, v) \u2194\n    { toProd := (a, b), adj := h }.toProd.1 = u \u2227 { toProd := (a, b), adj := h }.toProd.2 = v \u2228\n      { toProd := (a, b), adj := h }.toProd.1 = v \u2227 { toProd := (a, b), adj := h }.toProd.2 = u", "state_after": "case mk.mk\nV : Type u_1\nG : SimpleGraph V\na b : V\nh : G.Adj (a, b).1 (a, b).2\nu v : V\n\u22a2 { toProd := (a, b), adj := h }.toProd = (u, v) \u2228 { toProd := (a, b), adj := h }.toProd = (u, v).swap \u2194\n    { toProd := (a, b), adj := h }.toProd.1 = u \u2227 { toProd := (a, b), adj := h }.toProd.2 = v \u2228\n      { toProd := (a, b), adj := h }.toProd.1 = v \u2227 { toProd := (a, b), adj := h }.toProd.2 = u"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.mk\nV : Type u_1\nG : SimpleGraph V\na b : V\nh : G.Adj (a, b).1 (a, b).2\nu v : V\n\u22a2 { toProd := (a, b), adj := h }.toProd = (u, v) \u2228 { toProd := (a, b), adj := h }.toProd = (u, v).swap \u2194\n    { toProd := (a, b), adj := h }.toProd.1 = u \u2227 { toProd := (a, b), adj := h }.toProd.2 = v \u2228\n      { toProd := (a, b), adj := h }.toProd.1 = v \u2227 { toProd := (a, b), adj := h }.toProd.2 = u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.leadingCoeff_C", "start": [875, 1], "end": [876, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "full_name": "Submodule.comap_iInf_map_of_injective", "start": [365, 1], "end": [367, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Iterate.lean", "full_name": "Monotone.seq_lt_seq_of_lt_of_le", "start": [68, 1], "end": [71, 78], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx y : \u2115 \u2192 \u03b1\nhf : Monotone f\nn : \u2115\nh\u2080 : x 0 < y 0\nhx : \u2200 (k : \u2115), k < n \u2192 x (k + 1) < f (x k)\nhy : \u2200 (k : \u2115), k < n \u2192 f (y k) \u2264 y (k + 1)\n\u22a2 x n < y n", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx y : \u2115 \u2192 \u03b1\nhf : Monotone f\nh\u2080 : x 0 < y 0\nhx : \u2200 (k : \u2115), k < 0 \u2192 x (k + 1) < f (x k)\nhy : \u2200 (k : \u2115), k < 0 \u2192 f (y k) \u2264 y (k + 1)\n\u22a2 x 0 < y 0\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx y : \u2115 \u2192 \u03b1\nhf : Monotone f\nh\u2080 : x 0 < y 0\nn\u271d : \u2115\nhx : \u2200 (k : \u2115), k < n\u271d + 1 \u2192 x (k + 1) < f (x k)\nhy : \u2200 (k : \u2115), k < n\u271d + 1 \u2192 f (y k) \u2264 y (k + 1)\n\u22a2 x (n\u271d + 1) < y (n\u271d + 1)"}, {"tactic": "exacts [h\u2080, hf.seq_pos_lt_seq_of_lt_of_le (Nat.zero_lt_succ _) h\u2080.le hx hy]", "annotated_tactic": ["exacts [h\u2080, hf.seq_pos_lt_seq_of_lt_of_le (<a>Nat.zero_lt_succ</a> _) h\u2080.le hx hy]", [{"full_name": "Nat.zero_lt_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1671, 9], "def_end_pos": [1671, 25]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx y : \u2115 \u2192 \u03b1\nhf : Monotone f\nh\u2080 : x 0 < y 0\nhx : \u2200 (k : \u2115), k < 0 \u2192 x (k + 1) < f (x k)\nhy : \u2200 (k : \u2115), k < 0 \u2192 f (y k) \u2264 y (k + 1)\n\u22a2 x 0 < y 0\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx y : \u2115 \u2192 \u03b1\nhf : Monotone f\nh\u2080 : x 0 < y 0\nn\u271d : \u2115\nhx : \u2200 (k : \u2115), k < n\u271d + 1 \u2192 x (k + 1) < f (x k)\nhy : \u2200 (k : \u2115), k < n\u271d + 1 \u2192 f (y k) \u2264 y (k + 1)\n\u22a2 x (n\u271d + 1) < y (n\u271d + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_le_nat_iff", "start": [1349, 1], "end": [1350, 37], "traced_tactics": [{"tactic": "rw [\u2190 lift_natCast.{v,u}, lift_le]", "annotated_tactic": ["rw [\u2190 <a>lift_natCast</a>.{v,u}, <a>lift_le</a>]", [{"full_name": "Cardinal.lift_natCast", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1307, 9], "def_end_pos": [1307, 21]}, {"full_name": "Cardinal.lift_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}]], "state_before": "\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nn : \u2115\n\u22a2 lift.{v, u} a \u2264 \u2191n \u2194 a \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "continuousAt_extChartAt_symm''", "start": [1264, 1], "end": [1266, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "full_name": "Profinite.NobelingProof.GoodProducts.finsupp_sum_mem_span_eval", "start": [581, 1], "end": [597, 54], "traced_tactics": [{"tactic": "apply Submodule.finsupp_sum_mem", "annotated_tactic": ["apply <a>Submodule.finsupp_sum_mem</a>", [{"full_name": "Submodule.finsupp_sum_mem", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [1269, 19], "def_end_pos": [1269, 44]}]], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\n\u22a2 (c.sum fun a_1 b => e (\u03c0 C fun x => x \u2208 s) a * b \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) a_1) \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\n\u22a2 \u2200 (c_1 : Products I),\n    c c_1 \u2260 0 \u2192\n      e (\u03c0 C fun x => x \u2208 s) a * c c_1 \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) c_1 \u2208\n        Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\n\u22a2 \u2200 (c_1 : Products I),\n    c c_1 \u2260 0 \u2192\n      e (\u03c0 C fun x => x \u2208 s) a * c c_1 \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) c_1 \u2208\n        Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * c m \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})"}, {"tactic": "dsimp at hsm", "annotated_tactic": ["dsimp at hsm", []], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    (LinearMap.mulLeft \u2124 (e (\u03c0 C fun x => x \u2208 s) a)) (c \u2022 x) = c \u2022 (LinearMap.mulLeft \u2124 (e (\u03c0 C fun x => x \u2208 s) a)) x\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * c m \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * c m \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})"}, {"tactic": "rw [hsm]", "annotated_tactic": ["rw [hsm]", []], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * c m \u2022 Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 c m \u2022 (e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m) \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})"}, {"tactic": "apply Submodule.smul_mem", "annotated_tactic": ["apply <a>Submodule.smul_mem</a>", [{"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 c m \u2022 (e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m) \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})"}, {"tactic": "apply Submodule.subset_span", "annotated_tactic": ["apply <a>Submodule.subset_span</a>", [{"full_name": "Submodule.subset_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [78, 9], "def_end_pos": [78, 20]}]], "state_before": "case h.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Submodule.span \u2124 (Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as})", "state_after": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as}"}, {"tactic": "have hmas : m.val \u2264 as := by\n  apply hc\n  simpa only [Finset.mem_coe, Finsupp.mem_support_iff] using hm", "annotated_tactic": ["have hmas : m.val \u2264 as := by\n    apply hc\n    simpa only [<a>Finset.mem_coe</a>, <a>Finsupp.mem_support_iff</a>] using hm", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [171, 9], "def_end_pos": [171, 24]}]], "state_before": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as}", "state_after": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\nhmas : \u2191m \u2264 as\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as}"}, {"tactic": "refine \u27e8\u27e8a :: m.val, ha.cons_of_le m.prop hmas\u27e9, \u27e8List.cons_le_cons a hmas, ?_\u27e9\u27e9", "annotated_tactic": ["refine \u27e8\u27e8a :: m.val, ha.cons_of_le m.prop hmas\u27e9, \u27e8<a>List.cons_le_cons</a> a hmas, ?_\u27e9\u27e9", [{"full_name": "List.cons_le_cons", "def_path": "Mathlib/Data/List/Lex.lean", "def_pos": [235, 9], "def_end_pos": [235, 21]}]], "state_before": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\nhmas : \u2191m \u2264 as\n\u22a2 e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m \u2208\n    Products.eval (\u03c0 C fun x => x \u2208 s) '' {m | \u2191m \u2264 a :: as}", "state_after": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\nhmas : \u2191m \u2264 as\n\u22a2 Products.eval (\u03c0 C fun x => x \u2208 s) \u27e8a :: \u2191m, \u22ef\u27e9 = e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m"}, {"tactic": "simp only [Products.eval, List.map, List.prod_cons]", "annotated_tactic": ["simp only [<a>Products.eval</a>, <a>List.map</a>, <a>List.prod_cons</a>]", [{"full_name": "Profinite.NobelingProof.Products.eval", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [322, 5], "def_end_pos": [322, 9]}, {"full_name": "List.map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [361, 19], "def_end_pos": [361, 22]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [95, 9], "def_end_pos": [95, 18]}]], "state_before": "case h.h.a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\nhmas : \u2191m \u2264 as\n\u22a2 Products.eval (\u03c0 C fun x => x \u2208 s) \u27e8a :: \u2191m, \u22ef\u27e9 = e (\u03c0 C fun x => x \u2208 s) a * Products.eval (\u03c0 C fun x => x \u2208 s) m", "state_after": "no goals"}, {"tactic": "apply hc", "annotated_tactic": ["apply hc", []], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 \u2191m \u2264 as", "state_after": "case a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 m \u2208 \u2191c.support"}, {"tactic": "simpa only [Finset.mem_coe, Finsupp.mem_support_iff] using hm", "annotated_tactic": ["simpa only [<a>Finset.mem_coe</a>, <a>Finsupp.mem_support_iff</a>] using hm", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [171, 9], "def_end_pos": [171, 24]}]], "state_before": "case a\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\ns : Finset I\na : I\nas : List I\nha : List.Chain' (fun x x_1 => x > x_1) (a :: as)\nc : Products I \u2192\u2080 \u2124\nhc : \u2191c.support \u2286 {m | \u2191m \u2264 as}\nm : Products I\nhm : c m \u2260 0\nhsm :\n  \u2200 (c : \u2124) (x : LocallyConstant \u2191(\u03c0 C fun x => x \u2208 s) \u2124),\n    e (\u03c0 C fun x => x \u2208 s) a * c \u2022 x = c \u2022 (e (\u03c0 C fun x => x \u2208 s) a * x)\n\u22a2 m \u2208 \u2191c.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Inverse.lean", "full_name": "FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1", "start": [379, 1], "end": [435, 44], "traced_tactics": [{"tactic": "simp_rw [mul_sum]", "annotated_tactic": ["simp_rw [<a>mul_sum</a>]", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [61, 7], "def_end_pos": [61, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 k \u2208 Ico 2 (n + 1), a ^ k * \u2211 c \u2208 {c | 1 < c.length}.toFinset, r ^ c.length * \u220f j : Fin c.length, p (c.blocksFun j) =\n    \u2211 k \u2208 Ico 2 (n + 1),\n      \u2211 c \u2208 {c | 1 < c.length}.toFinset, \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x \u2208 Ico 2 (n + 1),\n      \u2211 i \u2208 {c | 1 < c.length}.toFinset, a ^ x * (r ^ i.length * \u220f j : Fin i.length, p (i.blocksFun j)) =\n    \u2211 k \u2208 Ico 2 (n + 1),\n      \u2211 c \u2208 {c | 1 < c.length}.toFinset, \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))"}, {"tactic": "congr! with k _ c", "annotated_tactic": ["congr! with k _ c", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x \u2208 Ico 2 (n + 1),\n      \u2211 i \u2208 {c | 1 < c.length}.toFinset, a ^ x * (r ^ i.length * \u220f j : Fin i.length, p (i.blocksFun j)) =\n    \u2211 k \u2208 Ico 2 (n + 1),\n      \u2211 c \u2208 {c | 1 < c.length}.toFinset, \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))", "state_after": "case a.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\na\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\na\u271d : c \u2208 {c | 1 < c.length}.toFinset\n\u22a2 a ^ k * (r ^ c.length * \u220f j : Fin c.length, p (c.blocksFun j)) =\n    \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))"}, {"tactic": "rw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, Composition.sum_blocksFun,\n  prod_const, card_fin]", "annotated_tactic": ["rw [<a>prod_mul_distrib</a>, <a>prod_mul_distrib</a>, <a>prod_pow_eq_pow_sum</a>, <a>Composition.sum_blocksFun</a>,\n        <a>prod_const</a>, <a>card_fin</a>]", [{"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}, {"full_name": "Finset.prod_pow_eq_pow_sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1768, 7], "def_end_pos": [1768, 26]}, {"full_name": "Composition.sum_blocksFun", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [160, 9], "def_end_pos": [160, 22]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1737, 9], "def_end_pos": [1737, 19]}, {"full_name": "Finset.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [324, 9], "def_end_pos": [324, 24]}]], "state_before": "case a.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\na\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\na\u271d : c \u2208 {c | 1 < c.length}.toFinset\n\u22a2 a ^ k * (r ^ c.length * \u220f j : Fin c.length, p (c.blocksFun j)) =\n    \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))", "state_after": "case a.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\na\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\na\u271d : c \u2208 {c | 1 < c.length}.toFinset\n\u22a2 a ^ k * (r ^ c.length * \u220f j : Fin c.length, p (c.blocksFun j)) =\n    r ^ c.length * (a ^ k * \u220f x : Fin c.length, p (c.blocksFun x))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case a.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\na\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\na\u271d : c \u2208 {c | 1 < c.length}.toFinset\n\u22a2 a ^ k * (r ^ c.length * \u220f j : Fin c.length, p (c.blocksFun j)) =\n    r ^ c.length * (a ^ k * \u220f x : Fin c.length, p (c.blocksFun x))", "state_after": "no goals"}, {"tactic": "rw [sum_sigma']", "annotated_tactic": ["rw [<a>sum_sigma'</a>]", [{"full_name": "Finset.sum_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [633, 3], "def_end_pos": [633, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 k \u2208 Ico 2 (n + 1),\n      \u2211 c \u2208 {c | 1 < c.length}.toFinset, \u220f j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j)) \u2264\n    \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x \u2208 (Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset,\n      \u220f j : Fin x.snd.length, r * (a ^ x.snd.blocksFun j * p (x.snd.blocksFun j)) \u2264\n    \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))"}, {"tactic": "refine\n  sum_le_sum_of_subset_of_nonneg ?_ fun x _ _ =>\n    prod_nonneg fun j _ => mul_nonneg hr (mul_nonneg (pow_nonneg ha _) (hp _))", "annotated_tactic": ["refine\n        <a>sum_le_sum_of_subset_of_nonneg</a> ?_ fun x _ _ =>\n          <a>prod_nonneg</a> fun j _ => <a>mul_nonneg</a> hr (<a>mul_nonneg</a> (<a>pow_nonneg</a> ha _) (hp _))", [{"full_name": "Finset.sum_le_sum_of_subset_of_nonneg", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [154, 15], "def_end_pos": [154, 45]}, {"full_name": "Finset.prod_nonneg", "def_path": "Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean", "def_pos": [29, 7], "def_end_pos": [29, 18]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x \u2208 (Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset,\n      \u220f j : Fin x.snd.length, r * (a ^ x.snd.blocksFun j * p (x.snd.blocksFun j)) \u2264\n    \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 ((Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset) \u2286 compPartialSumTarget 2 (n + 1) n"}, {"tactic": "rintro \u27e8k, c\u27e9 hd", "annotated_tactic": ["rintro \u27e8k, c\u27e9 hd", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 ((Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset) \u2286 compPartialSumTarget 2 (n + 1) n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : \u27e8k, c\u27e9 \u2208 (Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset\n\u22a2 \u27e8k, c\u27e9 \u2208 compPartialSumTarget 2 (n + 1) n"}, {"tactic": "simp only [Set.mem_toFinset (s := {c | 1 < Composition.length c}), mem_Ico, mem_sigma,\n  Set.mem_setOf_eq] at hd", "annotated_tactic": ["simp only [<a>Set.mem_toFinset</a> (s := {c | 1 < <a>Composition.length</a> c}), <a>mem_Ico</a>, <a>mem_sigma</a>,\n        <a>Set.mem_setOf_eq</a>] at hd", [{"full_name": "Set.mem_toFinset", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 21]}, {"full_name": "Composition.length", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [143, 8], "def_end_pos": [143, 14]}, {"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : \u27e8k, c\u27e9 \u2208 (Ico 2 (n + 1)).sigma fun k => {c | 1 < c.length}.toFinset\n\u22a2 \u27e8k, c\u27e9 \u2208 compPartialSumTarget 2 (n + 1) n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\n\u22a2 \u27e8k, c\u27e9 \u2208 compPartialSumTarget 2 (n + 1) n"}, {"tactic": "simp only [mem_compPartialSumTarget_iff]", "annotated_tactic": ["simp only [<a>mem_compPartialSumTarget_iff</a>]", [{"full_name": "FormalMultilinearSeries.mem_compPartialSumTarget_iff", "def_path": "Mathlib/Analysis/Analytic/Composition.lean", "def_pos": [626, 9], "def_end_pos": [626, 37]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\n\u22a2 \u27e8k, c\u27e9 \u2208 compPartialSumTarget 2 (n + 1) n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\n\u22a2 2 \u2264 c.length \u2227 c.length < n + 1 \u2227 \u2200 (j : Fin c.length), c.blocksFun j < n"}, {"tactic": "refine \u27e8hd.2, c.length_le.trans_lt hd.1.2, fun j => ?_\u27e9", "annotated_tactic": ["refine \u27e8hd.2, c.length_le.trans_lt hd.1.2, fun j => ?_\u27e9", []], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\n\u22a2 2 \u2264 c.length \u2227 c.length < n + 1 \u2227 \u2200 (j : Fin c.length), c.blocksFun j < n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\n\u22a2 c.blocksFun j < n"}, {"tactic": "have : c \u2260 Composition.single k (zero_lt_two.trans_le hd.1.1) := by\n  simp [Composition.eq_single_iff_length, ne_of_gt hd.2]", "annotated_tactic": ["have : c \u2260 <a>Composition.single</a> k (zero_lt_two.trans_le hd.1.1) := by\n        simp [<a>Composition.eq_single_iff_length</a>, <a>ne_of_gt</a> hd.2]", [{"full_name": "Composition.single", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [551, 5], "def_end_pos": [551, 11]}, {"full_name": "Composition.eq_single_iff_length", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [577, 9], "def_end_pos": [577, 29]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\n\u22a2 c.blocksFun j < n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\nthis : c \u2260 Composition.single k \u22ef\n\u22a2 c.blocksFun j < n"}, {"tactic": "rw [Composition.ne_single_iff] at this", "annotated_tactic": ["rw [<a>Composition.ne_single_iff</a>] at this", [{"full_name": "Composition.ne_single_iff", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [591, 9], "def_end_pos": [591, 22]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\nthis : c \u2260 Composition.single k \u22ef\n\u22a2 c.blocksFun j < n", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\nthis : \u2200 (i : Fin c.length), c.blocksFun i < k\n\u22a2 c.blocksFun j < n"}, {"tactic": "exact (this j).trans_le (Nat.lt_succ_iff.mp hd.1.2)", "annotated_tactic": ["exact (this j).<a>trans_le</a> (Nat.lt_succ_iff.mp hd.1.2)", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\nthis : \u2200 (i : Fin c.length), c.blocksFun i < k\n\u22a2 c.blocksFun j < n", "state_after": "no goals"}, {"tactic": "simp [Composition.eq_single_iff_length, ne_of_gt hd.2]", "annotated_tactic": ["simp [<a>Composition.eq_single_iff_length</a>, <a>ne_of_gt</a> hd.2]", [{"full_name": "Composition.eq_single_iff_length", "def_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "def_pos": [577, 9], "def_end_pos": [577, 29]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < c.length\nj : Fin c.length\n\u22a2 c \u2260 Composition.single k \u22ef", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j)) =\n    \u2211 e \u2208 compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 e \u2208 compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j)) =\n    \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))"}, {"tactic": "apply compChangeOfVariables_sum", "annotated_tactic": ["apply <a>compChangeOfVariables_sum</a>", [{"full_name": "FormalMultilinearSeries.compChangeOfVariables_sum", "def_path": "Mathlib/Analysis/Analytic/Composition.lean", "def_pos": [637, 9], "def_end_pos": [637, 34]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 e \u2208 compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j)) =\n    \u2211 d \u2208 compPartialSumTarget 2 (n + 1) n, \u220f j : Fin d.snd.length, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2200 (e : (n : \u2115) \u00d7 (Fin n \u2192 \u2115)) (he : e \u2208 compPartialSumSource 2 (n + 1) n),\n    \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j)) =\n      \u220f j : Fin (compChangeOfVariables 2 (n + 1) n e he).snd.length,\n        r *\n          (a ^ (compChangeOfVariables 2 (n + 1) n e he).snd.blocksFun j *\n            p ((compChangeOfVariables 2 (n + 1) n e he).snd.blocksFun j))"}, {"tactic": "rintro \u27e8k, blocks_fun\u27e9 H", "annotated_tactic": ["rintro \u27e8k, blocks_fun\u27e9 H", []], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2200 (e : (n : \u2115) \u00d7 (Fin n \u2192 \u2115)) (he : e \u2208 compPartialSumSource 2 (n + 1) n),\n    \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j)) =\n      \u220f j : Fin (compChangeOfVariables 2 (n + 1) n e he).snd.length,\n        r *\n          (a ^ (compChangeOfVariables 2 (n + 1) n e he).snd.blocksFun j *\n            p ((compChangeOfVariables 2 (n + 1) n e he).snd.blocksFun j))", "state_after": "case h.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\n\u22a2 \u220f j : Fin \u27e8k, blocks_fun\u27e9.fst, r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    \u220f j : Fin (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length,\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))"}, {"tactic": "have K : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k := by simp", "annotated_tactic": ["have K : (<a>compChangeOfVariables</a> 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k := by simp", [{"full_name": "FormalMultilinearSeries.compChangeOfVariables", "def_path": "Mathlib/Analysis/Analytic/Composition.lean", "def_pos": [567, 5], "def_end_pos": [567, 26]}]], "state_before": "case h.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\n\u22a2 \u220f j : Fin \u27e8k, blocks_fun\u27e9.fst, r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    \u220f j : Fin (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length,\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))", "state_after": "case h.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 \u220f j : Fin \u27e8k, blocks_fun\u27e9.fst, r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    \u220f j : Fin (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length,\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))"}, {"tactic": "congr 2 <;> try rw [K]", "annotated_tactic": ["congr 2 <;> try rw [K]", []], "state_before": "case h.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 \u220f j : Fin \u27e8k, blocks_fun\u27e9.fst, r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    \u220f j : Fin (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length,\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))", "state_after": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 HEq (fun j => r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j))) fun j =>\n    r *\n      (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n        p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))"}, {"tactic": "rw [Fin.heq_fun_iff K.symm]", "annotated_tactic": ["rw [<a>Fin.heq_fun_iff</a> K.symm]", [{"full_name": "Fin.heq_fun_iff", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [196, 19], "def_end_pos": [196, 30]}]], "state_before": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 HEq (fun j => r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j))) fun j =>\n    r *\n      (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j *\n        p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun j))", "state_after": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 \u2200 (i : Fin k),\n    r * (a ^ \u27e8k, blocks_fun\u27e9.snd i * p (\u27e8k, blocks_fun\u27e9.snd i)) =\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191i, \u22ef\u27e9 *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191i, \u22ef\u27e9))"}, {"tactic": "intro j", "annotated_tactic": ["intro j", []], "state_before": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 \u2200 (i : Fin k),\n    r * (a ^ \u27e8k, blocks_fun\u27e9.snd i * p (\u27e8k, blocks_fun\u27e9.snd i)) =\n      r *\n        (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191i, \u22ef\u27e9 *\n          p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191i, \u22ef\u27e9))", "state_after": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\nj : Fin k\n\u22a2 r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    r *\n      (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191j, \u22ef\u27e9 *\n        p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191j, \u22ef\u27e9))"}, {"tactic": "rw [compChangeOfVariables_blocksFun]", "annotated_tactic": ["rw [<a>compChangeOfVariables_blocksFun</a>]", [{"full_name": "FormalMultilinearSeries.compChangeOfVariables_blocksFun", "def_path": "Mathlib/Analysis/Analytic/Composition.lean", "def_pos": [586, 9], "def_end_pos": [586, 40]}]], "state_before": "case h.mk.h.e_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\nj : Fin k\n\u22a2 r * (a ^ \u27e8k, blocks_fun\u27e9.snd j * p (\u27e8k, blocks_fun\u27e9.snd j)) =\n    r *\n      (a ^ (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191j, \u22ef\u27e9 *\n        p ((compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.blocksFun \u27e8\u2191j, \u22ef\u27e9))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\n\u22a2 (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k", "state_after": "no goals"}, {"tactic": "rw [K]", "annotated_tactic": ["rw [K]", []], "state_before": "case h.mk.h.e_4.e_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : \u27e8k, blocks_fun\u27e9 \u2208 compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k\n\u22a2 HEq (Fin.fintype \u27e8k, blocks_fun\u27e9.fst) (Fin.fintype (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length)", "state_after": "no goals"}, {"tactic": "rw [compPartialSumSource,\n  \u2190 sum_sigma' (Ico 2 (n + 1))\n    (fun k : \u2115 => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k \u2192 \u2115)))\n    (fun n e => \u220f j : Fin n, r * (a ^ e j * p (e j)))]", "annotated_tactic": ["rw [<a>compPartialSumSource</a>,\n        \u2190 <a>sum_sigma'</a> (<a>Ico</a> 2 (n + 1))\n          (fun k : \u2115 => (<a>Fintype.piFinset</a> fun _ : <a>Fin</a> k => <a>Ico</a> 1 n : <a>Finset</a> (<a>Fin</a> k \u2192 \u2115)))\n          (fun n e => \u220f j : <a>Fin</a> n, r * (a ^ e j * p (e j)))]", [{"full_name": "FormalMultilinearSeries.compPartialSumSource", "def_path": "Mathlib/Analysis/Analytic/Composition.lean", "def_pos": [553, 5], "def_end_pos": [553, 25]}, {"full_name": "Finset.sum_sigma'", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [633, 3], "def_end_pos": [633, 14]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 e \u2208 compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ e.snd j * p (e.snd j)) =\n    \u2211 j \u2208 Ico 2 (n + 1), r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 a_1 \u2208 Ico 2 (n + 1), \u2211 s \u2208 Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin a_1, r * (a ^ s j * p (s j)) =\n    \u2211 j \u2208 Ico 2 (n + 1), r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j"}, {"tactic": "congr! with j", "annotated_tactic": ["congr! with j", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 a_1 \u2208 Ico 2 (n + 1), \u2211 s \u2208 Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin a_1, r * (a ^ s j * p (s j)) =\n    \u2211 j \u2208 Ico 2 (n + 1), r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2211 s \u2208 Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin j, r * (a ^ s j * p (s j)) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j"}, {"tactic": "simp only [\u2190 @MultilinearMap.mkPiAlgebra_apply \u211d (Fin j) _ \u211d]", "annotated_tactic": ["simp only [\u2190 @<a>MultilinearMap.mkPiAlgebra_apply</a> \u211d (<a>Fin</a> j) _ \u211d]", [{"full_name": "MultilinearMap.mkPiAlgebra_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [1185, 9], "def_end_pos": [1185, 26]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2211 s \u2208 Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin j, r * (a ^ s j * p (s j)) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 (\u2211 x \u2208 Fintype.piFinset fun x => Ico 1 n, (MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun j => r * (a ^ x j * p (x j))) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j"}, {"tactic": "simp only [\u2190\n  MultilinearMap.map_sum_finset (MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun _ (m : \u2115) =>\n    r * (a ^ m * p m)]", "annotated_tactic": ["simp only [\u2190\n        <a>MultilinearMap.map_sum_finset</a> (<a>MultilinearMap.mkPiAlgebra</a> \u211d (<a>Fin</a> j) \u211d) fun _ (m : \u2115) =>\n          r * (a ^ m * p m)]", [{"full_name": "MultilinearMap.map_sum_finset", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [642, 9], "def_end_pos": [642, 23]}, {"full_name": "MultilinearMap.mkPiAlgebra", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [1176, 15], "def_end_pos": [1176, 26]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 (\u2211 x \u2208 Fintype.piFinset fun x => Ico 1 n, (MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun j => r * (a ^ x j * p (x j))) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 ((MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun i => \u2211 j \u2208 Ico 1 n, r * (a ^ j * p j)) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j"}, {"tactic": "simp only [MultilinearMap.mkPiAlgebra_apply]", "annotated_tactic": ["simp only [<a>MultilinearMap.mkPiAlgebra_apply</a>]", [{"full_name": "MultilinearMap.mkPiAlgebra_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [1185, 9], "def_end_pos": [1185, 26]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 ((MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun i => \u2211 j \u2208 Ico 1 n, r * (a ^ j * p j)) =\n    r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u220f i : Fin j, \u2211 j \u2208 Ico 1 n, r * (a ^ j * p j) = r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j"}, {"tactic": "simp [prod_const, \u2190 mul_sum, mul_pow]", "annotated_tactic": ["simp [<a>prod_const</a>, \u2190 <a>mul_sum</a>, <a>mul_pow</a>]", [{"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1737, 9], "def_end_pos": [1737, 19]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [61, 7], "def_end_pos": [61, 14]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [310, 32], "def_end_pos": [310, 39]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\na\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u220f i : Fin j, \u2211 j \u2208 Ico 1 n, r * (a ^ j * p j) = r ^ j * (\u2211 k \u2208 Ico 1 n, a ^ k * p k) ^ j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_succ_self", "start": [1326, 1], "end": [1327, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "full_name": "Matrix.det_succ_row_zero", "start": [765, 1], "end": [770, 72], "traced_tactics": [{"tactic": "rw [\u2190 det_transpose A, det_succ_column_zero]", "annotated_tactic": ["rw [\u2190 <a>det_transpose</a> A, <a>det_succ_column_zero</a>]", [{"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"full_name": "Matrix.det_succ_column_zero", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [733, 9], "def_end_pos": [733, 29]}]], "state_before": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\n\u22a2 A.det = \u2211 j : Fin n.succ, (-1) ^ \u2191j * A 0 j * (A.submatrix Fin.succ j.succAbove).det", "state_after": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\n\u22a2 \u2211 i : Fin n.succ, (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ).det =\n    \u2211 j : Fin n.succ, (-1) ^ \u2191j * A 0 j * (A.submatrix Fin.succ j.succAbove).det"}, {"tactic": "refine Finset.sum_congr rfl fun i _ => ?_", "annotated_tactic": ["refine <a>Finset.sum_congr</a> <a>rfl</a> fun i _ => ?_", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\n\u22a2 \u2211 i : Fin n.succ, (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ).det =\n    \u2211 j : Fin n.succ, (-1) ^ \u2191j * A 0 j * (A.submatrix Fin.succ j.succAbove).det", "state_after": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ).det =\n    (-1) ^ \u2191i * A 0 i * (A.submatrix Fin.succ i.succAbove).det"}, {"tactic": "rw [\u2190 det_transpose]", "annotated_tactic": ["rw [\u2190 <a>det_transpose</a>]", [{"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}]], "state_before": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ).det =\n    (-1) ^ \u2191i * A 0 i * (A.submatrix Fin.succ i.succAbove).det", "state_after": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ)\u1d40.det =\n    (-1) ^ \u2191i * A 0 i * (A.submatrix Fin.succ i.succAbove).det"}, {"tactic": "simp only [transpose_apply, transpose_submatrix, transpose_transpose]", "annotated_tactic": ["simp only [<a>transpose_apply</a>, <a>transpose_submatrix</a>, <a>transpose_transpose</a>]", [{"full_name": "Matrix.transpose_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 24]}, {"full_name": "Matrix.transpose_submatrix", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2626, 9], "def_end_pos": [2626, 28]}, {"full_name": "Matrix.transpose_transpose", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2063, 9], "def_end_pos": [2063, 28]}]], "state_before": "m : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni : Fin n.succ\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * (A\u1d40.submatrix i.succAbove Fin.succ)\u1d40.det =\n    (-1) ^ \u2191i * A 0 i * (A.submatrix Fin.succ i.succAbove).det", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EssentialImage.lean", "full_name": "CategoryTheory.Functor.essSurj_of_iso", "start": [174, 1], "end": [175, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cos_sq_le_one", "start": [948, 1], "end": [949, 72], "traced_tactics": [{"tactic": "rw [\u2190 sin_sq_add_cos_sq x]", "annotated_tactic": ["rw [\u2190 <a>sin_sq_add_cos_sq</a> x]", [{"full_name": "Real.sin_sq_add_cos_sq", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [936, 16], "def_end_pos": [936, 33]}]], "state_before": "x y : \u211d\n\u22a2 cos x ^ 2 \u2264 1", "state_after": "x y : \u211d\n\u22a2 cos x ^ 2 \u2264 sin x ^ 2 + cos x ^ 2"}, {"tactic": "exact le_add_of_nonneg_left (sq_nonneg _)", "annotated_tactic": ["exact <a>le_add_of_nonneg_left</a> (<a>sq_nonneg</a> _)", [{"full_name": "le_add_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [409, 15], "def_end_pos": [409, 36]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}]], "state_before": "x y : \u211d\n\u22a2 cos x ^ 2 \u2264 sin x ^ 2 + cos x ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Finset.lean", "full_name": "Finset.gcd_union", "start": [184, 1], "end": [186, 79], "traced_tactics": [{"tactic": "rw [empty_union, gcd_empty, gcd_zero_left, normalize_gcd]", "annotated_tactic": ["rw [<a>empty_union</a>, <a>gcd_empty</a>, <a>gcd_zero_left</a>, <a>normalize_gcd</a>]", [{"full_name": "Finset.empty_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1505, 9], "def_end_pos": [1505, 20]}, {"full_name": "Finset.gcd_empty", "def_path": "Mathlib/Algebra/GCDMonoid/Finset.lean", "def_pos": [147, 9], "def_end_pos": [147, 18]}, {"full_name": "gcd_zero_left", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [385, 9], "def_end_pos": [385, 22]}, {"full_name": "Finset.normalize_gcd", "def_path": "Mathlib/Algebra/GCDMonoid/Finset.lean", "def_pos": [181, 9], "def_end_pos": [181, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\n\u22a2 (\u2205 \u222a s\u2082).gcd f = GCDMonoid.gcd (\u2205.gcd f) (s\u2082.gcd f)", "state_after": "no goals"}, {"tactic": "rw [insert_union, gcd_insert, gcd_insert, ih, gcd_assoc]", "annotated_tactic": ["rw [<a>insert_union</a>, <a>gcd_insert</a>, <a>gcd_insert</a>, ih, <a>gcd_assoc</a>]", [{"full_name": "Finset.insert_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 21]}, {"full_name": "Finset.gcd_insert", "def_path": "Mathlib/Algebra/GCDMonoid/Finset.lean", "def_pos": [166, 9], "def_end_pos": [166, 19]}, {"full_name": "Finset.gcd_insert", "def_path": "Mathlib/Algebra/GCDMonoid/Finset.lean", "def_pos": [166, 9], "def_end_pos": [166, 19]}, {"full_name": "gcd_assoc", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 18]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b2\ns : Finset \u03b2\nx\u271d : a \u2209 s\nih : (s \u222a s\u2082).gcd f = GCDMonoid.gcd (s.gcd f) (s\u2082.gcd f)\n\u22a2 (insert a s \u222a s\u2082).gcd f = GCDMonoid.gcd ((insert a s).gcd f) (s\u2082.gcd f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.prod_quotient_preimage_eq_image", "start": [593, 1], "end": [602, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "full_name": "Ordinal.IsFundamentalSequence.strict_mono", "start": [560, 11], "end": [562, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean", "full_name": "LeftInvariantDerivation.lift_zero", "start": [174, 1], "end": [176, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Sqrt.lean", "full_name": "Real.sqrt_sq", "start": [205, 1], "end": [205, 74], "traced_tactics": [{"tactic": "rw [sq, sqrt_mul_self h]", "annotated_tactic": ["rw [<a>sq</a>, <a>sqrt_mul_self</a> h]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [684, 41], "def_end_pos": [684, 43]}, {"full_name": "Real.sqrt_mul_self", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [171, 9], "def_end_pos": [171, 22]}]], "state_before": "x y : \u211d\nh : 0 \u2264 x\n\u22a2 \u221a(x ^ 2) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_sdiff_cancel", "start": [2248, 1], "end": [2249, 73], "traced_tactics": [{"tactic": "rw [insert_sdiff_of_not_mem _ ha, Finset.sdiff_self, insert_emptyc_eq]", "annotated_tactic": ["rw [<a>insert_sdiff_of_not_mem</a> _ ha, <a>Finset.sdiff_self</a>, <a>insert_emptyc_eq</a>]", [{"full_name": "Finset.insert_sdiff_of_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2237, 9], "def_end_pos": [2237, 32]}, {"full_name": "Finset.sdiff_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2148, 19], "def_end_pos": [2148, 29]}, {"full_name": "LawfulSingleton.insert_emptyc_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [474, 3], "def_end_pos": [474, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : a \u2209 s\n\u22a2 insert a s \\ s = {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Instances/Sphere.lean", "full_name": "ContMDiff.codRestrict_sphere", "start": [465, 1], "end": [486, 61], "traced_tactics": [{"tactic": "rw [contMDiff_iff_target]", "annotated_tactic": ["rw [<a>contMDiff_iff_target</a>]", [{"full_name": "contMDiff_iff_target", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [621, 9], "def_end_pos": [621, 29]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\n\u22a2 ContMDiff I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m (Set.codRestrict f (sphere 0 1) hf')", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\n\u22a2 Continuous (Set.codRestrict f (sphere 0 1) hf') \u2227\n    \u2200 (y : \u2191(sphere 0 1)),\n      ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n        (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y) \u2218 Set.codRestrict f (sphere 0 1) hf')\n        (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y).source)"}, {"tactic": "refine \u27e8continuous_induced_rng.2 hf.continuous, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>continuous_induced_rng</a>.2 hf.continuous, ?_\u27e9", [{"full_name": "continuous_induced_rng", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [702, 9], "def_end_pos": [702, 31]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\n\u22a2 Continuous (Set.codRestrict f (sphere 0 1) hf') \u2227\n    \u2200 (y : \u2191(sphere 0 1)),\n      ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n        (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y) \u2218 Set.codRestrict f (sphere 0 1) hf')\n        (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\n\u22a2 \u2200 (y : \u2191(sphere 0 1)),\n    ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n      (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y) \u2218 Set.codRestrict f (sphere 0 1) hf')\n      (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y).source)"}, {"tactic": "intro v", "annotated_tactic": ["intro v", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\n\u22a2 \u2200 (y : \u2191(sphere 0 1)),\n    ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n      (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y) \u2218 Set.codRestrict f (sphere 0 1) hf')\n      (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) y).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)"}, {"tactic": "let U : _ \u2243\u2097\u1d62[\u211d] _ :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton\n      n (ne_zero_of_mem_unit_sphere (-v))).repr", "annotated_tactic": ["let U : _ \u2243\u2097\u1d62[\u211d] _ :=\n    (-- Again, partially removing type ascription... Weird that this helps!\n        <a>OrthonormalBasis.fromOrthogonalSpanSingleton</a>\n        n (<a>ne_zero_of_mem_unit_sphere</a> (-v))).<a>repr</a>", [{"full_name": "OrthonormalBasis.fromOrthogonalSpanSingleton", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [957, 5], "def_end_pos": [957, 49]}, {"full_name": "ne_zero_of_mem_unit_sphere", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [690, 15], "def_end_pos": [690, 41]}, {"full_name": "OrthonormalBasis.repr", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [341, 3], "def_end_pos": [341, 7]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)"}, {"tactic": "have h : ContDiffOn \u211d \u22a4 _ Set.univ := U.contDiff.contDiffOn", "annotated_tactic": ["have h : <a>ContDiffOn</a> \u211d \u22a4 _ <a>Set.univ</a> := U.contDiff.contDiffOn", [{"full_name": "ContDiffOn", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [640, 5], "def_end_pos": [640, 15]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)"}, {"tactic": "have H\u2081 := (h.comp' contDiffOn_stereoToFun).contMDiffOn", "annotated_tactic": ["have H\u2081 := (h.comp' <a>contDiffOn_stereoToFun</a>).<a>contMDiffOn</a>", [{"full_name": "contDiffOn_stereoToFun", "def_path": "Mathlib/Geometry/Manifold/Instances/Sphere.lean", "def_pos": [98, 9], "def_end_pos": [98, 31]}, {"full_name": "ContDiffOn.contMDiffOn", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean", "def_pos": [77, 32], "def_end_pos": [77, 54]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)"}, {"tactic": "have H\u2082 : ContMDiffOn _ _ _ _ Set.univ := hf.contMDiffOn", "annotated_tactic": ["have H\u2082 : <a>ContMDiffOn</a> _ _ _ _ <a>Set.univ</a> := hf.contMDiffOn", [{"full_name": "ContMDiffOn", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [208, 5], "def_end_pos": [208, 16]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)"}, {"tactic": "convert (H\u2081.of_le le_top).comp' H\u2082 using 1", "annotated_tactic": ["convert (H\u2081.of_le <a>le_top</a>).<a>comp'</a> H\u2082 using 1", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}, {"full_name": "ContMDiffOn.comp'", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Basic.lean", "def_pos": [107, 9], "def_end_pos": [107, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\n\u22a2 ContMDiffOn I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) m\n    (\u2191(extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v) \u2218 Set.codRestrict f (sphere 0 1) hf')\n    (Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source)", "state_after": "case h.e'_23\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\n\u22a2 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source =\n    Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_23\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\n\u22a2 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source =\n    Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)", "state_after": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source \u2194\n    x \u2208 Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)"}, {"tactic": "have hfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1 := by\n  have hfx : \u2016f x\u2016 = 1 := by simpa using hf' x\n  rw [inner_eq_one_iff_of_norm_one hfx]\n  exact norm_eq_of_mem_sphere (-v)", "annotated_tactic": ["have hfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1 := by\n    have hfx : \u2016f x\u2016 = 1 := by simpa using hf' x\n    rw [<a>inner_eq_one_iff_of_norm_one</a> hfx]\n    exact <a>norm_eq_of_mem_sphere</a> (-v)", [{"full_name": "inner_eq_one_iff_of_norm_one", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1703, 9], "def_end_pos": [1703, 37]}, {"full_name": "norm_eq_of_mem_sphere", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [678, 30], "def_end_pos": [678, 51]}]], "state_before": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source \u2194\n    x \u2208 Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)", "state_after": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source \u2194\n    x \u2208 Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)"}, {"tactic": "dsimp [chartAt, Set.codRestrict, ChartedSpace.chartAt]", "annotated_tactic": ["dsimp [<a>chartAt</a>, <a>Set.codRestrict</a>, <a>ChartedSpace.chartAt</a>]", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}, {"full_name": "Set.codRestrict", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [148, 5], "def_end_pos": [148, 16]}, {"full_name": "ChartedSpace.chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [578, 13], "def_end_pos": [578, 20]}]], "state_before": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (extChartAt \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) v).source \u2194\n    x \u2208 Set.univ \u2229 f \u207b\u00b9' ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)", "state_after": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (stereographic' n (-v)).source \u2229 Set.univ \u2194\n    x \u2208 Set.univ \u2229 ({a | \u00ac\u27ea-\u2191v, f a\u27eb_\u211d = 1} \u2229 Set.univ)"}, {"tactic": "simp [not_iff_not, Subtype.ext_iff, hfxv, real_inner_comm]", "annotated_tactic": ["simp [<a>not_iff_not</a>, <a>Subtype.ext_iff</a>, hfxv, <a>real_inner_comm</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [447, 9], "def_end_pos": [447, 20]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "real_inner_comm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 24]}]], "state_before": "case h.e'_23.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfxv : f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1\n\u22a2 x \u2208 Set.codRestrict f (sphere 0 1) hf' \u207b\u00b9' (stereographic' n (-v)).source \u2229 Set.univ \u2194\n    x \u2208 Set.univ \u2229 ({a | \u00ac\u27ea-\u2191v, f a\u27eb_\u211d = 1} \u2229 Set.univ)", "state_after": "no goals"}, {"tactic": "have hfx : \u2016f x\u2016 = 1 := by simpa using hf' x", "annotated_tactic": ["have hfx : \u2016f x\u2016 = 1 := by simpa using hf' x", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\n\u22a2 f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfx : \u2016f x\u2016 = 1\n\u22a2 f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1"}, {"tactic": "rw [inner_eq_one_iff_of_norm_one hfx]", "annotated_tactic": ["rw [<a>inner_eq_one_iff_of_norm_one</a> hfx]", [{"full_name": "inner_eq_one_iff_of_norm_one", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1703, 9], "def_end_pos": [1703, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfx : \u2016f x\u2016 = 1\n\u22a2 f x = -\u2191v \u2194 \u27eaf x, -\u2191v\u27eb_\u211d = 1", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfx : \u2016f x\u2016 = 1\n\u22a2 \u2016-\u2191v\u2016 = 1"}, {"tactic": "exact norm_eq_of_mem_sphere (-v)", "annotated_tactic": ["exact <a>norm_eq_of_mem_sphere</a> (-v)", [{"full_name": "norm_eq_of_mem_sphere", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [678, 30], "def_end_pos": [678, 51]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\nhfx : \u2016f x\u2016 = 1\n\u22a2 \u2016-\u2191v\u2016 = 1", "state_after": "no goals"}, {"tactic": "simpa using hf' x", "annotated_tactic": ["simpa using hf' x", []], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d F H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nn : \u2115\ninst\u271d : Fact (finrank \u211d E = n + 1)\nm : \u2115\u221e\nf : M \u2192 E\nhf : ContMDiff I \ud835\udcd8(\u211d, E) m f\nhf' : \u2200 (x : M), f x \u2208 sphere 0 1\nv : \u2191(sphere 0 1)\nU : \u21a5(Submodule.span \u211d {\u2191(-v)})\u15ee \u2243\u2097\u1d62[\u211d] EuclideanSpace \u211d (Fin n) :=\n  (OrthonormalBasis.fromOrthogonalSpanSingleton n \u22ef).repr\nh : ContDiffOn \u211d \u22a4 (\u21d1U) Set.univ\nH\u2081 :\n  ContMDiffOn \ud835\udcd8(\u211d, E) \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) \u22a4 (\u21d1U \u2218 stereoToFun \u2191(-v))\n    ({x | ((innerSL \u211d) \u2191(-v)) x \u2260 1} \u2229 stereoToFun \u2191(-v) \u207b\u00b9' Set.univ)\nH\u2082 : ContMDiffOn I \ud835\udcd8(\u211d, E) m f Set.univ\nx : M\n\u22a2 \u2016f x\u2016 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.blsub_pos", "start": [1889, 1], "end": [1890, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Semisimple.lean", "full_name": "Module.End.IsSemisimple.minpoly_squarefree", "start": [141, 1], "end": [143, 99], "traced_tactics": [{"tactic": "rw [isRadical_iff_span_singleton, span_minpoly_eq_annihilator]", "annotated_tactic": ["rw [<a>isRadical_iff_span_singleton</a>, <a>span_minpoly_eq_annihilator</a>]", [{"full_name": "isRadical_iff_span_singleton", "def_path": "Mathlib/RingTheory/Nilpotent/Lemmas.lean", "def_pos": [32, 9], "def_end_pos": [32, 37]}, {"full_name": "Polynomial.span_minpoly_eq_annihilator", "def_path": "Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean", "def_pos": [179, 9], "def_end_pos": [179, 36]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nK : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Module K M\nf g : End K M\ninst\u271d : FiniteDimensional K M\nhf : f.IsSemisimple\n\u22a2 IsRadical (minpoly K f)", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nK : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Module K M\nf g : End K M\ninst\u271d : FiniteDimensional K M\nhf : f.IsSemisimple\n\u22a2 (annihilator K[X] (AEval' f)).IsRadical"}, {"tactic": "exact hf.annihilator_isRadical", "annotated_tactic": ["exact hf.annihilator_isRadical", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nK : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Module K M\nf g : End K M\ninst\u271d : FiniteDimensional K M\nhf : f.IsSemisimple\n\u22a2 (annihilator K[X] (AEval' f)).IsRadical", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Antisymmetrization.lean", "full_name": "AntisymmRel.image", "start": [135, 1], "end": [137, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/FourFunctions.lean", "full_name": "four_functions_theorem", "start": [267, 1], "end": [300, 94], "traced_tactics": [{"tactic": "set L : Sublattice \u03b1 := \u27e8latticeClosure (s \u222a t), isSublattice_latticeClosure.1,\n  isSublattice_latticeClosure.2\u27e9", "annotated_tactic": ["set L : <a>Sublattice</a> \u03b1 := \u27e8<a>latticeClosure</a> (s \u222a t), <a>isSublattice_latticeClosure</a>.1,\n    <a>isSublattice_latticeClosure</a>.2\u27e9", [{"full_name": "Sublattice", "def_path": "Mathlib/Order/Sublattice.lean", "def_pos": [27, 11], "def_end_pos": [27, 21]}, {"full_name": "latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [388, 5], "def_end_pos": [388, 19]}, {"full_name": "isSublattice_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [393, 15], "def_end_pos": [393, 42]}, {"full_name": "isSublattice_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [393, 15], "def_end_pos": [393, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "have : Finite L := (s.finite_toSet.union t.finite_toSet).latticeClosure.to_subtype", "annotated_tactic": ["have : <a>Finite</a> L := (s.finite_toSet.union t.finite_toSet).latticeClosure.to_subtype", [{"full_name": "Finite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [81, 17], "def_end_pos": [81, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "set s' : Finset L := s.preimage (\u2191) Subtype.coe_injective.injOn", "annotated_tactic": ["set s' : <a>Finset</a> L := s.preimage (\u2191) Subtype.coe_injective.injOn", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "set t' : Finset L := t.preimage (\u2191) Subtype.coe_injective.injOn", "annotated_tactic": ["set t' : <a>Finset</a> L := t.preimage (\u2191) Subtype.coe_injective.injOn", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "have hs' : s'.map \u27e8L.subtype, Subtype.coe_injective\u27e9 = s := by\n  simp [s', map_eq_image, image_preimage, filter_eq_self]\n  exact fun a ha \u21a6 subset_latticeClosure <| Set.subset_union_left ha", "annotated_tactic": ["have hs' : s'.map \u27e8L.subtype, <a>Subtype.coe_injective</a>\u27e9 = s := by\n    simp [s', <a>map_eq_image</a>, <a>image_preimage</a>, <a>filter_eq_self</a>]\n    exact fun a ha \u21a6 <a>subset_latticeClosure</a> <| <a>Set.subset_union_left</a> ha", [{"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [369, 9], "def_end_pos": [369, 21]}, {"full_name": "Finset.image_preimage", "def_path": "Mathlib/Data/Finset/Preimage.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Finset.filter_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2634, 7], "def_end_pos": [2634, 21]}, {"full_name": "subset_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [391, 15], "def_end_pos": [391, 36]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "have ht' : t'.map \u27e8L.subtype, Subtype.coe_injective\u27e9 = t := by\n  simp [t', map_eq_image, image_preimage, filter_eq_self]\n  exact fun a ha \u21a6 subset_latticeClosure <| Set.subset_union_right ha", "annotated_tactic": ["have ht' : t'.map \u27e8L.subtype, <a>Subtype.coe_injective</a>\u27e9 = t := by\n    simp [t', <a>map_eq_image</a>, <a>image_preimage</a>, <a>filter_eq_self</a>]\n    exact fun a ha \u21a6 <a>subset_latticeClosure</a> <| <a>Set.subset_union_right</a> ha", [{"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [369, 9], "def_end_pos": [369, 21]}, {"full_name": "Finset.image_preimage", "def_path": "Mathlib/Data/Finset/Preimage.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Finset.filter_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2634, 7], "def_end_pos": [2634, 21]}, {"full_name": "subset_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [391, 15], "def_end_pos": [391, 36]}, {"full_name": "Set.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "clear_value s' t'", "annotated_tactic": ["clear_value s' t'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "obtain \u27e8\u03b2, _, _, g, hg\u27e9 := exists_birkhoff_representation L", "annotated_tactic": ["obtain \u27e8\u03b2, _, _, g, hg\u27e9 := <a>exists_birkhoff_representation</a> L", [{"full_name": "exists_birkhoff_representation", "def_path": "Mathlib/Order/Birkhoff.lean", "def_pos": [277, 7], "def_end_pos": [277, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a"}, {"tactic": "have := four_functions_theorem_aux (extend g (f\u2081 \u2218 (\u2191)) 0) (extend g (f\u2082 \u2218 (\u2191)) 0)\n  (extend g (f\u2083 \u2218 (\u2191)) 0) (extend g (f\u2084 \u2218 (\u2191)) 0) (extend_nonneg (fun _ \u21a6 h\u2081 _) le_rfl)\n  (extend_nonneg (fun _ \u21a6 h\u2082 _) le_rfl) (extend_nonneg (fun _ \u21a6 h\u2083 _) le_rfl)\n  (extend_nonneg (fun _ \u21a6 h\u2084 _) le_rfl) ?_ (s'.map \u27e8g, hg\u27e9) (t'.map \u27e8g, hg\u27e9)", "annotated_tactic": ["have := <a>four_functions_theorem_aux</a> (<a>extend</a> g (f\u2081 \u2218 (\u2191)) 0) (<a>extend</a> g (f\u2082 \u2218 (\u2191)) 0)\n    (<a>extend</a> g (f\u2083 \u2218 (\u2191)) 0) (<a>extend</a> g (f\u2084 \u2218 (\u2191)) 0) (<a>extend_nonneg</a> (fun _ \u21a6 h\u2081 _) <a>le_rfl</a>)\n    (<a>extend_nonneg</a> (fun _ \u21a6 h\u2082 _) <a>le_rfl</a>) (<a>extend_nonneg</a> (fun _ \u21a6 h\u2083 _) <a>le_rfl</a>)\n    (<a>extend_nonneg</a> (fun _ \u21a6 h\u2084 _) <a>le_rfl</a>) ?_ (s'.map \u27e8g, hg\u27e9) (t'.map \u27e8g, hg\u27e9)", [{"full_name": "_private.Mathlib.Combinatorics.SetFamily.FourFunctions.0.four_functions_theorem_aux", "def_path": "Mathlib/Combinatorics/SetFamily/FourFunctions.lean", "def_pos": [256, 15], "def_end_pos": [256, 41]}, {"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [717, 5], "def_end_pos": [717, 11]}, {"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [717, 5], "def_end_pos": [717, 11]}, {"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [717, 5], "def_end_pos": [717, 11]}, {"full_name": "Function.extend", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [717, 5], "def_end_pos": [717, 11]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "case intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis\u271d : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nthis :\n  (\u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s', extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s) *\n      \u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } t', extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 s \u2264\n    (\u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s' \u22bc map { toFun := \u21d1g, inj' := hg } t',\n        extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 s) *\n      \u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s' \u22bb map { toFun := \u21d1g, inj' := hg } t', extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a\n\ncase intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\n\u22a2 \u2200 (s t : Finset \u03b2),\n    extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n      extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)"}, {"tactic": "rintro s t", "annotated_tactic": ["rintro s t", []], "state_before": "case intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\n\u22a2 \u2200 (s t : Finset \u03b2),\n    extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n      extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)", "state_after": "case intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns\u271d t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s\u271d \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\u271d\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\ns t : Finset \u03b2\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)"}, {"tactic": "simp [s', map_eq_image, image_preimage, filter_eq_self]", "annotated_tactic": ["simp [s', <a>map_eq_image</a>, <a>image_preimage</a>, <a>filter_eq_self</a>]", [{"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [369, 9], "def_end_pos": [369, 21]}, {"full_name": "Finset.image_preimage", "def_path": "Mathlib/Data/Finset/Preimage.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Finset.filter_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2634, 7], "def_end_pos": [2634, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\n\u22a2 map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\n\u22a2 \u2200 x \u2208 s, x \u2208 L"}, {"tactic": "exact fun a ha \u21a6 subset_latticeClosure <| Set.subset_union_left ha", "annotated_tactic": ["exact fun a ha \u21a6 <a>subset_latticeClosure</a> <| <a>Set.subset_union_left</a> ha", [{"full_name": "subset_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [391, 15], "def_end_pos": [391, 36]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\n\u22a2 \u2200 x \u2208 s, x \u2208 L", "state_after": "no goals"}, {"tactic": "simp [t', map_eq_image, image_preimage, filter_eq_self]", "annotated_tactic": ["simp [t', <a>map_eq_image</a>, <a>image_preimage</a>, <a>filter_eq_self</a>]", [{"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [369, 9], "def_end_pos": [369, 21]}, {"full_name": "Finset.image_preimage", "def_path": "Mathlib/Data/Finset/Preimage.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "Finset.filter_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2634, 7], "def_end_pos": [2634, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 \u2200 x \u2208 t, x \u2208 L"}, {"tactic": "exact fun a ha \u21a6 subset_latticeClosure <| Set.subset_union_right ha", "annotated_tactic": ["exact fun a ha \u21a6 <a>subset_latticeClosure</a> <| <a>Set.subset_union_right</a> ha", [{"full_name": "subset_latticeClosure", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [391, 15], "def_end_pos": [391, 36]}, {"full_name": "Set.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\nf f\u2081 f\u2082 f\u2083 f\u2084 g \u03bc : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\ns' : Finset \u21a5L := s.preimage Subtype.val \u22ef\nt' : Finset \u21a5L := t.preimage Subtype.val \u22ef\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u22a2 \u2200 x \u2208 t, x \u2208 L", "state_after": "no goals"}, {"tactic": "simpa only [\u2190 hs', \u2190 ht', \u2190 map_sups, \u2190 map_infs, sum_map, Embedding.coeFn_mk, hg.extend_apply]\n  using this", "annotated_tactic": ["simpa only [\u2190 hs', \u2190 ht', \u2190 <a>map_sups</a>, \u2190 <a>map_infs</a>, <a>sum_map</a>, <a>Embedding.coeFn_mk</a>, hg.extend_apply]\n      using this", [{"full_name": "Finset.map_sups", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [185, 7], "def_end_pos": [185, 15]}, {"full_name": "Finset.map_infs", "def_path": "Mathlib/Data/Finset/Sups.lean", "def_pos": [368, 7], "def_end_pos": [368, 15]}, {"full_name": "Finset.sum_map", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [407, 3], "def_end_pos": [407, 14]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 17]}]], "state_before": "case intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis\u271d : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nthis :\n  (\u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s', extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s) *\n      \u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } t', extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 s \u2264\n    (\u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s' \u22bc map { toFun := \u21d1g, inj' := hg } t',\n        extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 s) *\n      \u2211 s \u2208 map { toFun := \u21d1g, inj' := hg } s' \u22bb map { toFun := \u21d1g, inj' := hg } t', extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 s\n\u22a2 (\u2211 a \u2208 s, f\u2081 a) * \u2211 a \u2208 t, f\u2082 a \u2264 (\u2211 a \u2208 s \u22bc t, f\u2083 a) * \u2211 a \u2208 s \u22bb t, f\u2084 a", "state_after": "no goals"}, {"tactic": "obtain \u27e8a, rfl\u27e9 | hs := em (\u2203 a, g a = s)", "annotated_tactic": ["obtain \u27e8a, rfl\u27e9 | hs := <a>em</a> (\u2203 a, g a = s)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}]], "state_before": "case intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns\u271d t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s\u271d \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\u271d\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\ns t : Finset \u03b2\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)", "state_after": "case intro.intro.intro.intro.refine_1.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nt : Finset \u03b2\na : \u21a5L\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a t)\n\ncase intro.intro.intro.intro.refine_1.inr\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns\u271d t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s\u271d \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\u271d\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\ns t : Finset \u03b2\nhs : \u00ac\u2203 a, g a = s\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)"}, {"tactic": "obtain \u27e8b, rfl\u27e9 | ht := em (\u2203 b, g b = t)", "annotated_tactic": ["obtain \u27e8b, rfl\u27e9 | ht := <a>em</a> (\u2203 b, g b = t)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}]], "state_before": "case intro.intro.intro.intro.refine_1.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nt : Finset \u03b2\na : \u21a5L\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a t)", "state_after": "case intro.intro.intro.intro.refine_1.inl.intro.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\na b : \u21a5L\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 (g b) \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 g b) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a g b)\n\ncase intro.intro.intro.intro.refine_1.inl.intro.inr\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nt : Finset \u03b2\na : \u21a5L\nht : \u00ac\u2203 b, g b = t\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a t)"}, {"tactic": "simp_rw [\u2190 sup_eq_union, \u2190 inf_eq_inter, \u2190 map_sup, \u2190 map_inf, hg.extend_apply]", "annotated_tactic": ["simp_rw [\u2190 <a>sup_eq_union</a>, \u2190 <a>inf_eq_inter</a>, \u2190 <a>map_sup</a>, \u2190 <a>map_inf</a>, hg.extend_apply]", [{"full_name": "Finset.sup_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 21]}, {"full_name": "Finset.inf_eq_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1379, 9], "def_end_pos": [1379, 21]}, {"full_name": "SupHomClass.map_sup", "def_path": "Mathlib/Order/Hom/Lattice.lean", "def_pos": [104, 3], "def_end_pos": [104, 10]}, {"full_name": "InfHomClass.map_inf", "def_path": "Mathlib/Order/Hom/Lattice.lean", "def_pos": [112, 3], "def_end_pos": [112, 10]}]], "state_before": "case intro.intro.intro.intro.refine_1.inl.intro.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\na b : \u21a5L\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 (g b) \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 g b) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a g b)", "state_after": "case intro.intro.intro.intro.refine_1.inl.intro.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\na b : \u21a5L\n\u22a2 (f\u2081 \u2218 Subtype.val) a * (f\u2082 \u2218 Subtype.val) b \u2264 (f\u2083 \u2218 Subtype.val) (a \u2293 b) * (f\u2084 \u2218 Subtype.val) (a \u2294 b)"}, {"tactic": "exact h _ _", "annotated_tactic": ["exact h _ _", []], "state_before": "case intro.intro.intro.intro.refine_1.inl.intro.inl.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\na b : \u21a5L\n\u22a2 (f\u2081 \u2218 Subtype.val) a * (f\u2082 \u2218 Subtype.val) b \u2264 (f\u2083 \u2218 Subtype.val) (a \u2293 b) * (f\u2084 \u2218 Subtype.val) (a \u2294 b)", "state_after": "no goals"}, {"tactic": "simpa [extend_apply' _ _ _ ht] using mul_nonneg\n  (extend_nonneg (fun a : L \u21a6 h\u2083 a) le_rfl _) (extend_nonneg (fun a : L \u21a6 h\u2084 a) le_rfl _)", "annotated_tactic": ["simpa [<a>extend_apply'</a> _ _ _ ht] using <a>mul_nonneg</a>\n        (<a>extend_nonneg</a> (fun a : L \u21a6 h\u2083 a) <a>le_rfl</a> _) (<a>extend_nonneg</a> (fun a : L \u21a6 h\u2084 a) <a>le_rfl</a> _)", [{"full_name": "Function.extend_apply'", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [749, 9], "def_end_pos": [749, 22]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case intro.intro.intro.intro.refine_1.inl.intro.inr\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\nt : Finset \u03b2\na : \u21a5L\nht : \u00ac\u2203 b, g b = t\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 (g a) * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (g a \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (g a \u222a t)", "state_after": "no goals"}, {"tactic": "simpa [extend_apply' _ _ _ hs] using mul_nonneg\n  (extend_nonneg (fun a : L \u21a6 h\u2083 a) le_rfl _) (extend_nonneg (fun a : L \u21a6 h\u2084 a) le_rfl _)", "annotated_tactic": ["simpa [<a>extend_apply'</a> _ _ _ hs] using <a>mul_nonneg</a>\n      (<a>extend_nonneg</a> (fun a : L \u21a6 h\u2083 a) <a>le_rfl</a> _) (<a>extend_nonneg</a> (fun a : L \u21a6 h\u2084 a) <a>le_rfl</a> _)", [{"full_name": "Function.extend_apply'", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [749, 9], "def_end_pos": [749, 22]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Function.extend_nonneg", "def_path": "Mathlib/Algebra/Order/Pi.lean", "def_pos": [148, 15], "def_end_pos": [148, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case intro.intro.intro.intro.refine_1.inr\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b3 : DistribLattice \u03b1\ninst\u271d\u00b2 : LinearOrderedCommSemiring \u03b2\u271d\ninst\u271d\u00b9 : ExistsAddOfLE \u03b2\u271d\nf f\u2081 f\u2082 f\u2083 f\u2084 g\u271d \u03bc : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\nh\u2081 : 0 \u2264 f\u2081\nh\u2082 : 0 \u2264 f\u2082\nh\u2083 : 0 \u2264 f\u2083\nh\u2084 : 0 \u2264 f\u2084\nh : \u2200 (a b : \u03b1), f\u2081 a * f\u2082 b \u2264 f\u2083 (a \u2293 b) * f\u2084 (a \u2294 b)\ns\u271d t\u271d : Finset \u03b1\nL : Sublattice \u03b1 := { carrier := latticeClosure (\u2191s\u271d \u222a \u2191t\u271d), supClosed' := \u22ef, infClosed' := \u22ef }\nthis : Finite \u21a5L\nt' : Finset \u21a5L\nht' : map { toFun := \u21d1L.subtype, inj' := \u22ef } t' = t\u271d\ns' : Finset \u21a5L\nhs' : map { toFun := \u21d1L.subtype, inj' := \u22ef } s' = s\u271d\n\u03b2 : Type u_1\nw\u271d\u00b9 : DecidableEq \u03b2\nw\u271d : Fintype \u03b2\ng : LatticeHom (\u21a5L) (Finset \u03b2)\nhg : Injective \u21d1g\ns t : Finset \u03b2\nhs : \u00ac\u2203 a, g a = s\n\u22a2 extend (\u21d1g) (f\u2081 \u2218 Subtype.val) 0 s * extend (\u21d1g) (f\u2082 \u2218 Subtype.val) 0 t \u2264\n    extend (\u21d1g) (f\u2083 \u2218 Subtype.val) 0 (s \u2229 t) * extend (\u21d1g) (f\u2084 \u2218 Subtype.val) 0 (s \u222a t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "full_name": "ConvexCone.mem_inf", "start": [121, 1], "end": [122, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.graphOn_singleton", "start": [817, 1], "end": [819, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/LocallyBijective.lean", "full_name": "CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.mk'", "start": [135, 1], "end": [145, 45], "traced_tactics": [{"tactic": "rw [W_iff, \u2190 Sheaf.isLocallyBijective_iff_isIso,\n  \u2190 Presheaf.isLocallyInjective_comp_iff J f (CategoryTheory.toSheafify J Q),\n  \u2190 Presheaf.isLocallySurjective_comp_iff J f (CategoryTheory.toSheafify J Q),\n  CategoryTheory.toSheafify_naturality, Presheaf.comp_isLocallyInjective_iff,\n  Presheaf.comp_isLocallySurjective_iff]", "annotated_tactic": ["rw [<a>W_iff</a>, \u2190 <a>Sheaf.isLocallyBijective_iff_isIso</a>,\n      \u2190 <a>Presheaf.isLocallyInjective_comp_iff</a> J f (<a>CategoryTheory.toSheafify</a> J Q),\n      \u2190 <a>Presheaf.isLocallySurjective_comp_iff</a> J f (<a>CategoryTheory.toSheafify</a> J Q),\n      <a>CategoryTheory.toSheafify_naturality</a>, <a>Presheaf.comp_isLocallyInjective_iff</a>,\n      <a>Presheaf.comp_isLocallySurjective_iff</a>]", [{"full_name": "CategoryTheory.GrothendieckTopology.W_iff", "def_path": "Mathlib/CategoryTheory/Sites/Localization.lean", "def_pos": [73, 7], "def_end_pos": [73, 12]}, {"full_name": "CategoryTheory.Sheaf.isLocallyBijective_iff_isIso", "def_path": "Mathlib/CategoryTheory/Sites/LocallyBijective.lean", "def_pos": [78, 7], "def_end_pos": [78, 35]}, {"full_name": "CategoryTheory.Presheaf.isLocallyInjective_comp_iff", "def_path": "Mathlib/CategoryTheory/Sites/LocallyInjective.lean", "def_pos": [157, 7], "def_end_pos": [157, 34]}, {"full_name": "CategoryTheory.toSheafify", "def_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "def_pos": [84, 22], "def_end_pos": [84, 32]}, {"full_name": "CategoryTheory.Presheaf.isLocallySurjective_comp_iff", "def_path": "Mathlib/CategoryTheory/Sites/LocallySurjective.lean", "def_pos": [253, 7], "def_end_pos": [253, 35]}, {"full_name": "CategoryTheory.toSheafify", "def_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "def_pos": [84, 22], "def_end_pos": [84, 32]}, {"full_name": "CategoryTheory.toSheafify_naturality", "def_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "def_pos": [105, 9], "def_end_pos": [105, 30]}, {"full_name": "CategoryTheory.Presheaf.comp_isLocallyInjective_iff", "def_path": "Mathlib/CategoryTheory/Sites/LocallySurjective.lean", "def_pos": [243, 7], "def_end_pos": [243, 34]}, {"full_name": "CategoryTheory.Presheaf.comp_isLocallySurjective_iff", "def_path": "Mathlib/CategoryTheory/Sites/LocallySurjective.lean", "def_pos": [179, 7], "def_end_pos": [179, 35]}]], "state_before": "C : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2076 : Category.{v', u'} A\ninst\u271d\u2075 : ConcreteCategory A\ninst\u271d\u2074 : HasWeakSheafify J A\ninst\u271d\u00b3 : (forget A).ReflectsIsomorphisms\ninst\u271d\u00b2 : J.HasSheafCompose (forget A)\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 A), Presheaf.IsLocallyInjective J (CategoryTheory.toSheafify J P)\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 A), Presheaf.IsLocallySurjective J (CategoryTheory.toSheafify J P)\nP Q : C\u1d52\u1d56 \u2964 A\nf : P \u27f6 Q\n\u22a2 J.W f \u2194 Presheaf.IsLocallyInjective J f \u2227 Presheaf.IsLocallySurjective J f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean", "full_name": "LinearMap.BilinForm.isCompl_orthogonal_of_restrict_nondegenerate", "start": [371, 1], "end": [385, 20], "traced_tactics": [{"tactic": "have : W \u2293 B.orthogonal W = \u22a5 := by\n  rw [eq_bot_iff]\n  intro x hx\n  obtain \u27e8hx\u2081, hx\u2082\u27e9 := mem_inf.1 hx\n  refine Subtype.mk_eq_mk.1 (b\u2082 \u27e8x, hx\u2081\u27e9 ?_)\n  rintro \u27e8n, hn\u27e9\n  simp only [restrict_apply, domRestrict_apply]\n  exact b\u2081 n x (b\u2081 x n (b\u2081 n x (hx\u2082 n hn)))", "annotated_tactic": ["have : W \u2293 B.orthogonal W = \u22a5 := by\n    rw [<a>eq_bot_iff</a>]\n    intro x hx\n    obtain \u27e8hx\u2081, hx\u2082\u27e9 := <a>mem_inf</a>.1 hx\n    refine <a>Subtype.mk_eq_mk</a>.1 (b\u2082 \u27e8x, hx\u2081\u27e9 ?_)\n    rintro \u27e8n, hn\u27e9\n    simp only [<a>restrict_apply</a>, <a>domRestrict_apply</a>]\n    exact b\u2081 n x (b\u2081 x n (b\u2081 n x (hx\u2082 n hn)))", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}, {"full_name": "Submodule.mem_inf", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [232, 9], "def_end_pos": [232, 16]}, {"full_name": "Subtype.mk_eq_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [116, 9], "def_end_pos": [116, 17]}, {"full_name": "LinearMap.BilinForm.restrict_apply", "def_path": "Mathlib/LinearAlgebra/BilinearForm/Basic.lean", "def_pos": [226, 10], "def_end_pos": [226, 15]}, {"full_name": "LinearMap.domRestrict_apply", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [140, 9], "def_end_pos": [140, 26]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\n\u22a2 IsCompl W (B.orthogonal W)", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 IsCompl W (B.orthogonal W)"}, {"tactic": "refine IsCompl.of_eq this (eq_top_of_finrank_eq <| (finrank_le _).antisymm ?_)", "annotated_tactic": ["refine <a>IsCompl.of_eq</a> this (<a>eq_top_of_finrank_eq</a> <| (<a>finrank_le</a> _).<a>antisymm</a> ?_)", [{"full_name": "IsCompl.of_eq", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [512, 9], "def_end_pos": [512, 14]}, {"full_name": "Submodule.eq_top_of_finrank_eq", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [225, 9], "def_end_pos": [225, 46]}, {"full_name": "Submodule.finrank_le", "def_path": "Mathlib/LinearAlgebra/Dimension/Constructions.lean", "def_pos": [406, 9], "def_end_pos": [406, 29]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [123, 7], "def_end_pos": [123, 21]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 IsCompl W (B.orthogonal W)", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K \u21a5(W \u2294 B.orthogonal W)"}, {"tactic": "conv_rhs => rw [\u2190 add_zero (finrank K _)]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>add_zero</a> (<a>finrank</a> K _)]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K \u21a5(W \u2294 B.orthogonal W)", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K \u21a5(W \u2294 B.orthogonal W) + 0"}, {"tactic": "rw [\u2190 finrank_bot K V, \u2190 this, finrank_sup_add_finrank_inf_eq,\n  finrank_add_finrank_orthogonal b\u2081]", "annotated_tactic": ["rw [\u2190 <a>finrank_bot</a> K V, \u2190 this, <a>finrank_sup_add_finrank_inf_eq</a>,\n    <a>finrank_add_finrank_orthogonal</a> b\u2081]", [{"full_name": "finrank_bot", "def_path": "Mathlib/LinearAlgebra/Dimension/Finite.lean", "def_pos": [399, 9], "def_end_pos": [399, 20]}, {"full_name": "Submodule.finrank_sup_add_finrank_inf_eq", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [483, 9], "def_end_pos": [483, 39]}, {"full_name": "LinearMap.BilinForm.finrank_add_finrank_orthogonal", "def_path": "Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean", "def_pos": [330, 9], "def_end_pos": [330, 39]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K \u21a5(W \u2294 B.orthogonal W) + 0", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K V + finrank K \u21a5(W \u2293 B.orthogonal \u22a4)"}, {"tactic": "exact le_self_add", "annotated_tactic": ["exact <a>le_self_add</a>", [{"full_name": "le_self_add", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [141, 3], "def_end_pos": [141, 14]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nthis : W \u2293 B.orthogonal W = \u22a5\n\u22a2 finrank K V \u2264 finrank K V + finrank K \u21a5(W \u2293 B.orthogonal \u22a4)", "state_after": "no goals"}, {"tactic": "rw [eq_bot_iff]", "annotated_tactic": ["rw [<a>eq_bot_iff</a>]", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\n\u22a2 W \u2293 B.orthogonal W = \u22a5", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\n\u22a2 W \u2293 B.orthogonal W \u2264 \u22a5"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\n\u22a2 W \u2293 B.orthogonal W \u2264 \u22a5", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\n\u22a2 x \u2208 \u22a5"}, {"tactic": "obtain \u27e8hx\u2081, hx\u2082\u27e9 := mem_inf.1 hx", "annotated_tactic": ["obtain \u27e8hx\u2081, hx\u2082\u27e9 := <a>mem_inf</a>.1 hx", [{"full_name": "Submodule.mem_inf", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [232, 9], "def_end_pos": [232, 16]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\n\u22a2 x \u2208 \u22a5", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\n\u22a2 x \u2208 \u22a5"}, {"tactic": "refine Subtype.mk_eq_mk.1 (b\u2082 \u27e8x, hx\u2081\u27e9 ?_)", "annotated_tactic": ["refine <a>Subtype.mk_eq_mk</a>.1 (b\u2082 \u27e8x, hx\u2081\u27e9 ?_)", [{"full_name": "Subtype.mk_eq_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [116, 9], "def_end_pos": [116, 17]}]], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\n\u22a2 x \u2208 \u22a5", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\n\u22a2 \u2200 (n : \u21a5W), ((B.restrict W) \u27e8x, hx\u2081\u27e9) n = 0"}, {"tactic": "rintro \u27e8n, hn\u27e9", "annotated_tactic": ["rintro \u27e8n, hn\u27e9", []], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\n\u22a2 \u2200 (n : \u21a5W), ((B.restrict W) \u27e8x, hx\u2081\u27e9) n = 0", "state_after": "case intro.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\nn : V\nhn : n \u2208 W\n\u22a2 ((B.restrict W) \u27e8x, hx\u2081\u27e9) \u27e8n, hn\u27e9 = 0"}, {"tactic": "simp only [restrict_apply, domRestrict_apply]", "annotated_tactic": ["simp only [<a>restrict_apply</a>, <a>domRestrict_apply</a>]", [{"full_name": "LinearMap.BilinForm.restrict_apply", "def_path": "Mathlib/LinearAlgebra/BilinearForm/Basic.lean", "def_pos": [226, 10], "def_end_pos": [226, 15]}, {"full_name": "LinearMap.domRestrict_apply", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [140, 9], "def_end_pos": [140, 26]}]], "state_before": "case intro.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\nn : V\nhn : n \u2208 W\n\u22a2 ((B.restrict W) \u27e8x, hx\u2081\u27e9) \u27e8n, hn\u27e9 = 0", "state_after": "case intro.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\nn : V\nhn : n \u2208 W\n\u22a2 (B x) n = 0"}, {"tactic": "exact b\u2081 n x (b\u2081 x n (b\u2081 n x (hx\u2082 n hn)))", "annotated_tactic": ["exact b\u2081 n x (b\u2081 x n (b\u2081 n x (hx\u2082 n hn)))", []], "state_before": "case intro.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u2078 : CommRing R\u2081\ninst\u271d\u2077 : AddCommGroup M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nM\u2082' : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb\u2081 : B.IsRefl\nb\u2082 : (B.restrict W).Nondegenerate\nx : V\nhx : x \u2208 W \u2293 B.orthogonal W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 B.orthogonal W\nn : V\nhn : n \u2208 W\n\u22a2 (B x) n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Support.lean", "full_name": "tsupport_smul_subset_left", "start": [101, 1], "end": [103, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "full_name": "WeierstrassCurve.Affine.equation_negAdd", "start": [582, 1], "end": [588, 39], "traced_tactics": [{"tactic": "rw [equation_add_iff, addPolynomial_slope h\u2081 h\u2082 hxy]", "annotated_tactic": ["rw [<a>equation_add_iff</a>, <a>addPolynomial_slope</a> h\u2081 h\u2082 hxy]", [{"full_name": "WeierstrassCurve.Affine.equation_add_iff", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [442, 7], "def_end_pos": [442, 23]}, {"full_name": "WeierstrassCurve.Affine.addPolynomial_slope", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [553, 7], "def_end_pos": [553, 26]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 W.Equation (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) (W.negAddY x\u2081 x\u2082 y\u2081 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 eval (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))\n      (-((X - C x\u2081) * (X - C x\u2082) * (X - C (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))))) =\n    0"}, {"tactic": "eval_simp", "annotated_tactic": ["eval_simp", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 eval (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))\n      (-((X - C x\u2081) * (X - C x\u2082) * (X - C (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))))) =\n    0", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 -((W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - x\u2081) * (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - x\u2082) *\n        (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))) =\n    0"}, {"tactic": "rw [neg_eq_zero, sub_self, mul_zero]", "annotated_tactic": ["rw [<a>neg_eq_zero</a>, <a>sub_self</a>, <a>mul_zero</a>]", [{"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 -((W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - x\u2081) * (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - x\u2082) *\n        (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) - W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))) =\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Functor/EpiMono.lean", "full_name": "CategoryTheory.Functor.reflectsMonomorphisms.iso_iff", "start": [156, 1], "end": [158, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "mul_eq_of_eq_inv_mul\u2080", "start": [330, 1], "end": [332, 61], "traced_tactics": [{"tactic": "rwa [\u2190 eq_inv_mul_iff_mul_eq\u2080]", "annotated_tactic": ["rwa [\u2190 <a>eq_inv_mul_iff_mul_eq\u2080</a>]", [{"full_name": "eq_inv_mul_iff_mul_eq\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [302, 7], "def_end_pos": [302, 29]}]], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : GroupWithZero G\u2080\na b c d : G\u2080\nm n : \u2115\nhb : b \u2260 0\nh : b = a\u207b\u00b9 * c\n\u22a2 a * b = c", "state_after": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : GroupWithZero G\u2080\na b c d : G\u2080\nm n : \u2115\nhb : b \u2260 0\nh : b = a\u207b\u00b9 * c\n\u22a2 a \u2260 0"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : GroupWithZero G\u2080\na b c d : G\u2080\nm n : \u2115\nhb : b \u2260 0\nh : b = a\u207b\u00b9 * c\n\u22a2 a \u2260 0", "state_after": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : GroupWithZero G\u2080\nb c d : G\u2080\nm n : \u2115\nhb : b \u2260 0\nh : b = 0\u207b\u00b9 * c\n\u22a2 False"}, {"tactic": "simp [hb] at h", "annotated_tactic": ["simp [hb] at h", []], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : GroupWithZero G\u2080\nb c d : G\u2080\nm n : \u2115\nhb : b \u2260 0\nh : b = 0\u207b\u00b9 * c\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "full_name": "RingHom.rangeS_top_iff_surjective", "start": [1135, 1], "end": [1137, 93], "traced_tactics": [{"tactic": "rw [coe_rangeS, coe_top]", "annotated_tactic": ["rw [<a>coe_rangeS</a>, <a>coe_top</a>]", [{"full_name": "RingHom.coe_rangeS", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [516, 9], "def_end_pos": [516, 19]}, {"full_name": "Subsemiring.coe_top", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [424, 9], "def_end_pos": [424, 16]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u2077 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u2076 : NonAssocSemiring S\ninst\u271d\u2075 inst\u271d\u2074 : NonAssocSemiring T\ns : Subsemiring R\n\u03c3R : Type u_1\n\u03c3S : Type u_2\ninst\u271d\u00b3 : SetLike \u03c3R R\ninst\u271d\u00b2 : SetLike \u03c3S S\ninst\u271d\u00b9 : SubsemiringClass \u03c3R R\ninst\u271d : SubsemiringClass \u03c3S S\nf : R \u2192+* S\n\u22a2 \u2191f.rangeS = \u2191\u22a4 \u2194 Set.range \u21d1f = Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Real.rpow_add_nat'", "start": [442, 1], "end": [443, 36], "traced_tactics": [{"tactic": "rw [rpow_add' hx h, rpow_natCast]", "annotated_tactic": ["rw [<a>rpow_add'</a> hx h, <a>rpow_natCast</a>]", [{"full_name": "Real.rpow_add'", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [205, 9], "def_end_pos": [205, 18]}, {"full_name": "Real.rpow_natCast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [73, 9], "def_end_pos": [73, 21]}]], "state_before": "x y z : \u211d\nn : \u2115\nhx : 0 \u2264 x\nh : y + \u2191n \u2260 0\n\u22a2 x ^ (y + \u2191n) = x ^ y * x ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean", "full_name": "conformal_const_smul", "start": [124, 1], "end": [125, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "Filter.cocompact_eq_bot", "start": [861, 1], "end": [862, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Real.continuousOn_exp", "start": [140, 1], "end": [141, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/PGroup.lean", "full_name": "IsPGroup.comap_of_ker_isPGroup", "start": [285, 1], "end": [292, 63], "traced_tactics": [{"tactic": "intro g", "annotated_tactic": ["intro g", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\n\u22a2 IsPGroup p \u21a5(Subgroup.comap \u03d5 H)", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\n\u22a2 \u2203 k, g ^ p ^ k = 1"}, {"tactic": "obtain \u27e8j, hj\u27e9 := hH \u27e8\u03d5 g.1, g.2\u27e9", "annotated_tactic": ["obtain \u27e8j, hj\u27e9 := hH \u27e8\u03d5 g.1, g.2\u27e9", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\n\u22a2 \u2203 k, g ^ p ^ k = 1", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u27e8\u03d5 \u2191g, \u22ef\u27e9 ^ p ^ j = 1\n\u22a2 \u2203 k, g ^ p ^ k = 1"}, {"tactic": "rw [Subtype.ext_iff, H.coe_pow, Subtype.coe_mk, \u2190 \u03d5.map_pow] at hj", "annotated_tactic": ["rw [<a>Subtype.ext_iff</a>, H.coe_pow, <a>Subtype.coe_mk</a>, \u2190 \u03d5.map_pow] at hj", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}]], "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u27e8\u03d5 \u2191g, \u22ef\u27e9 ^ p ^ j = 1\n\u22a2 \u2203 k, g ^ p ^ k = 1", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\n\u22a2 \u2203 k, g ^ p ^ k = 1"}, {"tactic": "obtain \u27e8k, hk\u27e9 := h\u03d5 \u27e8g.1 ^ p ^ j, hj\u27e9", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := h\u03d5 \u27e8g.1 ^ p ^ j, hj\u27e9", []], "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\n\u22a2 \u2203 k, g ^ p ^ k = 1", "state_after": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\nk : \u2115\nhk : \u27e8\u2191g ^ p ^ j, hj\u27e9 ^ p ^ k = 1\n\u22a2 \u2203 k, g ^ p ^ k = 1"}, {"tactic": "rw [Subtype.ext_iff, \u03d5.ker.coe_pow, Subtype.coe_mk, \u2190 pow_mul, \u2190 pow_add] at hk", "annotated_tactic": ["rw [<a>Subtype.ext_iff</a>, \u03d5.ker.coe_pow, <a>Subtype.coe_mk</a>, \u2190 <a>pow_mul</a>, \u2190 <a>pow_add</a>] at hk", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}]], "state_before": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\nk : \u2115\nhk : \u27e8\u2191g ^ p ^ j, hj\u27e9 ^ p ^ k = 1\n\u22a2 \u2203 k, g ^ p ^ k = 1", "state_after": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\nk : \u2115\nhk : \u2191g ^ p ^ (j + k) = \u21911\n\u22a2 \u2203 k, g ^ p ^ k = 1"}, {"tactic": "exact \u27e8j + k, by rwa [Subtype.ext_iff, (H.comap \u03d5).coe_pow]\u27e9", "annotated_tactic": ["exact \u27e8j + k, by rwa [<a>Subtype.ext_iff</a>, (H.comap \u03d5).<a>coe_pow</a>]\u27e9", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subgroup.coe_pow", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 16]}]], "state_before": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\nk : \u2115\nhk : \u2191g ^ p ^ (j + k) = \u21911\n\u22a2 \u2203 k, g ^ p ^ k = 1", "state_after": "no goals"}, {"tactic": "rwa [Subtype.ext_iff, (H.comap \u03d5).coe_pow]", "annotated_tactic": ["rwa [<a>Subtype.ext_iff</a>, (H.comap \u03d5).<a>coe_pow</a>]", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subgroup.coe_pow", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 16]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nhH : IsPGroup p \u21a5H\nK : Type u_2\ninst\u271d : Group K\n\u03d5 : K \u2192* G\nh\u03d5 : IsPGroup p \u21a5\u03d5.ker\ng : \u21a5(Subgroup.comap \u03d5 H)\nj : \u2115\nhj : \u03d5 (\u2191g ^ p ^ j) = \u21911\nk : \u2115\nhk : \u2191g ^ p ^ (j + k) = \u21911\n\u22a2 g ^ p ^ (j + k) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.piCongr'_symm_apply_symm_apply", "start": [1970, 1], "end": [1972, 39], "traced_tactics": [{"tactic": "simp [piCongr', piCongr_apply_apply]", "annotated_tactic": ["simp [<a>piCongr'</a>, <a>piCongr_apply_apply</a>]", [{"full_name": "Equiv.piCongr'", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1955, 5], "def_end_pos": [1955, 13]}, {"full_name": "Equiv.piCongr_apply_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1941, 9], "def_end_pos": [1941, 28]}]], "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\nW : \u03b1 \u2192 Sort w\nZ : \u03b2 \u2192 Sort z\nh\u2081 : \u03b1 \u2243 \u03b2\nh\u2082 : (b : \u03b2) \u2192 W (h\u2081.symm b) \u2243 Z b\nf : (b : \u03b2) \u2192 Z b\nb : \u03b2\n\u22a2 (h\u2081.piCongr' h\u2082).symm f (h\u2081.symm b) = (h\u2082 b).symm (f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.Perm.nil_eq", "start": [117, 1], "end": [117, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Composition.lean", "full_name": "FormalMultilinearSeries.compPartialSumTarget_tendsto_atTop", "start": [678, 1], "end": [692, 73], "traced_tactics": [{"tactic": "apply Monotone.tendsto_atTop_finset", "annotated_tactic": ["apply <a>Monotone.tendsto_atTop_finset</a>", [{"full_name": "Monotone.tendsto_atTop_finset", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1519, 7], "def_end_pos": [1519, 43]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\n\u22a2 Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\n\u22a2 Monotone fun N => compPartialSumTarget 0 N N\n\ncase h'\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\n\u22a2 \u2200 (x : (n : \u2115) \u00d7 Composition n), \u2203 n, x \u2208 compPartialSumTarget 0 n n"}, {"tactic": "intro m n hmn a ha", "annotated_tactic": ["intro m n hmn a ha", []], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\n\u22a2 Monotone fun N => compPartialSumTarget 0 N N", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nm n : \u2115\nhmn : m \u2264 n\na : (n : \u2115) \u00d7 Composition n\nha : a \u2208 (fun N => compPartialSumTarget 0 N N) m\n\u22a2 a \u2208 (fun N => compPartialSumTarget 0 N N) n"}, {"tactic": "have : \u2200 i, i < m \u2192 i < n := fun i hi => lt_of_lt_of_le hi hmn", "annotated_tactic": ["have : \u2200 i, i < m \u2192 i < n := fun i hi => <a>lt_of_lt_of_le</a> hi hmn", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nm n : \u2115\nhmn : m \u2264 n\na : (n : \u2115) \u00d7 Composition n\nha : a \u2208 (fun N => compPartialSumTarget 0 N N) m\n\u22a2 a \u2208 (fun N => compPartialSumTarget 0 N N) n", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nm n : \u2115\nhmn : m \u2264 n\na : (n : \u2115) \u00d7 Composition n\nha : a \u2208 (fun N => compPartialSumTarget 0 N N) m\nthis : \u2200 i < m, i < n\n\u22a2 a \u2208 (fun N => compPartialSumTarget 0 N N) n"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nm n : \u2115\nhmn : m \u2264 n\na : (n : \u2115) \u00d7 Composition n\nha : a \u2208 (fun N => compPartialSumTarget 0 N N) m\nthis : \u2200 i < m, i < n\n\u22a2 a \u2208 (fun N => compPartialSumTarget 0 N N) n", "state_after": "no goals"}, {"tactic": "rintro \u27e8n, c\u27e9", "annotated_tactic": ["rintro \u27e8n, c\u27e9", []], "state_before": "case h'\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\n\u22a2 \u2200 (x : (n : \u2115) \u00d7 Composition n), \u2203 n, x \u2208 compPartialSumTarget 0 n n", "state_after": "case h'.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : 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i) Finset.univ)\nj : Fin c.length\n\u22a2 c.blocksFun j \u2264 n"}, {"tactic": "apply hn", "annotated_tactic": ["apply hn", []], "state_before": "case h'.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nn\u271d : \u2115\nc : Composition n\u271d\nn : \u2115\nhn : n \u2208 upperBounds \u2191(Finset.image (fun i => c.blocksFun i) Finset.univ)\nj : Fin c.length\n\u22a2 c.blocksFun j \u2264 n", "state_after": "case h'.mk.intro.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u2078 : NontriviallyNormedField 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goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EqToHom.lean", "full_name": "CategoryTheory.congrArg_mpr_hom_right", "start": [138, 1], "end": [141, 7], "traced_tactics": [{"tactic": "cases q", "annotated_tactic": ["cases q", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nX Y Z : C\np : X \u27f6 Y\nq : Z = Y\n\u22a2 \u22ef.mpr p = p \u226b eqToHom \u22ef", "state_after": "case refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nX Y : C\np : X \u27f6 Y\n\u22a2 \u22ef.mpr p = p \u226b eqToHom \u22ef"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nX Y : C\np : X \u27f6 Y\n\u22a2 \u22ef.mpr p = p \u226b eqToHom \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Irrational.neg", "start": [304, 11], "end": [305, 29], "traced_tactics": [{"tactic": "rwa [neg_neg]", "annotated_tactic": ["rwa [<a>neg_neg</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}]], "state_before": "q : \u211a\nx y : \u211d\nh : Irrational x\n\u22a2 Irrational (- -x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "full_name": "CategoryTheory.shift_shiftFunctorCompIsoId_inv_app", "start": [504, 1], "end": [511, 6], "traced_tactics": [{"tactic": "rw [\u2190 neg_eq_of_add_eq_zero_left h, add_right_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_eq_of_add_eq_zero_left</a> h, <a>add_right_neg</a>]", [{"full_name": "neg_eq_of_add_eq_zero_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1164, 3], "def_end_pos": [1164, 14]}, {"full_name": "add_right_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1237, 3], "def_end_pos": [1237, 14]}]], "state_before": "C : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 m + n = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 cancel_mono (((shiftFunctorCompIsoId C n m h).hom.app X)\u27e6n\u27e7'),\n  \u2190 Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id,\n  shift_shiftFunctorCompIsoId_hom_app, Iso.inv_hom_id_app]", "annotated_tactic": ["rw [\u2190 <a>cancel_mono</a> (((<a>shiftFunctorCompIsoId</a> C n m h).hom.app X)\u27e6n\u27e7'),\n    \u2190 <a>Functor.map_comp</a>, <a>Iso.inv_hom_id_app</a>, <a>Functor.map_id</a>,\n    <a>shift_shiftFunctorCompIsoId_hom_app</a>, <a>Iso.inv_hom_id_app</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "CategoryTheory.shiftFunctorCompIsoId", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [417, 5], "def_end_pos": [417, 26]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}, {"full_name": "CategoryTheory.shift_shiftFunctorCompIsoId_hom_app", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [493, 9], "def_end_pos": [493, 44]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}]], "state_before": "C : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 (shiftFunctor C n).map ((shiftFunctorCompIsoId C n m h).inv.app X) =\n    (shiftFunctorCompIsoId C m n \u22ef).inv.app ((shiftFunctor C n).obj X)", "state_after": "C : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 \ud835\udfd9 ((shiftFunctor C n).obj ((\ud835\udfed C).obj X)) = \ud835\udfd9 ((\ud835\udfed C).obj ((shiftFunctor C n).obj X))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 \ud835\udfd9 ((shiftFunctor C n).obj ((\ud835\udfed C).obj X)) = \ud835\udfd9 ((\ud835\udfed C).obj ((shiftFunctor C n).obj X))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.reverseAux_append", "start": [334, 11], "end": [339, 49], "traced_tactics": [{"tactic": "induction p with\n| nil => rfl\n| cons h _ ih => simp [ih (cons (G.symm h) q)]", "annotated_tactic": ["induction p with\n  | <a>nil</a> => rfl\n  | <a>cons</a> h _ ih => simp [ih (<a>cons</a> (G.symm h) q)]", [{"full_name": "SimpleGraph.Walk.nil", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [87, 5], "def_end_pos": [87, 8]}, {"full_name": "SimpleGraph.Walk.cons", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [88, 5], "def_end_pos": [88, 9]}, {"full_name": "SimpleGraph.Walk.cons", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [88, 5], "def_end_pos": [88, 9]}]], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\np : G.Walk u v\nq : G.Walk u w\nr : G.Walk w x\n\u22a2 (p.reverseAux q).append r = p.reverseAux (q.append r)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : G.Walk w x\nu\u271d : V\nq : G.Walk u\u271d w\n\u22a2 (nil.reverseAux q).append r = nil.reverseAux (q.append r)", "state_after": "no goals"}, {"tactic": "simp [ih (cons (G.symm h) q)]", "annotated_tactic": ["simp [ih (<a>cons</a> (G.symm h) q)]", [{"full_name": "SimpleGraph.Walk.cons", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [88, 5], "def_end_pos": [88, 9]}]], "state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : G.Walk w x\nu\u271d v\u271d w\u271d : V\nh : G.Adj u\u271d v\u271d\np\u271d : G.Walk v\u271d w\u271d\nih : \u2200 (q : G.Walk v\u271d w), (p\u271d.reverseAux q).append r = p\u271d.reverseAux (q.append r)\nq : G.Walk u\u271d w\n\u22a2 ((cons h p\u271d).reverseAux q).append r = (cons h p\u271d).reverseAux (q.append r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Basic.lean", "full_name": "ULift.down_algebraMap", "start": [69, 1], "end": [70, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Meromorphic.lean", "full_name": "MeromorphicAt.pow", "start": [135, 1], "end": [138, 42], "traced_tactics": [{"tactic": "induction' n with m hm", "annotated_tactic": ["induction' n with m hm", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 \ud835\udd5c\nx : \ud835\udd5c\nhf : MeromorphicAt f x\nn : \u2115\n\u22a2 MeromorphicAt (f ^ n) x", "state_after": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 \ud835\udd5c\nx : \ud835\udd5c\nhf : MeromorphicAt f x\n\u22a2 MeromorphicAt (f ^ 0) x\n\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 \ud835\udd5c\nx : \ud835\udd5c\nhf : MeromorphicAt f x\nm : \u2115\nhm : MeromorphicAt (f ^ m) x\n\u22a2 MeromorphicAt (f ^ (m + 1)) x"}, {"tactic": "simpa only [Nat.zero_eq, pow_zero] using MeromorphicAt.const 1 x", "annotated_tactic": ["simpa only [<a>Nat.zero_eq</a>, <a>pow_zero</a>] using <a>MeromorphicAt.const</a> 1 x", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "MeromorphicAt.const", "def_path": "Mathlib/Analysis/Analytic/Meromorphic.lean", "def_pos": [40, 7], "def_end_pos": [40, 12]}]], "state_before": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 \ud835\udd5c\nx : \ud835\udd5c\nhf : MeromorphicAt f x\n\u22a2 MeromorphicAt (f ^ 0) x", "state_after": "no goals"}, {"tactic": "simpa only [pow_succ] using hm.mul hf", "annotated_tactic": ["simpa only [<a>pow_succ</a>] using hm.mul hf", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 \ud835\udd5c\nx : \ud835\udd5c\nhf : MeromorphicAt f x\nm : \u2115\nhm : MeromorphicAt (f ^ m) x\n\u22a2 MeromorphicAt (f ^ (m + 1)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.hasSum_integral_iUnion", "start": [296, 1], "end": [301, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/SurjOn.lean", "full_name": "surjOn_Iic_of_monotone_surjective", "start": [83, 1], "end": [85, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean", "full_name": "UniqueMDiffOn.smooth_bundle_preimage", "start": [141, 1], "end": [143, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "Submodule.sup_orthogonal_of_completeSpace", "start": [796, 1], "end": [798, 7], "traced_tactics": [{"tactic": "convert Submodule.sup_orthogonal_inf_of_completeSpace (le_top : K \u2264 \u22a4) using 2", "annotated_tactic": ["convert <a>Submodule.sup_orthogonal_inf_of_completeSpace</a> (<a>le_top</a> : K \u2264 \u22a4) using 2", [{"full_name": "Submodule.sup_orthogonal_inf_of_completeSpace", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [781, 9], "def_end_pos": [781, 54]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K \u2294 K\u15ee = \u22a4", "state_after": "case h.e'_2.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee = K\u15ee \u2293 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee = K\u15ee \u2293 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Order.lean", "full_name": "Finsupp.support_mono", "start": [214, 1], "end": [214, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Cast/Defs.lean", "full_name": "Rat.cast_div_of_ne_zero", "start": [212, 1], "end": [219, 28], "traced_tactics": [{"tactic": "rw [div_def', cast_divInt_of_ne_zero, cast_def, cast_def, div_eq_mul_inv (_ / _), inv_div,\n  (Int.commute_cast _ _).div_mul_div_comm (Nat.commute_cast _ _)]", "annotated_tactic": ["rw [<a>div_def'</a>, <a>cast_divInt_of_ne_zero</a>, <a>cast_def</a>, <a>cast_def</a>, <a>div_eq_mul_inv</a> (_ / _), <a>inv_div</a>,\n    (<a>Int.commute_cast</a> _ _).<a>div_mul_div_comm</a> (<a>Nat.commute_cast</a> _ _)]", [{"full_name": "Rat.div_def'", "def_path": "Mathlib/Data/Rat/Defs.lean", "def_pos": [299, 7], "def_end_pos": [299, 15]}, {"full_name": "Rat.cast_divInt_of_ne_zero", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [155, 7], "def_end_pos": [155, 29]}, {"full_name": "Rat.cast_def", "def_path": "Mathlib/Algebra/Field/Defs.lean", "def_pos": [212, 7], "def_end_pos": [212, 15]}, {"full_name": "Rat.cast_def", "def_path": "Mathlib/Algebra/Field/Defs.lean", "def_pos": [212, 7], "def_end_pos": [212, 15]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "inv_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}, {"full_name": "Int.commute_cast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [119, 7], "def_end_pos": [119, 19]}, {"full_name": "Commute.div_mul_div_comm", "def_path": "Mathlib/Algebra/Group/Commute/Basic.lean", "def_pos": [40, 19], "def_end_pos": [40, 35]}, {"full_name": "Nat.commute_cast", "def_path": "Mathlib/Data/Nat/Cast/Commute.lean", "def_pos": [37, 9], "def_end_pos": [37, 21]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191(p / q) = \u2191p / \u2191q", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191(p.num * \u2191q.den) / \u2191(\u2191p.den * q.num) = \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num)\n\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191(\u2191p.den * q.num) \u2260 0"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191(p.num * \u2191q.den) / \u2191(\u2191p.den * q.num) = \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num)", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num) = \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num) = \u2191p.num * \u2191q.den / (\u2191p.den * \u2191q.num)", "state_after": "no goals"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191(\u2191p.den * q.num) \u2260 0", "state_after": "case b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191p.den * \u2191q.num \u2260 0"}, {"tactic": "exact mul_ne_zero hp hq", "annotated_tactic": ["exact <a>mul_ne_zero</a> hp hq", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}]], "state_before": "case b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\np q : \u211a\nhp : \u2191p.den \u2260 0\nhq : \u2191q.num \u2260 0\n\u22a2 \u2191p.den * \u2191q.num \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Complex.abs_cpow_of_ne_zero", "start": [310, 1], "end": [313, 39], "traced_tactics": [{"tactic": "rw [cpow_def_of_ne_zero hz, abs_exp, mul_re, log_re, log_im, Real.exp_sub,\n  Real.rpow_def_of_pos (abs.pos hz)]", "annotated_tactic": ["rw [<a>cpow_def_of_ne_zero</a> hz, <a>abs_exp</a>, <a>mul_re</a>, <a>log_re</a>, <a>log_im</a>, <a>Real.exp_sub</a>,\n    <a>Real.rpow_def_of_pos</a> (abs.pos hz)]", [{"full_name": "Complex.cpow_def_of_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [40, 9], "def_end_pos": [40, 28]}, {"full_name": "Complex.abs_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 16]}, {"full_name": "Complex.mul_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [247, 9], "def_end_pos": [247, 15]}, {"full_name": "Complex.log_re", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": [33, 9], "def_end_pos": [33, 15]}, {"full_name": "Complex.log_im", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": [36, 9], "def_end_pos": [36, 15]}, {"full_name": "Real.exp_sub", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [866, 9], "def_end_pos": [866, 16]}, {"full_name": "Real.rpow_def_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [56, 9], "def_end_pos": [56, 24]}]], "state_before": "z : \u2102\nhz : z \u2260 0\nw : \u2102\n\u22a2 abs (z ^ w) = abs z ^ w.re / rexp (z.arg * w.im)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.coe_toSubmodule_eq_iff", "start": [147, 1], "end": [148, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff", "start": [212, 1], "end": [216, 43], "traced_tactics": [{"tactic": "simp [\u2190 hfg.sum_smul_comp_perm_eq_sum_smul_iff h\u03c3, lt_iff_le_and_ne, eq_comm,\n  hfg.sum_smul_le_sum_smul_comp_perm h\u03c3]", "annotated_tactic": ["simp [\u2190 hfg.sum_smul_comp_perm_eq_sum_smul_iff h\u03c3, <a>lt_iff_le_and_ne</a>, <a>eq_comm</a>,\n    hfg.sum_smul_le_sum_smul_comp_perm h\u03c3]", [{"full_name": "lt_iff_le_and_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : AntivaryOn f g \u2191s\nh\u03c3 : {x | \u03c3 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i \u2208 s, f i \u2022 g i < \u2211 i \u2208 s, f i \u2022 g (\u03c3 i) \u2194 \u00acAntivaryOn f (g \u2218 \u21d1\u03c3) \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean", "full_name": "ContinuousLinearMap.nndist_le_opNNNorm", "start": [117, 1], "end": [118, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/CatCommSq.lean", "full_name": "CategoryTheory.CatCommSq.hInv_hInv", "start": [75, 1], "end": [85, 6], "traced_tactics": [{"tactic": "ext X", "annotated_tactic": ["ext X", []], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\n\u22a2 hInv T.symm R L B.symm (hInv T L R B h) = h", "state_after": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 iso'.hom.app X = iso'.hom.app X"}, {"tactic": "erw [\u2190 cancel_mono (B.functor.map (L.map (T.unitIso.hom.app X))),\n  \u2190 h.iso'.hom.naturality (T.unitIso.hom.app X), hInv_iso'_hom_app, hInv_iso'_inv_app]", "annotated_tactic": ["erw [\u2190 <a>cancel_mono</a> (B.functor.map (L.map (T.unitIso.hom.app X))),\n    \u2190 h.iso'.hom.naturality (T.unitIso.hom.app X), <a>hInv_iso'_hom_app</a>, <a>hInv_iso'_inv_app</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "CategoryTheory.CatCommSq.hInv_iso'_hom_app", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [67, 10], "def_end_pos": [67, 22]}, {"full_name": "CategoryTheory.CatCommSq.hInv_iso'_inv_app", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [67, 23], "def_end_pos": [67, 35]}]], "state_before": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 iso'.hom.app X = iso'.hom.app X", "state_after": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 (B.symm.unitIso.hom.app (R.obj (T.symm.inverse.obj X)) \u226b\n        B.symm.inverse.map\n            (B.inverse.map (R.map (T.counitIso.inv.app (T.symm.inverse.obj X))) \u226b\n              B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.symm.inverse.obj X))) \u226b\n                B.unitIso.inv.app (L.obj (T.inverse.obj (T.symm.inverse.obj X)))) \u226b\n          B.symm.inverse.map (L.map (T.symm.counitIso.hom.app X))) \u226b\n      B.functor.map (L.map (T.unitIso.hom.app X)) =\n    (T.functor \u22d9 R).map (T.unitIso.hom.app X) \u226b iso'.hom.app ((T.functor \u22d9 T.inverse).obj X)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 (B.symm.unitIso.hom.app (R.obj (T.symm.inverse.obj X)) \u226b\n        B.symm.inverse.map\n            (B.inverse.map (R.map (T.counitIso.inv.app (T.symm.inverse.obj X))) \u226b\n              B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.symm.inverse.obj X))) \u226b\n                B.unitIso.inv.app (L.obj (T.inverse.obj (T.symm.inverse.obj X)))) \u226b\n          B.symm.inverse.map (L.map (T.symm.counitIso.hom.app X))) \u226b\n      B.functor.map (L.map (T.unitIso.hom.app X)) =\n    (T.functor \u22d9 R).map (T.unitIso.hom.app X) \u226b iso'.hom.app ((T.functor \u22d9 T.inverse).obj X)", "state_after": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 (B.counitIso.inv.app (R.obj (T.functor.obj X)) \u226b\n        B.functor.map\n            (B.inverse.map (R.map (T.counitIso.inv.app (T.functor.obj X))) \u226b\n              B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X))) \u226b\n                B.unitIso.inv.app (L.obj (T.inverse.obj (T.functor.obj X)))) \u226b\n          B.functor.map (L.map (T.unitIso.inv.app X))) \u226b\n      B.functor.map (L.map (T.unitIso.hom.app X)) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))"}, {"tactic": "simp only [Functor.comp_obj, assoc, \u2190 Functor.map_comp, Iso.inv_hom_id_app,\n  Equivalence.counitInv_app_functor, Functor.map_id]", "annotated_tactic": ["simp only [<a>Functor.comp_obj</a>, <a>assoc</a>, \u2190 <a>Functor.map_comp</a>, <a>Iso.inv_hom_id_app</a>,\n    <a>Equivalence.counitInv_app_functor</a>, <a>Functor.map_id</a>]", [{"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "CategoryTheory.Equivalence.counitInv_app_functor", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [166, 9], "def_end_pos": [166, 30]}, {"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}]], "state_before": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 (B.counitIso.inv.app (R.obj (T.functor.obj X)) \u226b\n        B.functor.map\n            (B.inverse.map (R.map (T.counitIso.inv.app (T.functor.obj X))) \u226b\n              B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X))) \u226b\n                B.unitIso.inv.app (L.obj (T.inverse.obj (T.functor.obj X)))) \u226b\n          B.functor.map (L.map (T.unitIso.inv.app X))) \u226b\n      B.functor.map (L.map (T.unitIso.hom.app X)) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))", "state_after": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 B.counitIso.inv.app (R.obj (T.functor.obj X)) \u226b\n      B.functor.map\n        (B.inverse.map (R.map (T.functor.map (T.unit.app X))) \u226b\n          B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X))) \u226b\n            B.unitIso.inv.app (L.obj (T.inverse.obj (T.functor.obj X))) \u226b \ud835\udfd9 (L.obj (T.inverse.obj (T.functor.obj X)))) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))"}, {"tactic": "simp only [Functor.map_comp, Equivalence.fun_inv_map, assoc,\n  Equivalence.counitInv_functor_comp, comp_id, Iso.inv_hom_id_app_assoc]", "annotated_tactic": ["simp only [<a>Functor.map_comp</a>, <a>Equivalence.fun_inv_map</a>, <a>assoc</a>,\n    <a>Equivalence.counitInv_functor_comp</a>, <a>comp_id</a>, <a>Iso.inv_hom_id_app_assoc</a>]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Equivalence.fun_inv_map", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [228, 9], "def_end_pos": [228, 20]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Equivalence.counitInv_functor_comp", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [159, 9], "def_end_pos": [159, 31]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [70, 3], "def_end_pos": [70, 25]}]], "state_before": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 B.counitIso.inv.app (R.obj (T.functor.obj X)) \u226b\n      B.functor.map\n        (B.inverse.map (R.map (T.functor.map (T.unit.app X))) \u226b\n          B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X))) \u226b\n            B.unitIso.inv.app (L.obj (T.inverse.obj (T.functor.obj X))) \u226b \ud835\udfd9 (L.obj (T.inverse.obj (T.functor.obj X)))) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))", "state_after": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 R.map (T.functor.map (T.unit.app X)) \u226b (iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X)) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case iso'.w.w.h\nC\u2081 : Type u_1\nC\u2082 : Type u_2\nC\u2083 : Type u_3\nC\u2084 : Type u_4\nC\u2085 : Type u_5\nC\u2086 : Type u_6\ninst\u271d\u2075 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2074 : Category.{u_9, u_2} C\u2082\ninst\u271d\u00b3 : Category.{u_10, u_3} C\u2083\ninst\u271d\u00b2 : Category.{u_7, u_4} C\u2084\ninst\u271d\u00b9 : Category.{?u.17460, u_5} C\u2085\ninst\u271d : Category.{?u.17464, u_6} C\u2086\nT\u271d : C\u2081 \u2964 C\u2082\nL\u271d : C\u2081 \u2964 C\u2083\nR\u271d : C\u2082 \u2964 C\u2084\nB\u271d : C\u2083 \u2964 C\u2084\nT : C\u2081 \u224c C\u2082\nL : C\u2081 \u2964 C\u2083\nR : C\u2082 \u2964 C\u2084\nB : C\u2083 \u224c C\u2084\nh : CatCommSq T.functor L R B.functor\nX : C\u2081\n\u22a2 R.map (T.functor.map (T.unit.app X)) \u226b (iso T.functor L R B.functor).hom.app (T.inverse.obj (T.functor.obj X)) =\n    R.map (T.functor.map (T.unitIso.hom.app X)) \u226b iso'.hom.app (T.inverse.obj (T.functor.obj X))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsGreatest.isLeast_image2_of_isLeast", "start": [1563, 1], "end": [1566, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean", "full_name": "Matrix.charmatrix_apply_natDegree", "start": [49, 1], "end": [51, 88], "traced_tactics": [{"tactic": "by_cases h : i = j <;> simp [h, \u2190 degree_eq_iff_natDegree_eq_of_pos (Nat.succ_pos 0)]", "annotated_tactic": ["by_cases h : i = j <;> simp [h, \u2190 <a>degree_eq_iff_natDegree_eq_of_pos</a> (<a>Nat.succ_pos</a> 0)]", [{"full_name": "Polynomial.degree_eq_iff_natDegree_eq_of_pos", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [150, 9], "def_end_pos": [150, 42]}, {"full_name": "Nat.succ_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 21]}]], "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\ninst\u271d : Nontrivial R\ni j : n\n\u22a2 (M.charmatrix i j).natDegree = if i = j then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Diffeomorph.lean", "full_name": "ModelWithCorners.extChartAt_transDiffeomorph_target", "start": [547, 1], "end": [549, 63], "traced_tactics": [{"tactic": "simp only [e.range_comp, preimage_preimage, mfld_simps]", "annotated_tactic": ["simp only [e.range_comp, <a>preimage_preimage</a>, mfld_simps]", [{"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [162, 9], "def_end_pos": [162, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E'\nF : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nH : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace H\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nG : Type u_7\ninst\u271d\u2079 : TopologicalSpace G\nG' : Type u_8\ninst\u271d\u2078 : TopologicalSpace G'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\nJ : ModelWithCorners \ud835\udd5c F G\nJ' : ModelWithCorners \ud835\udd5c F G'\nM : Type u_9\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\nM' : Type u_10\ninst\u271d\u2075 : TopologicalSpace M'\ninst\u271d\u2074 : ChartedSpace H' M'\nN : Type u_11\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : ChartedSpace G N\nN' : Type u_12\ninst\u271d\u00b9 : TopologicalSpace N'\ninst\u271d : ChartedSpace G' N'\nn : \u2115\u221e\ne : E \u2243\u2098\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E'\nx : M\n\u22a2 (extChartAt (I.transDiffeomorph e) x).target = \u21d1e.symm \u207b\u00b9' (extChartAt I x).target", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E'\nF : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nH : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace H\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nG : Type u_7\ninst\u271d\u2079 : TopologicalSpace G\nG' : Type u_8\ninst\u271d\u2078 : TopologicalSpace G'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\nJ : ModelWithCorners \ud835\udd5c F G\nJ' : ModelWithCorners \ud835\udd5c F G'\nM : Type u_9\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\nM' : Type u_10\ninst\u271d\u2075 : TopologicalSpace M'\ninst\u271d\u2074 : ChartedSpace H' M'\nN : Type u_11\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : ChartedSpace G N\nN' : Type u_12\ninst\u271d\u00b9 : TopologicalSpace N'\ninst\u271d : ChartedSpace G' N'\nn : \u2115\u221e\ne : E \u2243\u2098\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E'\nx : M\n\u22a2 \u21d1e.symm \u207b\u00b9' range \u2191I \u2229 \u2191I.symm \u2218 \u21d1e.symm \u207b\u00b9' (chartAt H x).target =\n    \u21d1e.symm \u207b\u00b9' range \u2191I \u2229 (fun x => \u2191I.symm (e.symm x)) \u207b\u00b9' (chartAt H x).target"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E'\nF : Type u_4\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nH : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace H\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nG : Type u_7\ninst\u271d\u2079 : TopologicalSpace G\nG' : Type u_8\ninst\u271d\u2078 : TopologicalSpace G'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\nJ : ModelWithCorners \ud835\udd5c F G\nJ' : ModelWithCorners \ud835\udd5c F G'\nM : Type u_9\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\nM' : Type u_10\ninst\u271d\u2075 : TopologicalSpace M'\ninst\u271d\u2074 : ChartedSpace H' M'\nN : Type u_11\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : ChartedSpace G N\nN' : Type u_12\ninst\u271d\u00b9 : TopologicalSpace N'\ninst\u271d : ChartedSpace G' N'\nn : \u2115\u221e\ne : E \u2243\u2098\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E'\nx : M\n\u22a2 \u21d1e.symm \u207b\u00b9' range \u2191I \u2229 \u2191I.symm \u2218 \u21d1e.symm \u207b\u00b9' (chartAt H x).target =\n    \u21d1e.symm \u207b\u00b9' range \u2191I \u2229 (fun x => \u2191I.symm (e.symm x)) \u207b\u00b9' (chartAt H x).target", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Split.lean", "full_name": "BoxIntegral.Prepartition.splitMany_le_split", "start": [259, 1], "end": [261, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.Measure.measurePreserving_div_left", "start": [505, 1], "end": [508, 72], "traced_tactics": [{"tactic": "simp_rw [div_eq_mul_inv]", "annotated_tactic": ["simp_rw [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : MeasurableSpace H\ninst\u271d\u2074 : DivisionMonoid G\ninst\u271d\u00b3 : MeasurableMul G\ninst\u271d\u00b2 : MeasurableInv G\n\u03bc\u271d \u03bc : Measure G\ninst\u271d\u00b9 : \u03bc.IsInvInvariant\ninst\u271d : \u03bc.IsMulLeftInvariant\ng : G\n\u22a2 MeasurePreserving (fun t => g / t) \u03bc \u03bc", "state_after": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : MeasurableSpace H\ninst\u271d\u2074 : DivisionMonoid G\ninst\u271d\u00b3 : MeasurableMul G\ninst\u271d\u00b2 : MeasurableInv G\n\u03bc\u271d \u03bc : Measure G\ninst\u271d\u00b9 : \u03bc.IsInvInvariant\ninst\u271d : \u03bc.IsMulLeftInvariant\ng : G\n\u22a2 MeasurePreserving (fun t => g * t\u207b\u00b9) \u03bc \u03bc"}, {"tactic": "exact (measurePreserving_mul_left \u03bc g).comp (measurePreserving_inv \u03bc)", "annotated_tactic": ["exact (<a>measurePreserving_mul_left</a> \u03bc g).<a>comp</a> (<a>measurePreserving_inv</a> \u03bc)", [{"full_name": "MeasureTheory.measurePreserving_mul_left", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [128, 9], "def_end_pos": [128, 35]}, {"full_name": "MeasureTheory.MeasurePreserving.comp", "def_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean", "def_pos": [102, 19], "def_end_pos": [102, 23]}, {"full_name": "MeasureTheory.Measure.measurePreserving_inv", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [432, 9], "def_end_pos": [432, 30]}]], "state_before": "\ud835\udd5c : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : MeasurableSpace H\ninst\u271d\u2074 : DivisionMonoid G\ninst\u271d\u00b3 : MeasurableMul G\ninst\u271d\u00b2 : MeasurableInv G\n\u03bc\u271d \u03bc : Measure G\ninst\u271d\u00b9 : \u03bc.IsInvInvariant\ninst\u271d : \u03bc.IsMulLeftInvariant\ng : G\n\u22a2 MeasurePreserving (fun t => g * t\u207b\u00b9) \u03bc \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "full_name": "CochainComplex.HomComplex.Cochain.congr_v", "start": [88, 1], "end": [89, 51], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m : \u2124\nz\u2081 z\u2082 : Cochain F G n\nh : z\u2081 = z\u2082\np q : \u2124\nhpq : p + n = q\n\u22a2 z\u2081.v p q hpq = z\u2082.v p q hpq", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m : \u2124\nz\u2081 : Cochain F G n\np q : \u2124\nhpq : p + n = q\n\u22a2 z\u2081.v p q hpq = z\u2081.v p q hpq"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m : \u2124\nz\u2081 : Cochain F G n\np q : \u2124\nhpq : p + n = q\n\u22a2 z\u2081.v p q hpq = z\u2081.v p q hpq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean", "full_name": "SimpleGraph.sum_degrees_eq_twice_card_edges", "start": [111, 1], "end": [112, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Defs.lean", "full_name": "Rat.intCast_eq_one", "start": [88, 1], "end": [88, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "full_name": "IsClosed.vadd_right_of_isCompact", "start": [324, 1], "end": [334, 56], "traced_tactics": [{"tactic": "refine IsSeqClosed.isClosed (fun u p husv hup \u21a6 ?_)", "annotated_tactic": ["refine <a>IsSeqClosed.isClosed</a> (fun u p husv hup \u21a6 ?_)", [{"full_name": "IsSeqClosed.isClosed", "def_path": "Mathlib/Topology/Defs/Sequences.lean", "def_pos": [101, 19], "def_end_pos": [101, 39]}]], "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\n\u22a2 IsClosed (s +\u1d65 t)", "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhusv : \u2200 (n : \u2115), u n \u2208 s +\u1d65 t\nhup : Tendsto u atTop (\ud835\udcdd p)\n\u22a2 p \u2208 s +\u1d65 t"}, {"tactic": "choose! a ha v hv hav using husv", "annotated_tactic": ["choose! a ha v hv hav using husv", []], "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhusv : \u2200 (n : \u2115), u n \u2208 s +\u1d65 t\nhup : Tendsto u atTop (\ud835\udcdd p)\n\u22a2 p \u2208 s +\u1d65 t", "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\n\u22a2 p \u2208 s +\u1d65 t"}, {"tactic": "rcases ht.isSeqCompact hv with \u27e8q, hqt, \u03c6, \u03c6_mono, h\u03c6q\u27e9", "annotated_tactic": ["rcases ht.isSeqCompact hv with \u27e8q, hqt, \u03c6, \u03c6_mono, h\u03c6q\u27e9", []], "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\n\u22a2 p \u2208 s +\u1d65 t", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\n\u22a2 p \u2208 s +\u1d65 t"}, {"tactic": "refine \u27e8p -\u1d65 q, hs.mem_of_tendsto ((hup.comp \u03c6_mono.tendsto_atTop).vsub h\u03c6q)\n  (eventually_of_forall fun n \u21a6 ?_), q, hqt, vsub_vadd _ _\u27e9", "annotated_tactic": ["refine \u27e8p -\u1d65 q, hs.mem_of_tendsto ((hup.comp \u03c6_mono.tendsto_atTop).<a>vsub</a> h\u03c6q)\n    (<a>eventually_of_forall</a> fun n \u21a6 ?_), q, hqt, <a>vsub_vadd</a> _ _\u27e9", [{"full_name": "Filter.Tendsto.vsub", "def_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "def_pos": [270, 9], "def_end_pos": [270, 28]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 29]}, {"full_name": "vsub_vadd", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [85, 9], "def_end_pos": [85, 18]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\n\u22a2 p \u2208 s +\u1d65 t", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 v \u2218 \u03c6) n \u2208 s"}, {"tactic": "convert ha (\u03c6 n) using 1", "annotated_tactic": ["convert ha (\u03c6 n) using 1", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 v \u2218 \u03c6) n \u2208 s", "state_after": "case h.e'_4\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nha : \u2200 (n : \u2115), a n \u2208 s\nv : \u2115 \u2192 P\nhv : \u2200 (n : \u2115), v n \u2208 t\nhav : \u2200 (n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 v \u2218 \u03c6) n = a (\u03c6 n)"}, {"tactic": "exact (eq_vadd_iff_vsub_eq _ _ _).mp (hav (\u03c6 n)).symm", "annotated_tactic": ["exact (<a>eq_vadd_iff_vsub_eq</a> _ _ _).<a>mp</a> (hav (\u03c6 n)).<a>symm</a>", 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(n : \u2115), (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto (v \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 v \u2218 \u03c6) n = a (\u03c6 n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/ConcreteCategory.lean", "full_name": "CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists", "start": [130, 1], "end": [134, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.add_contLinear", "start": [129, 1], "end": [130, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", 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AffineIndependent k (p \u2218 \u21d1e)\nthis : p = p \u2218 \u21d1e \u2218 \u21d1e.symm.toEmbedding\n\u22a2 AffineIndependent k (p \u2218 \u21d1e \u2218 \u21d1e.symm.toEmbedding)", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u21d1e)\n\u22a2 p = p \u2218 \u21d1e \u2218 \u21d1e.symm.toEmbedding", "state_after": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u21d1e)\nx\u271d : \u03b9'\n\u22a2 p x\u271d = (p \u2218 \u21d1e \u2218 \u21d1e.symm.toEmbedding) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u21d1e)\nx\u271d : \u03b9'\n\u22a2 p x\u271d = (p \u2218 \u21d1e \u2218 \u21d1e.symm.toEmbedding) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Intervals.lean", "full_name": "List.Ico.eq_nil_of_le", "start": [72, 1], "end": [73, 43], "traced_tactics": [{"tactic": "simp [Ico, Nat.sub_eq_zero_iff_le.mpr h]", "annotated_tactic": ["simp [<a>Ico</a>, Nat.sub_eq_zero_iff_le.mpr h]", [{"full_name": "List.Ico", "def_path": "Mathlib/Data/List/Intervals.lean", "def_pos": [36, 5], "def_end_pos": [36, 8]}]], "state_before": "n m : \u2115\nh : m \u2264 n\n\u22a2 Ico n m = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/SeparableDegree.lean", "full_name": "Polynomial.natSepDegree_X", "start": [280, 1], "end": [281, 89], "traced_tactics": [{"tactic": "simp only [natSepDegree, aroots_X, Multiset.toFinset_singleton, Finset.card_singleton]", "annotated_tactic": ["simp only [<a>natSepDegree</a>, <a>aroots_X</a>, <a>Multiset.toFinset_singleton</a>, <a>Finset.card_singleton</a>]", [{"full_name": "Polynomial.natSepDegree", "def_path": "Mathlib/FieldTheory/SeparableDegree.lean", "def_pos": [267, 5], "def_end_pos": [267, 17]}, {"full_name": "Polynomial.aroots_X", "def_path": "Mathlib/Algebra/Polynomial/Roots.lean", "def_pos": [442, 9], "def_end_pos": [442, 17]}, {"full_name": "Multiset.toFinset_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3115, 9], "def_end_pos": [3115, 27]}, {"full_name": "Finset.card_singleton", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [88, 9], "def_end_pos": [88, 23]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\nK : Type w\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nf : F[X]\n\u22a2 X.natSepDegree = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.extendDomain_pow", "start": [352, 1], "end": [354, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Separable.lean", "full_name": "Polynomial.isUnit_of_self_mul_dvd_separable", "start": [175, 1], "end": [186, 70], "traced_tactics": [{"tactic": "obtain \u27e8p, rfl\u27e9 := hq", "annotated_tactic": ["obtain \u27e8p, rfl\u27e9 := hq", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhp : p.Separable\nhq : q * q \u2223 p\n\u22a2 IsUnit q", "state_after": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsUnit q"}, {"tactic": "apply isCoprime_self.mp", "annotated_tactic": ["apply isCoprime_self.mp", []], "state_before": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsUnit q", "state_after": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsCoprime q q"}, {"tactic": "have : IsCoprime (q * (q * p))\n    (q * (derivative q * p + derivative q * p + q * derivative p)) := by\n  simp only [\u2190 mul_assoc, mul_add]\n  dsimp only [Separable] at hp\n  convert hp using 1\n  rw [derivative_mul, derivative_mul]\n  ring", "annotated_tactic": ["have : <a>IsCoprime</a> (q * (q * p))\n      (q * (<a>derivative</a> q * p + <a>derivative</a> q * p + q * <a>derivative</a> p)) := by\n    simp only [\u2190 <a>mul_assoc</a>, <a>mul_add</a>]\n    dsimp only [<a>Separable</a>] at hp\n    convert hp using 1\n    rw [<a>derivative_mul</a>, <a>derivative_mul</a>]\n    ring", [{"full_name": "IsCoprime", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Polynomial.derivative", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [40, 5], "def_end_pos": [40, 15]}, {"full_name": "Polynomial.derivative", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [40, 5], "def_end_pos": [40, 15]}, {"full_name": "Polynomial.derivative", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [40, 5], "def_end_pos": [40, 15]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "Polynomial.Separable", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "Polynomial.derivative_mul", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}, {"full_name": "Polynomial.derivative_mul", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}]], "state_before": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsCoprime q q", "state_after": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\nthis : IsCoprime (q * (q * p)) (q * (derivative q * p + derivative q * p + q * derivative p))\n\u22a2 IsCoprime q q"}, {"tactic": "exact IsCoprime.of_mul_right_left (IsCoprime.of_mul_left_left this)", "annotated_tactic": ["exact <a>IsCoprime.of_mul_right_left</a> (<a>IsCoprime.of_mul_left_left</a> this)", [{"full_name": "IsCoprime.of_mul_right_left", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [149, 9], "def_end_pos": [149, 36]}, {"full_name": "IsCoprime.of_mul_left_left", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [139, 9], "def_end_pos": [139, 35]}]], "state_before": "case intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\nthis : IsCoprime (q * (q * p)) (q * (derivative q * p + derivative q * p + q * derivative p))\n\u22a2 IsCoprime q q", "state_after": "no goals"}, {"tactic": "simp only [\u2190 mul_assoc, mul_add]", "annotated_tactic": ["simp only [\u2190 <a>mul_assoc</a>, <a>mul_add</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsCoprime (q * (q * p)) (q * (derivative q * p + derivative q * p + q * derivative p))", "state_after": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsCoprime (q * q * p) (q * derivative q * p + q * derivative q * p + q * q * derivative p)"}, {"tactic": "dsimp only [Separable] at hp", "annotated_tactic": ["dsimp only [<a>Separable</a>] at hp", [{"full_name": "Polynomial.Separable", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : (q * q * p).Separable\n\u22a2 IsCoprime (q * q * p) (q * derivative q * p + q * derivative q * p + q * q * derivative p)", "state_after": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 IsCoprime (q * q * p) (q * derivative q * p + q * derivative q * p + q * q * derivative p)"}, {"tactic": "convert hp using 1", "annotated_tactic": ["convert hp using 1", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 IsCoprime (q * q * p) (q * derivative q * p + q * derivative q * p + q * q * derivative p)", "state_after": "case h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 q * derivative q * p + q * derivative q * p + q * q * derivative p = derivative (q * q * p)"}, {"tactic": "rw [derivative_mul, derivative_mul]", "annotated_tactic": ["rw [<a>derivative_mul</a>, <a>derivative_mul</a>]", [{"full_name": "Polynomial.derivative_mul", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}, {"full_name": "Polynomial.derivative_mul", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}]], "state_before": "case h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 q * derivative q * p + q * derivative q * p + q * q * derivative p = derivative (q * q * p)", "state_after": "case h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 q * derivative q * p + q * derivative q * p + q * q * derivative p =\n    (derivative q * q + q * derivative q) * p + q * q * derivative p"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (derivative (q * q * p))\n\u22a2 q * derivative q * p + q * derivative q * p + q * q * derivative p =\n    (derivative q * q + q * derivative q) * p + q * q * derivative p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Disjoint.le_of_codisjoint", "start": [459, 1], "end": [461, 39], "traced_tactics": [{"tactic": "rw [\u2190 @inf_top_eq _ _ _ a, \u2190 @bot_sup_eq _ _ _ c, \u2190 hab.eq_bot, \u2190 hbc.eq_top, sup_inf_right]", "annotated_tactic": ["rw [\u2190 @<a>inf_top_eq</a> _ _ _ a, \u2190 @<a>bot_sup_eq</a> _ _ _ c, \u2190 hab.eq_bot, \u2190 hbc.eq_top, <a>sup_inf_right</a>]", [{"full_name": "inf_top_eq", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [446, 7], "def_end_pos": [446, 17]}, {"full_name": "bot_sup_eq", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "sup_inf_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [722, 9], "def_end_pos": [722, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b c : \u03b1\nhab : Disjoint a b\nhbc : Codisjoint b c\n\u22a2 a \u2264 c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b c : \u03b1\nhab : Disjoint a b\nhbc : Codisjoint b c\n\u22a2 a \u2293 (b \u2294 c) \u2264 (a \u2294 c) \u2293 (b \u2294 c)"}, {"tactic": "exact inf_le_inf_right _ le_sup_left", "annotated_tactic": ["exact <a>inf_le_inf_right</a> _ <a>le_sup_left</a>", [{"full_name": "inf_le_inf_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [447, 9], "def_end_pos": [447, 25]}, {"full_name": "le_sup_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b c : \u03b1\nhab : Disjoint a b\nhbc : Codisjoint b c\n\u22a2 a \u2293 (b \u2294 c) \u2264 (a \u2294 c) \u2293 (b \u2294 c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "full_name": "MvPolynomial.totalDegree_finset_prod", "start": [489, 1], "end": [493, 6], "traced_tactics": [{"tactic": "refine le_trans (totalDegree_multiset_prod _) ?_", "annotated_tactic": ["refine <a>le_trans</a> (<a>totalDegree_multiset_prod</a> _) ?_", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MvPolynomial.totalDegree_multiset_prod", "def_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "def_pos": [482, 9], "def_end_pos": [482, 34]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 (s.prod f).totalDegree \u2264 \u2211 i \u2208 s, (f i).totalDegree", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 (Multiset.map totalDegree (Multiset.map f s.val)).sum \u2264 \u2211 i \u2208 s, (f i).totalDegree"}, {"tactic": "rw [Multiset.map_map]", "annotated_tactic": ["rw [<a>Multiset.map_map</a>]", [{"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 16]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 (Multiset.map totalDegree (Multiset.map f s.val)).sum \u2264 \u2211 i \u2208 s, (f i).totalDegree", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 (Multiset.map (totalDegree \u2218 f) s.val).sum \u2264 \u2211 i \u2208 s, (f i).totalDegree"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\n\u22a2 (Multiset.map (totalDegree \u2218 f) s.val).sum \u2264 \u2211 i \u2208 s, (f i).totalDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.directedOn_iUnion", "start": [2009, 1], "end": [2015, 27], "traced_tactics": [{"tactic": "simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]", "annotated_tactic": ["simp only [<a>DirectedOn</a>, <a>exists_prop</a>, <a>mem_iUnion</a>, <a>exists_imp</a>]", [{"full_name": "DirectedOn", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [52, 5], "def_end_pos": [52, 15]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "exists_imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [200, 9], "def_end_pos": [200, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) f\nh : \u2200 (x : \u03b9), DirectedOn r (f x)\n\u22a2 DirectedOn r (\u22c3 x, f x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) f\nh : \u2200 (x : \u03b9), DirectedOn r (f x)\n\u22a2 \u2200 (x : \u03b1) (x_1 : \u03b9), x \u2208 f x_1 \u2192 \u2200 (y : \u03b1) (x_2 : \u03b9), y \u2208 f x_2 \u2192 \u2203 z, (\u2203 i, z \u2208 f i) \u2227 r x z \u2227 r y z"}, {"tactic": "exact fun a\u2081 b\u2081 fb\u2081 a\u2082 b\u2082 fb\u2082 =>\n  let \u27e8z, zb\u2081, zb\u2082\u27e9 := hd b\u2081 b\u2082\n  let \u27e8x, xf, xa\u2081, xa\u2082\u27e9 := h z a\u2081 (zb\u2081 fb\u2081) a\u2082 (zb\u2082 fb\u2082)\n  \u27e8x, \u27e8z, xf\u27e9, xa\u2081, xa\u2082\u27e9", "annotated_tactic": ["exact fun a\u2081 b\u2081 fb\u2081 a\u2082 b\u2082 fb\u2082 =>\n    let \u27e8z, zb\u2081, zb\u2082\u27e9 := hd b\u2081 b\u2082\n    let \u27e8x, xf, xa\u2081, xa\u2082\u27e9 := h z a\u2081 (zb\u2081 fb\u2081) a\u2082 (zb\u2082 fb\u2082)\n    \u27e8x, \u27e8z, xf\u27e9, xa\u2081, xa\u2082\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) f\nh : \u2200 (x : \u03b9), DirectedOn r (f x)\n\u22a2 \u2200 (x : \u03b1) (x_1 : \u03b9), x \u2208 f x_1 \u2192 \u2200 (y : \u03b1) (x_2 : \u03b9), y \u2208 f x_2 \u2192 \u2203 z, (\u2203 i, z \u2208 f i) \u2227 r x z \u2227 r y z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Homology.lean", "full_name": "CategoryTheory.ShortComplex.HomologyMapData.congr_left_\u03c6H", "start": [102, 1], "end": [103, 42], "traced_tactics": [{"tactic": "rw [eq]", "annotated_tactic": ["rw [eq]", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.HomologyData\nh\u2082 : S\u2082.HomologyData\n\u03b3\u2081 \u03b3\u2082 : HomologyMapData \u03c6 h\u2081 h\u2082\neq : \u03b3\u2081 = \u03b3\u2082\n\u22a2 \u03b3\u2081.left.\u03c6H = \u03b3\u2082.left.\u03c6H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "full_name": "DirichletCharacter.conductor_dvd_level", "start": [190, 1], "end": [190, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.hasSum_expSeries_of_smul_comm", "start": [104, 1], "end": [111, 92], "traced_tactics": [{"tactic": "have : HasSum (fun n => fst (expSeries \ud835\udd5c (tsze R M) n fun _ => x)) e := by\n  simpa [fst_expSeries] using h", "annotated_tactic": ["have : <a>HasSum</a> (fun n => <a>fst</a> (<a>expSeries</a> \ud835\udd5c (tsze R M) n fun _ => x)) e := by\n    simpa [<a>fst_expSeries</a>] using h", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Defs.lean", "def_pos": [74, 3], "def_end_pos": [74, 14]}, {"full_name": "TrivSqZeroExt.fst", "def_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "def_pos": [92, 5], "def_end_pos": [92, 8]}, {"full_name": "NormedSpace.expSeries", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [101, 5], "def_end_pos": [101, 14]}, {"full_name": "TrivSqZeroExt.fst_expSeries", "def_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "def_pos": [70, 17], "def_end_pos": [70, 30]}]], "state_before": "\ud835\udd5c : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u2076 : Field \ud835\udd5c\ninst\u271d\u00b9\u2075 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u2074 : Ring R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Algebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2078 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op x.fst \u2022 x.snd = x.fst \u2022 x.snd\ne : R\nh : HasSum (fun n => (expSeries \ud835\udd5c R n) fun x_1 => x.fst) e\n\u22a2 HasSum (fun n => (expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) (inl e + inr (e \u2022 x.snd))", "state_after": "\ud835\udd5c : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u2076 : Field \ud835\udd5c\ninst\u271d\u00b9\u2075 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u2074 : Ring R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Algebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2078 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op x.fst \u2022 x.snd = x.fst \u2022 x.snd\ne : R\nh : HasSum (fun n => (expSeries \ud835\udd5c R n) fun x_1 => x.fst) e\nthis : HasSum (fun n => ((expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x).fst) e\n\u22a2 HasSum (fun n => (expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) (inl e + inr (e \u2022 x.snd))"}, {"tactic": "simpa only [inl_fst_add_inr_snd_eq] using\n  (hasSum_inl _ <| this).add (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm \ud835\udd5c x hx h)", "annotated_tactic": ["simpa only [<a>inl_fst_add_inr_snd_eq</a>] using\n    (<a>hasSum_inl</a> _ <| this).<a>add</a> (<a>hasSum_inr</a> _ <| <a>hasSum_snd_expSeries_of_smul_comm</a> \ud835\udd5c x hx h)", [{"full_name": "TrivSqZeroExt.inl_fst_add_inr_snd_eq", "def_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "def_pos": [387, 9], "def_end_pos": [387, 31]}, {"full_name": "TrivSqZeroExt.hasSum_inl", "def_path": "Mathlib/Topology/Instances/TrivSqZeroExt.lean", "def_pos": [147, 9], "def_end_pos": [147, 19]}, {"full_name": "HasSum.add", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [272, 3], "def_end_pos": [272, 14]}, {"full_name": "TrivSqZeroExt.hasSum_inr", "def_path": "Mathlib/Topology/Instances/TrivSqZeroExt.lean", "def_pos": [152, 9], "def_end_pos": [152, 19]}, {"full_name": "TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm", "def_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "def_pos": [91, 9], "def_end_pos": [91, 42]}]], "state_before": "\ud835\udd5c : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u2076 : Field \ud835\udd5c\ninst\u271d\u00b9\u2075 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u2074 : Ring R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Algebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2078 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op x.fst \u2022 x.snd = x.fst \u2022 x.snd\ne : R\nh : HasSum (fun n => (expSeries \ud835\udd5c R n) fun x_1 => x.fst) e\nthis : HasSum (fun n => ((expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x).fst) e\n\u22a2 HasSum (fun n => (expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) (inl e + inr (e \u2022 x.snd))", "state_after": "no goals"}, {"tactic": "simpa [fst_expSeries] using h", "annotated_tactic": ["simpa [<a>fst_expSeries</a>] using h", [{"full_name": "TrivSqZeroExt.fst_expSeries", "def_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "def_pos": [70, 17], "def_end_pos": [70, 30]}]], "state_before": "\ud835\udd5c : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u2076 : Field \ud835\udd5c\ninst\u271d\u00b9\u2075 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u2074 : Ring R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : Algebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2078 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op x.fst \u2022 x.snd = x.fst \u2022 x.snd\ne : R\nh : HasSum (fun n => (expSeries \ud835\udd5c R n) fun x_1 => x.fst) e\n\u22a2 HasSum (fun n => ((expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x).fst) e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Basic.lean", "full_name": "Interval.coe_coe", "start": [480, 1], "end": [481, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_hasDerivAt_of_tendsto_ae_right", "start": [751, 1], "end": [754, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "one_lt_finprod'", "start": [591, 1], "end": [596, 69], "traced_tactics": [{"tactic": "rcases h' with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases h' with \u27e8i, hi\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nh' : \u2203 i, 1 < f i\nhf : (mulSupport f).Finite\n\u22a2 1 < \u220f\u1da0 (i : \u03b9), f i", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 1 < \u220f\u1da0 (i : \u03b9), f i"}, {"tactic": "rw [finprod_eq_prod _ hf]", "annotated_tactic": ["rw [<a>finprod_eq_prod</a> _ hf]", [{"full_name": "finprod_eq_prod", "def_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 1 < \u220f\u1da0 (i : \u03b9), f i", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 1 < \u220f i \u2208 hf.toFinset, f i"}, {"tactic": "refine Finset.one_lt_prod' (fun i _ \u21a6 h i) \u27e8i, ?_, hi\u27e9", "annotated_tactic": ["refine <a>Finset.one_lt_prod'</a> (fun i _ \u21a6 h i) \u27e8i, ?_, hi\u27e9", [{"full_name": "Finset.one_lt_prod'", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [519, 9], "def_end_pos": [519, 21]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 1 < \u220f i \u2208 hf.toFinset, f i", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 i \u2208 hf.toFinset"}, {"tactic": "simpa only [Finite.mem_toFinset, mem_mulSupport] using ne_of_gt hi", "annotated_tactic": ["simpa only [<a>Finite.mem_toFinset</a>, <a>mem_mulSupport</a>] using <a>ne_of_gt</a> hi", [{"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [166, 19], "def_end_pos": [166, 31]}, {"full_name": "Function.mem_mulSupport", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [54, 9], "def_end_pos": [54, 23]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM\u271d : Type u_5\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\u271d\na b : \u03b1\ns t : Set \u03b1\nM : Type u_7\ninst\u271d : OrderedCancelCommMonoid M\nf : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), 1 \u2264 f i\nhf : (mulSupport f).Finite\ni : \u03b9\nhi : 1 < f i\n\u22a2 i \u2208 hf.toFinset", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.closedBall_smul_closedBall", "start": [947, 1], "end": [952, 33], "traced_tactics": [{"tactic": "simp only [smul_subset_iff, mem_closedBall_zero, mem_closedBall_zero_iff, map_smul_eq_mul]", "annotated_tactic": ["simp only [<a>smul_subset_iff</a>, <a>mem_closedBall_zero</a>, <a>mem_closedBall_zero_iff</a>, <a>map_smul_eq_mul</a>]", [{"full_name": "Set.smul_subset_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [183, 9], "def_end_pos": [183, 24]}, {"full_name": "Seminorm.mem_closedBall_zero", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [692, 9], "def_end_pos": [692, 28]}, {"full_name": "mem_closedBall_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [631, 3], "def_end_pos": [631, 14]}, {"full_name": "SeminormClass.map_smul_eq_mul", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [61, 3], "def_end_pos": [61, 18]}]], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\n\u22a2 Metric.closedBall 0 r\u2081 \u2022 p.closedBall 0 r\u2082 \u2286 p.closedBall 0 (r\u2081 * r\u2082)", "state_after": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\n\u22a2 \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 r\u2081 \u2192 \u2200 (b : E), p b \u2264 r\u2082 \u2192 \u2016a\u2016 * p b \u2264 r\u2081 * r\u2082"}, {"tactic": "intro a ha b hb", "annotated_tactic": ["intro a ha b hb", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\n\u22a2 \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 r\u2081 \u2192 \u2200 (b : E), p b \u2264 r\u2082 \u2192 \u2016a\u2016 * p b \u2264 r\u2081 * r\u2082", "state_after": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 r\u2081\nb : E\nhb : p b \u2264 r\u2082\n\u22a2 \u2016a\u2016 * p b \u2264 r\u2081 * r\u2082"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 r\u2081\nb : E\nhb : p b \u2264 r\u2082\n\u22a2 \u2016a\u2016 * p b \u2264 r\u2081 * r\u2082", "state_after": "case b0\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 r\u2081\nb : E\nhb : p b \u2264 r\u2082\n\u22a2 0 \u2264 r\u2081"}, {"tactic": "exact (norm_nonneg _).trans ha", "annotated_tactic": ["exact (<a>norm_nonneg</a> _).<a>trans</a> ha", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case b0\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 r\u2081\nb : E\nhb : p b \u2264 r\u2082\n\u22a2 0 \u2264 r\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "List.Perm.prod_eq", "start": [419, 1], "end": [419, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "full_name": "Mon_.Mon_tensor_one_mul", "start": [378, 1], "end": [386, 40], "traced_tactics": [{"tactic": "simp only [comp_whiskerRight_assoc]", "annotated_tactic": ["simp only [<a>comp_whiskerRight_assoc</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [252, 3], "def_end_pos": [252, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n    (\u03bb_ (M.X \u2297 N.X)).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b (M.one \u2297 N.one) \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n    (\u03bb_ (M.X \u2297 N.X)).hom"}, {"tactic": "slice_lhs 2 3 => rw [tensor_\u03bc_natural_left]", "annotated_tactic": ["slice_lhs 2 3 => rw [<a>tensor_\u03bc_natural_left</a>]", [{"full_name": "CategoryTheory.tensor_\u03bc_natural_left", "def_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "def_pos": [564, 9], "def_end_pos": [564, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b (M.one \u2297 N.one) \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n    (\u03bb_ (M.X \u2297 N.X)).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b (tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b (M.one \u25b7 M.X \u2297 N.one \u25b7 N.X)) \u226b (M.mul \u2297 N.mul) =\n    (\u03bb_ (M.X \u2297 N.X)).hom"}, {"tactic": "slice_lhs 3 4 => rw [\u2190 tensor_comp, one_mul M, one_mul N]", "annotated_tactic": ["slice_lhs 3 4 => rw [\u2190 <a>tensor_comp</a>, <a>one_mul</a> M, <a>one_mul</a> N]", [{"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [168, 3], "def_end_pos": [168, 14]}, {"full_name": "Mon_.one_mul", "def_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "def_pos": [39, 3], "def_end_pos": [39, 10]}, {"full_name": "Mon_.one_mul", "def_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "def_pos": [39, 3], "def_end_pos": [39, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b (tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b (M.one \u25b7 M.X \u2297 N.one \u25b7 N.X)) \u226b (M.mul \u2297 N.mul) =\n    (\u03bb_ (M.X \u2297 N.X)).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom) =\n    (\u03bb_ (M.X \u2297 N.X)).hom"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom) =\n    (\u03bb_ (M.X \u2297 N.X)).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (M.X \u2297 N.X)).hom =\n    (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom)"}, {"tactic": "exact tensor_left_unitality C M.X N.X", "annotated_tactic": ["exact <a>tensor_left_unitality</a> C M.X N.X", [{"full_name": "CategoryTheory.tensor_left_unitality", "def_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (M.X \u2297 N.X)).hom =\n    (\u03bb_ (\ud835\udfd9_ C)).inv \u25b7 (M.X \u2297 N.X) \u226b tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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1], "end": [42, 76], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_add, toNat_add_toNat_neg_eq_natAbs, NNReal.natCast_natAbs]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_add</a>, <a>toNat_add_toNat_neg_eq_natAbs</a>, <a>NNReal.natCast_natAbs</a>]", [{"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Int.toNat_add_toNat_neg_eq_natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [500, 17], "def_end_pos": [500, 46]}, {"full_name": "NNReal.natCast_natAbs", "def_path": "Mathlib/Analysis/Normed/Group/Int.lean", "def_pos": [36, 9], "def_end_pos": [36, 37]}]], "state_before": "n : \u2124\n\u22a2 \u2191n.toNat + \u2191(-n).toNat = \u2016n\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Star.lean", "full_name": 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MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set \u03b2\nht : (Measure.map f \u03bd) t = 0\n\u22a2 (Measure.map f \u03bc) t = 0"}, {"tactic": "rw [hf.map_apply] at ht \u22a2", "annotated_tactic": ["rw [hf.map_apply] at ht \u22a2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set \u03b2\nht : (Measure.map f \u03bd) t = 0\n\u22a2 (Measure.map f \u03bc) t = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set \u03b2\nht : \u03bd (f \u207b\u00b9' t) = 0\n\u22a2 \u03bc (f \u207b\u00b9' t) = 0"}, {"tactic": "exact h\u03bc\u03bd ht", "annotated_tactic": ["exact h\u03bc\u03bd ht", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set \u03b2\nht : \u03bd (f \u207b\u00b9' t) = 0\n\u22a2 \u03bc (f \u207b\u00b9' t) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/OuterMeasure/Operations.lean", "full_name": "MeasureTheory.OuterMeasure.sup_apply", "start": [211, 1], "end": [212, 89], "traced_tactics": 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"annotated_tactic": ["rwa [<a>iSup_bool_eq</a>, <a>iSup_bool_eq</a>] at this", [{"full_name": "iSup_bool_eq", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1481, 9], "def_end_pos": [1481, 21]}, {"full_name": "iSup_bool_eq", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1481, 9], "def_end_pos": [1481, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm m\u2081 m\u2082 : OuterMeasure \u03b1\ns : Set \u03b1\nthis : (\u2a06 i, bif i then m\u2081 else m\u2082) s = \u2a06 i, (bif i then m\u2081 else m\u2082) s\n\u22a2 (m\u2081 \u2294 m\u2082) s = m\u2081 s \u2294 m\u2082 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Basic.lean", "full_name": "LieModuleEquiv.symm_symm", "start": [1096, 1], "end": [1097, 6], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ne : M \u2243\u2097\u2045R,L\u2046 N\n\u22a2 e.symm.symm = e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "full_name": "LinearMap.coe_toAddHom", "start": [237, 1], "end": [237, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartialHomeomorph.lean", "full_name": "PartialHomeomorph.eventually_left_inverse'", "start": [256, 1], "end": [258, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Polynomial.lean", "full_name": "LinearMap.polyCharpolyAux_map_eq_charpoly", "start": [295, 1], "end": [300, 77], "traced_tactics": [{"tactic": "nontriviality R", "annotated_tactic": ["nontriviality R", []], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup L\ninst\u271d\u00b9\u2070 : Module R L\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2077 : Fintype \u03b9\ninst\u271d\u2076 : Fintype \u03b9'\ninst\u271d\u2075 : Fintype \u03b9M\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : DecidableEq \u03b9'\ninst\u271d\u00b2 : DecidableEq \u03b9M\nb : Basis \u03b9 R L\nb\u2098 : Basis \u03b9M R M\ninst\u271d\u00b9 : Module.Finite R M\ninst\u271d : Module.Free R M\nx : L\n\u22a2 Polynomial.map (MvPolynomial.eval \u21d1(b.repr x)) (\u03c6.polyCharpolyAux b b\u2098) = charpoly (\u03c6 x)", "state_after": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup L\ninst\u271d\u00b9\u2070 : Module R L\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2077 : Fintype \u03b9\ninst\u271d\u2076 : Fintype \u03b9'\ninst\u271d\u2075 : Fintype \u03b9M\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : DecidableEq \u03b9'\ninst\u271d\u00b2 : DecidableEq \u03b9M\nb : Basis \u03b9 R L\nb\u2098 : Basis \u03b9M R M\ninst\u271d\u00b9 : Module.Finite R M\ninst\u271d : Module.Free R M\nx : L\na\u271d : Nontrivial R\n\u22a2 Polynomial.map (MvPolynomial.eval \u21d1(b.repr x)) (\u03c6.polyCharpolyAux b b\u2098) = charpoly (\u03c6 x)"}, {"tactic": "rw [polyCharpolyAux_map_eq_toMatrix_charpoly, LinearMap.charpoly_toMatrix]", "annotated_tactic": ["rw [<a>polyCharpolyAux_map_eq_toMatrix_charpoly</a>, <a>LinearMap.charpoly_toMatrix</a>]", [{"full_name": "LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly", "def_path": "Mathlib/Algebra/Module/LinearMap/Polynomial.lean", "def_pos": [280, 7], "def_end_pos": [280, 47]}, {"full_name": "LinearMap.charpoly_toMatrix", "def_path": "Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean", "def_pos": [48, 9], "def_end_pos": [48, 26]}]], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup L\ninst\u271d\u00b9\u2070 : Module R L\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2077 : Fintype \u03b9\ninst\u271d\u2076 : Fintype \u03b9'\ninst\u271d\u2075 : Fintype \u03b9M\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : DecidableEq \u03b9'\ninst\u271d\u00b2 : DecidableEq \u03b9M\nb : Basis \u03b9 R L\nb\u2098 : Basis \u03b9M R M\ninst\u271d\u00b9 : Module.Finite R M\ninst\u271d : Module.Free R M\nx : L\na\u271d : Nontrivial R\n\u22a2 Polynomial.map (MvPolynomial.eval \u21d1(b.repr x)) (\u03c6.polyCharpolyAux b b\u2098) = charpoly (\u03c6 x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PNat/Basic.lean", "full_name": "Nat.succPNat_injective", "start": [103, 1], "end": [104, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Fin.lean", "full_name": "Fin.castOrderIso_toEquiv", "start": [217, 1], "end": [220, 15], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "m n : \u2115\np : Fin (n + 1)\ni j : Fin n\nh : n = m\n\u22a2 (castOrderIso h).toEquiv = Equiv.cast \u22ef", "state_after": "n : \u2115\np : Fin (n + 1)\ni j : Fin n\n\u22a2 (castOrderIso \u22ef).toEquiv = Equiv.cast \u22ef"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : \u2115\np : Fin (n + 1)\ni j : Fin n\n\u22a2 (castOrderIso \u22ef).toEquiv = Equiv.cast \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "full_name": "MeasurableSpace.generateFrom_iUnion_countablePartition", "start": [464, 1], "end": [469, 41], "traced_tactics": [{"tactic": "rw [countablePartition, generateFrom_iUnion_memPartition,\n  range_enumerateCountable_of_mem _ empty_mem_countableGeneratingSet,\n  generateFrom_countableGeneratingSet]", "annotated_tactic": ["rw [<a>countablePartition</a>, <a>generateFrom_iUnion_memPartition</a>,\n    <a>range_enumerateCountable_of_mem</a> _ <a>empty_mem_countableGeneratingSet</a>,\n    <a>generateFrom_countableGeneratingSet</a>]", [{"full_name": "MeasurableSpace.countablePartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [425, 5], "def_end_pos": [425, 23]}, {"full_name": "MeasurableSpace.generateFrom_iUnion_memPartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [371, 7], "def_end_pos": [371, 39]}, {"full_name": "Set.range_enumerateCountable_of_mem", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [107, 7], "def_end_pos": [107, 38]}, {"full_name": "MeasurableSpace.empty_mem_countableGeneratingSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [69, 7], "def_end_pos": [69, 39]}, {"full_name": "MeasurableSpace.generateFrom_countableGeneratingSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [65, 7], "def_end_pos": [65, 42]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\u271d\nh : CountablyGenerated \u03b1\u271d\n\u03b1 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d : CountablyGenerated \u03b1\n\u22a2 generateFrom (\u22c3 n, countablePartition \u03b1 n) = m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Holder.lean", "full_name": "holderOnWith_empty", "start": [62, 1], "end": [62, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.le_of_epi", "start": [590, 1], "end": [591, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/IsLUB.lean", "full_name": "IsLUB.exists_seq_monotone_tendsto", "start": [186, 1], "end": [192, 61], "traced_tactics": [{"tactic": "by_cases h : x \u2208 t", "annotated_tactic": ["by_cases h : x \u2208 t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2208 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2209 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t"}, {"tactic": "exact \u27e8fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h\u27e9", "annotated_tactic": ["exact \u27e8fun _ => x, <a>monotone_const</a>, fun n => <a>le_rfl</a>, <a>tendsto_const_nhds</a>, fun _ => h\u27e9", [{"full_name": "monotone_const", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [570, 9], "def_end_pos": [570, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [991, 9], "def_end_pos": [991, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2208 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t", "state_after": "no goals"}, {"tactic": "rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with \u27e8u, hu\u27e9", "annotated_tactic": ["rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with \u27e8u, hu\u27e9", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2209 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t", "state_after": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2209 t\nu : \u2115 \u2192 \u03b1\nhu : StrictMono u \u2227 (\u2200 (n : \u2115), u n < x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t"}, {"tactic": "exact \u27e8u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2\u27e9", "annotated_tactic": ["exact \u27e8u, hu.1.<a>monotone</a>, fun n => (hu.2.1 n).<a>le</a>, hu.2.2\u27e9", [{"full_name": "StrictMono.monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [512, 19], "def_end_pos": [512, 38]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b2\nt : Set \u03b1\nx : \u03b1\ninst\u271d : (\ud835\udcdd x).IsCountablyGenerated\nhtx : IsLUB t x\nht : t.Nonempty\nh : x \u2209 t\nu : \u2115 \u2192 \u03b1\nhu : StrictMono u \u2227 (\u2200 (n : \u2115), u n < x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t\n\u22a2 \u2203 u, Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 x) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 (n : \u2115), u n \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Gamma.lean", "full_name": "integral_exp_neg_mul_rpow", "start": [65, 1], "end": [69, 76], "traced_tactics": [{"tactic": "convert (integral_rpow_mul_exp_neg_mul_rpow hp neg_one_lt_zero hb) using 1", "annotated_tactic": ["convert (<a>integral_rpow_mul_exp_neg_mul_rpow</a> hp <a>neg_one_lt_zero</a> hb) using 1", [{"full_name": "integral_rpow_mul_exp_neg_mul_rpow", "def_path": "Mathlib/MeasureTheory/Integral/Gamma.lean", "def_pos": [39, 9], "def_end_pos": [39, 43]}, {"full_name": "neg_one_lt_zero", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1258, 7], "def_end_pos": [1258, 22]}]], "state_before": "p b : \u211d\nhp : 0 < p\nhb : 0 < b\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-b * x ^ p) = b ^ (-1 / p) * Gamma (1 / p + 1)", "state_after": "case h.e'_2\np b : \u211d\nhp : 0 < p\nhb : 0 < b\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-b * x ^ p) = \u222b (x : \u211d) in Ioi 0, x ^ 0 * rexp (-b * x ^ p)\n\ncase h.e'_3\np b : \u211d\nhp : 0 < p\nhb : 0 < b\n\u22a2 b ^ (-1 / p) * Gamma (1 / p + 1) = b ^ (-(0 + 1) / p) * (1 / p) * Gamma ((0 + 1) / p)"}, {"tactic": "simp_rw [rpow_zero, one_mul]", "annotated_tactic": ["simp_rw [<a>rpow_zero</a>, <a>one_mul</a>]", [{"full_name": "Real.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [125, 9], "def_end_pos": [125, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h.e'_2\np b : \u211d\nhp : 0 < p\nhb : 0 < b\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-b * x ^ p) = \u222b (x : \u211d) in Ioi 0, x ^ 0 * rexp (-b * x ^ p)", "state_after": "no goals"}, {"tactic": "rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp)), mul_assoc]", "annotated_tactic": ["rw [<a>zero_add</a>, <a>Gamma_add_one</a> (<a>one_div_ne_zero</a> (<a>ne_of_gt</a> hp)), <a>mul_assoc</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "Real.Gamma_add_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [524, 9], "def_end_pos": [524, 22]}, {"full_name": "one_div_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 24]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case h.e'_3\np b : \u211d\nhp : 0 < p\nhb : 0 < b\n\u22a2 b ^ (-1 / p) * Gamma (1 / p + 1) = b ^ (-(0 + 1) / p) * (1 / p) * Gamma ((0 + 1) / p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "ContDiffAt.comp", "start": [730, 8], "end": [732, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean", "full_name": "AlgebraCat.forget\u2082_map_associator_hom", "start": [58, 1], "end": [64, 6], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\ninst\u271d : CommRing R\nX Y Z : AlgebraCat R\n\u22a2 (forget\u2082 (AlgebraCat R) (ModuleCat R)).map (\u03b1_ X Y Z).hom =\n    (\u03b1_ ((forget\u2082 (AlgebraCat R) (ModuleCat R)).obj X) ((forget\u2082 (AlgebraCat R) (ModuleCat R)).obj Y)\n        ((forget\u2082 (AlgebraCat R) (ModuleCat R)).obj Z)).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/LiftingProperties/Basic.lean", "full_name": "CategoryTheory.HasLiftingProperty.iff_of_arrow_iso_right", "start": [141, 1], "end": [144, 63], "traced_tactics": [{"tactic": "constructor <;> intro", "annotated_tactic": ["constructor <;> intro", []], "state_before": "C : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nA\u271d B\u271d B' X\u271d Y\u271d Y'\u271d : C\ni\u271d : A\u271d \u27f6 B\u271d\ni' : B\u271d \u27f6 B'\np\u271d : X\u271d \u27f6 Y\u271d\np'\u271d : Y\u271d \u27f6 Y'\u271d\nA B X Y X' Y' : C\ni : A \u27f6 B\np : X \u27f6 Y\np' : X' \u27f6 Y'\ne : Arrow.mk p \u2245 Arrow.mk p'\n\u22a2 HasLiftingProperty i p \u2194 HasLiftingProperty i p'", "state_after": "case mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nA\u271d B\u271d B' X\u271d Y\u271d Y'\u271d : C\ni\u271d : A\u271d \u27f6 B\u271d\ni' : B\u271d \u27f6 B'\np\u271d : X\u271d \u27f6 Y\u271d\np'\u271d : Y\u271d \u27f6 Y'\u271d\nA B X Y X' Y' : C\ni : A \u27f6 B\np : X \u27f6 Y\np' : X' \u27f6 Y'\ne : Arrow.mk p \u2245 Arrow.mk p'\na\u271d : HasLiftingProperty i p\n\u22a2 HasLiftingProperty i p'\n\ncase mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nA\u271d B\u271d B' X\u271d Y\u271d Y'\u271d : C\ni\u271d : A\u271d \u27f6 B\u271d\ni' : B\u271d \u27f6 B'\np\u271d : X\u271d \u27f6 Y\u271d\np'\u271d : Y\u271d \u27f6 Y'\u271d\nA B X Y X' Y' : C\ni : A \u27f6 B\np : X \u27f6 Y\np' : X' \u27f6 Y'\ne : Arrow.mk p \u2245 Arrow.mk p'\na\u271d : HasLiftingProperty i p'\n\u22a2 HasLiftingProperty i p"}, {"tactic": "exacts [of_arrow_iso_right i e, of_arrow_iso_right i e.symm]", "annotated_tactic": ["exacts [<a>of_arrow_iso_right</a> i e, <a>of_arrow_iso_right</a> i e.symm]", [{"full_name": "CategoryTheory.HasLiftingProperty.of_arrow_iso_right", "def_path": "Mathlib/CategoryTheory/LiftingProperties/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 27]}, {"full_name": "CategoryTheory.HasLiftingProperty.of_arrow_iso_right", "def_path": "Mathlib/CategoryTheory/LiftingProperties/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 27]}]], "state_before": "case mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nA\u271d B\u271d B' X\u271d Y\u271d Y'\u271d : C\ni\u271d : A\u271d \u27f6 B\u271d\ni' : B\u271d \u27f6 B'\np\u271d : X\u271d \u27f6 Y\u271d\np'\u271d : Y\u271d \u27f6 Y'\u271d\nA B X Y X' Y' : C\ni : A \u27f6 B\np : X \u27f6 Y\np' : X' \u27f6 Y'\ne : Arrow.mk p \u2245 Arrow.mk p'\na\u271d : HasLiftingProperty i p\n\u22a2 HasLiftingProperty i p'\n\ncase mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nA\u271d B\u271d B' X\u271d Y\u271d Y'\u271d : C\ni\u271d : A\u271d \u27f6 B\u271d\ni' : B\u271d \u27f6 B'\np\u271d : X\u271d \u27f6 Y\u271d\np'\u271d : Y\u271d \u27f6 Y'\u271d\nA B X Y X' Y' : C\ni : A \u27f6 B\np : X \u27f6 Y\np' : X' \u27f6 Y'\ne : Arrow.mk p \u2245 Arrow.mk p'\na\u271d : HasLiftingProperty i p'\n\u22a2 HasLiftingProperty i p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": "SimpleGraph.singletonSubgraph_le_iff", "start": [845, 1], "end": [851, 31], "traced_tactics": [{"tactic": "refine \u27e8fun h \u21a6 h.1 (Set.mem_singleton v), ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 h.1 (<a>Set.mem_singleton</a> v), ?_\u27e9", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\n\u22a2 G.singletonSubgraph v \u2264 H \u2194 v \u2208 H.verts", "state_after": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\n\u22a2 v \u2208 H.verts \u2192 G.singletonSubgraph v \u2264 H"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\n\u22a2 v \u2208 H.verts \u2192 G.singletonSubgraph v \u2264 H", "state_after": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 G.singletonSubgraph v \u2264 H"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 G.singletonSubgraph v \u2264 H", "state_after": "case left\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 (G.singletonSubgraph v).verts \u2286 H.verts\n\ncase right\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 \u2200 \u2983v_1 w : V\u2984, (G.singletonSubgraph v).Adj v_1 w \u2192 H.Adj v_1 w"}, {"tactic": "rwa [singletonSubgraph_verts, Set.singleton_subset_iff]", "annotated_tactic": ["rwa [<a>singletonSubgraph_verts</a>, <a>Set.singleton_subset_iff</a>]", [{"full_name": "SimpleGraph.singletonSubgraph_verts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [74, 3], "def_end_pos": [74, 8]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 29]}]], "state_before": "case left\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 (G.singletonSubgraph v).verts \u2286 H.verts", "state_after": "no goals"}, {"tactic": "exact fun _ _ \u21a6 False.elim", "annotated_tactic": ["exact fun _ _ \u21a6 <a>False.elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case right\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : G.Subgraph\nh : v \u2208 H.verts\n\u22a2 \u2200 \u2983v_1 w : V\u2984, (G.singletonSubgraph v).Adj v_1 w \u2192 H.Adj v_1 w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Irreducible.lean", "full_name": "InfPrime.finset_inf_le", "start": [211, 1], "end": [212, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Group/Abs.lean", "full_name": "inv_lt_of_mabs_lt", "start": [315, 1], "end": [315, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Valuation/ValuationSubring.lean", "full_name": "ValuationSubring.coe_unitGroupMulEquiv_apply", "start": [492, 1], "end": [493, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Analytic.lean", "full_name": "HasFPowerSeriesAt.hasFDerivAt", "start": [47, 1], "end": [49, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.le_iff_count", "start": [2624, 1], "end": [2626, 92], "traced_tactics": [{"tactic": "rw [\u2190 (ext.2 fun a => by simp [max_eq_right (al a)] : s \u222a t = t)]", "annotated_tactic": ["rw [\u2190 (<a>ext</a>.2 fun a => by simp [<a>max_eq_right</a> (al a)] : s \u222a t = t)]", [{"full_name": "Multiset.ext", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2609, 9], "def_end_pos": [2609, 12]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d s t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\n\u22a2 s \u2264 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d s t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\n\u22a2 s \u2264 s \u222a t"}, {"tactic": "apply le_union_left", "annotated_tactic": ["apply <a>le_union_left</a>", [{"full_name": "Multiset.le_union_left", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1759, 9], "def_end_pos": [1759, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d s t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\n\u22a2 s \u2264 s \u222a t", "state_after": "no goals"}, {"tactic": "simp [max_eq_right (al a)]", "annotated_tactic": ["simp [<a>max_eq_right</a> (al a)]", [{"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d s t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\na : \u03b1\n\u22a2 count a (s \u222a t) = count a t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "full_name": "Fintype.prod_dvd_of_isRelPrime", "start": [277, 1], "end": [279, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.coeff_def", "start": [150, 1], "end": [151, 46], "traced_tactics": [{"tactic": "erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]", "annotated_tactic": ["erw [<a>coeff</a>, \u2190 h, \u2190 <a>Finsupp.unique_single</a> s]", [{"full_name": "PowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [139, 5], "def_end_pos": [139, 10]}, {"full_name": "Finsupp.unique_single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [432, 9], "def_end_pos": [432, 22]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\ns : Unit \u2192\u2080 \u2115\nn : \u2115\nh : s () = n\n\u22a2 coeff R n = MvPowerSeries.coeff R s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "strictMonoOn_dual_iff", "start": [273, 1], "end": [275, 66], "traced_tactics": [{"tactic": "rw [strictMonoOn_toDual_comp_iff, strictAntiOn_comp_ofDual_iff]", "annotated_tactic": ["rw [<a>strictMonoOn_toDual_comp_iff</a>, <a>strictAntiOn_comp_ofDual_iff</a>]", [{"full_name": "strictMonoOn_toDual_comp_iff", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 37]}, {"full_name": "strictAntiOn_comp_ofDual_iff", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 37]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 StrictMonoOn (\u21d1toDual \u2218 f \u2218 \u21d1ofDual) s \u2194 StrictMonoOn f s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Abel.lean", "full_name": "Mathlib.Tactic.Abel.term_smul", "start": [221, 1], "end": [224, 61], "traced_tactics": [{"tactic": "simp [h\u2082.symm, h\u2081.symm, term, smul, nsmul_add, mul_nsmul']", "annotated_tactic": ["simp [h\u2082.symm, h\u2081.symm, <a>term</a>, <a>smul</a>, <a>nsmul_add</a>, 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\u2203 t, x.IsAmalgamation t) \u2192\n    \u2200 (x : FamilyOfElements P R), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t", "state_after": "case mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\n\u22a2 \u2203 t, x.IsAmalgamation t"}, {"tactic": "apply Exists.imp _ (q _ hx.sieveExtend)", "annotated_tactic": ["apply <a>Exists.imp</a> _ (q _ hx.sieveExtend)", [{"full_name": "Exists.imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [193, 9], "def_end_pos": [193, 19]}]], "state_before": "case mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\n\u22a2 \u2203 t, x.IsAmalgamation t", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\n\u22a2 \u2200 (a : P.obj { unop := X }), x.sieveExtend.IsAmalgamation a \u2192 x.IsAmalgamation a"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\n\u22a2 \u2200 (a : P.obj { unop := X }), x.sieveExtend.IsAmalgamation a \u2192 x.IsAmalgamation a", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\nt : P.obj { unop := X }\nht : x.sieveExtend.IsAmalgamation t\n\u22a2 x.IsAmalgamation t"}, {"tactic": "simpa [hx] using isAmalgamation_restrict (le_generate R) _ _ ht", "annotated_tactic": ["simpa [hx] using <a>isAmalgamation_restrict</a> (<a>le_generate</a> R) _ _ ht", [{"full_name": "CategoryTheory.Presieve.isAmalgamation_restrict", "def_path": "Mathlib/CategoryTheory/Sites/IsSheafFor.lean", "def_pos": [394, 9], "def_end_pos": [394, 32]}, {"full_name": "CategoryTheory.Sieve.le_generate", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [434, 9], "def_end_pos": [434, 20]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d R : Presieve X\nq : \u2200 (x : FamilyOfElements P (generate R).arrows), x.Compatible \u2192 \u2203 t, x.IsAmalgamation t\nx : FamilyOfElements P R\nhx : x.Compatible\nt : P.obj { unop := X }\nht : x.sieveExtend.IsAmalgamation t\n\u22a2 x.IsAmalgamation t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Preimage.lean", "full_name": "Finset.preimage_compl", "start": [71, 1], "end": [74, 33], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b2\nhf : Injective f\n\u22a2 \u2191(s\u1d9c.preimage f \u22ef) = \u2191(s.preimage f \u22ef)\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean", "full_name": "AlgebraicGeometry.isOpenImmersion_isLocalAtTarget", "start": [53, 1], "end": [74, 21], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "X Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 PropertyIsLocalAtTarget @IsOpenImmersion", "state_after": "case RespectsIso\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 MorphismProperty.RespectsIso @IsOpenImmersion\n\ncase restrict\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) (U : Opens \u2191\u2191Y.toPresheafedSpace), IsOpenImmersion f \u2192 IsOpenImmersion (f \u2223_ U)\n\ncase of_openCover\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) (\ud835\udcb0 : Y.OpenCover), (\u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd) \u2192 IsOpenImmersion f"}, {"tactic": "exact isOpenImmersion_respectsIso", "annotated_tactic": ["exact <a>isOpenImmersion_respectsIso</a>", [{"full_name": "AlgebraicGeometry.isOpenImmersion_respectsIso", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean", "def_pos": [46, 9], "def_end_pos": [46, 36]}]], "state_before": "case RespectsIso\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 MorphismProperty.RespectsIso @IsOpenImmersion", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case restrict\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) (U : Opens \u2191\u2191Y.toPresheafedSpace), IsOpenImmersion f \u2192 IsOpenImmersion (f \u2223_ U)", "state_after": "case restrict\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nU\u271d : Opens \u2191\u2191Y\u271d.toPresheafedSpace\na\u271d : IsOpenImmersion f\u271d\n\u22a2 IsOpenImmersion (f\u271d \u2223_ U\u271d)"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case restrict\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nU\u271d : Opens \u2191\u2191Y\u271d.toPresheafedSpace\na\u271d : IsOpenImmersion f\u271d\n\u22a2 IsOpenImmersion (f\u271d \u2223_ U\u271d)", "state_after": "no goals"}, {"tactic": "intro X Y f \ud835\udcb0 H", "annotated_tactic": ["intro X Y f \ud835\udcb0 H", []], "state_before": "case of_openCover\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) (\ud835\udcb0 : Y.OpenCover), (\u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd) \u2192 IsOpenImmersion f", "state_after": "case of_openCover\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 IsOpenImmersion f"}, {"tactic": "rw [isOpenImmersion_iff_stalk]", "annotated_tactic": ["rw [<a>isOpenImmersion_iff_stalk</a>]", [{"full_name": "AlgebraicGeometry.isOpenImmersion_iff_stalk", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean", "def_pos": [34, 9], "def_end_pos": [34, 34]}]], "state_before": "case of_openCover\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 IsOpenImmersion f", "state_after": "case of_openCover\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 OpenEmbedding \u21d1f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case of_openCover\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 OpenEmbedding \u21d1f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 OpenEmbedding \u21d1f.val.base\n\ncase of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "apply (openEmbedding_iff_openEmbedding_of_iSup_eq_top \ud835\udcb0.iSup_opensRange f.1.base.2).mpr", "annotated_tactic": ["apply (<a>openEmbedding_iff_openEmbedding_of_iSup_eq_top</a> \ud835\udcb0.iSup_opensRange f.1.<a>base</a>.2).<a>mpr</a>", [{"full_name": "openEmbedding_iff_openEmbedding_of_iSup_eq_top", "def_path": "Mathlib/Topology/LocalAtTarget.lean", "def_pos": [159, 9], "def_end_pos": [159, 55]}, {"full_name": "AlgebraicGeometry.PresheafedSpace.Hom.base", "def_path": "Mathlib/Geometry/RingedSpace/PresheafedSpace.lean", "def_pos": [102, 3], "def_end_pos": [102, 7]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 OpenEmbedding \u21d1f.val.base", "state_after": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 \u2200 (i : \ud835\udcb0.J), OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 \u2200 (i : \ud835\udcb0.J), OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)", "state_after": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)"}, {"tactic": "have := ((isOpenImmersion_respectsIso.arrow_iso_iff\n  (morphismRestrictOpensRange f (\ud835\udcb0.map i))).mpr (H i)).1", "annotated_tactic": ["have := ((isOpenImmersion_respectsIso.arrow_iso_iff\n        (<a>morphismRestrictOpensRange</a> f (\ud835\udcb0.map i))).<a>mpr</a> (H i)).1", [{"full_name": "AlgebraicGeometry.morphismRestrictOpensRange", "def_path": "Mathlib/AlgebraicGeometry/Restrict.lean", "def_pos": [377, 5], "def_end_pos": [377, 31]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)", "state_after": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\nthis : OpenEmbedding \u21d1(Arrow.mk (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map i))).hom.val.base\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)"}, {"tactic": "erw [Arrow.mk_hom, morphismRestrict_val_base] at this", "annotated_tactic": ["erw [<a>Arrow.mk_hom</a>, <a>morphismRestrict_val_base</a>] at this", [{"full_name": "CategoryTheory.Arrow.mk_hom", "def_path": "Mathlib/CategoryTheory/Comma/Arrow.lean", "def_pos": [78, 3], "def_end_pos": [78, 8]}, {"full_name": "AlgebraicGeometry.morphismRestrict_val_base", "def_path": "Mathlib/AlgebraicGeometry/Restrict.lean", "def_pos": [314, 9], "def_end_pos": [314, 34]}]], "state_before": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\nthis : OpenEmbedding \u21d1(Arrow.mk (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map i))).hom.val.base\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)", "state_after": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\nthis : OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage \u21d1f.val.base)\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case of_openCover.left\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\ni : \ud835\udcb0.J\nthis : OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage \u21d1f.val.base)\n\u22a2 OpenEmbedding ((Scheme.Hom.opensRange (\ud835\udcb0.map i)).carrier.restrictPreimage f.val.base.toFun)", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "have := Arrow.iso_w (morphismRestrictStalkMap\n  f (Scheme.Hom.opensRange (\ud835\udcb0.map <| \ud835\udcb0.f <| f.1.base x)) \u27e8x, \ud835\udcb0.covers _\u27e9)", "annotated_tactic": ["have := <a>Arrow.iso_w</a> (<a>morphismRestrictStalkMap</a>\n        f (<a>Scheme.Hom.opensRange</a> (\ud835\udcb0.map <| \ud835\udcb0.f <| f.1.<a>base</a> x)) \u27e8x, \ud835\udcb0.covers _\u27e9)", [{"full_name": "CategoryTheory.Arrow.iso_w", "def_path": "Mathlib/CategoryTheory/Comma/Arrow.lean", "def_pos": [171, 9], "def_end_pos": [171, 14]}, {"full_name": "AlgebraicGeometry.morphismRestrictStalkMap", "def_path": "Mathlib/AlgebraicGeometry/Restrict.lean", "def_pos": [443, 5], "def_end_pos": [443, 29]}, {"full_name": "AlgebraicGeometry.Scheme.Hom.opensRange", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}, {"full_name": "AlgebraicGeometry.PresheafedSpace.Hom.base", "def_path": "Mathlib/Geometry/RingedSpace/PresheafedSpace.lean", "def_pos": [102, 3], "def_end_pos": [102, 7]}]], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  (Arrow.mk (PresheafedSpace.stalkMap f.val \u2191\u27e8x, \u22ef\u27e9)).hom =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      (Arrow.mk (PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9)).hom \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "dsimp only [Arrow.mk_hom] at this", "annotated_tactic": ["dsimp only [<a>Arrow.mk_hom</a>] at this", [{"full_name": "CategoryTheory.Arrow.mk_hom", "def_path": "Mathlib/CategoryTheory/Comma/Arrow.lean", "def_pos": [78, 3], "def_end_pos": [78, 8]}]], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  (Arrow.mk (PresheafedSpace.stalkMap f.val \u2191\u27e8x, \u22ef\u27e9)).hom =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      (Arrow.mk (PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9)).hom \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso\n    ((morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right)"}, {"tactic": "haveI : IsOpenImmersion (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map <| \ud835\udcb0.f <| f.1.base x)) :=\n  (isOpenImmersion_respectsIso.arrow_iso_iff\n    (morphismRestrictOpensRange f (\ud835\udcb0.map _))).mpr (H _)", "annotated_tactic": ["haveI : <a>IsOpenImmersion</a> (f \u2223_ <a>Scheme.Hom.opensRange</a> (\ud835\udcb0.map <| \ud835\udcb0.f <| f.1.<a>base</a> x)) :=\n        (isOpenImmersion_respectsIso.arrow_iso_iff\n          (<a>morphismRestrictOpensRange</a> f (\ud835\udcb0.map _))).<a>mpr</a> (H _)", [{"full_name": "AlgebraicGeometry.IsOpenImmersion", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [36, 8], "def_end_pos": [36, 23]}, {"full_name": "AlgebraicGeometry.Scheme.Hom.opensRange", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [85, 5], "def_end_pos": [85, 15]}, {"full_name": "AlgebraicGeometry.PresheafedSpace.Hom.base", "def_path": "Mathlib/Geometry/RingedSpace/PresheafedSpace.lean", "def_pos": [102, 3], "def_end_pos": [102, 7]}, {"full_name": "AlgebraicGeometry.morphismRestrictOpensRange", "def_path": "Mathlib/AlgebraicGeometry/Restrict.lean", "def_pos": [377, 5], "def_end_pos": [377, 31]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\n\u22a2 IsIso\n    ((morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right)", "state_after": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis\u271d :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\nthis : IsOpenImmersion (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x))))\n\u22a2 IsIso\n    ((morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right)"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case of_openCover.right\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nX Y : Scheme\nf : X \u27f6 Y\n\ud835\udcb0 : Y.OpenCover\nH : \u2200 (i : \ud835\udcb0.J), IsOpenImmersion pullback.snd\nx : \u2191\u2191X.toPresheafedSpace\nthis\u271d :\n  PresheafedSpace.stalkMap f.val x =\n    (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right\nthis : IsOpenImmersion (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x))))\n\u22a2 IsIso\n    ((morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).inv.left \u226b\n      PresheafedSpace.stalkMap (f \u2223_ Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))).val \u27e8x, \u22ef\u27e9 \u226b\n        (morphismRestrictStalkMap f (Scheme.Hom.opensRange (\ud835\udcb0.map (\ud835\udcb0.f (f.val.base x)))) \u27e8x, \u22ef\u27e9).hom.right)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Cylinders.lean", "full_name": "MeasureTheory.compl_mem_measurableCylinders", "start": [317, 1], "end": [322, 22], "traced_tactics": [{"tactic": "rw [mem_measurableCylinders] at hs \u22a2", "annotated_tactic": ["rw [<a>mem_measurableCylinders</a>] at hs \u22a2", [{"full_name": "MeasureTheory.mem_measurableCylinders", "def_path": "Mathlib/MeasureTheory/Constructions/Cylinders.lean", "def_pos": [273, 9], "def_end_pos": [273, 32]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\ns t : Set ((i : \u03b9) \u2192 \u03b1 i)\nhs : s \u2208 measurableCylinders \u03b1\n\u22a2 s\u1d9c \u2208 measurableCylinders \u03b1", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\ns t : Set ((i : \u03b9) \u2192 \u03b1 i)\nhs : \u2203 s_1 S, MeasurableSet S \u2227 s = cylinder s_1 S\n\u22a2 \u2203 s_1 S, MeasurableSet S \u2227 s\u1d9c = cylinder s_1 S"}, {"tactic": "obtain \u27e8s, S, hS, rfl\u27e9 := hs", "annotated_tactic": ["obtain \u27e8s, S, hS, rfl\u27e9 := hs", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\ns t : Set ((i : \u03b9) \u2192 \u03b1 i)\nhs : \u2203 s_1 S, MeasurableSet S \u2227 s = cylinder s_1 S\n\u22a2 \u2203 s_1 S, MeasurableSet S \u2227 s\u1d9c = cylinder s_1 S", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\nt : Set ((i : \u03b9) \u2192 \u03b1 i)\ns : Finset \u03b9\nS : Set ((i : { x // x \u2208 s }) \u2192 \u03b1 \u2191i)\nhS : MeasurableSet S\n\u22a2 \u2203 s_1 S_1, MeasurableSet S_1 \u2227 (cylinder s S)\u1d9c = cylinder s_1 S_1"}, {"tactic": "refine \u27e8s, S\u1d9c, hS.compl, ?_\u27e9", "annotated_tactic": ["refine \u27e8s, S\u1d9c, hS.compl, ?_\u27e9", []], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\nt : Set ((i : \u03b9) \u2192 \u03b1 i)\ns : Finset \u03b9\nS : Set ((i : { x // x \u2208 s }) \u2192 \u03b1 \u2191i)\nhS : MeasurableSet S\n\u22a2 \u2203 s_1 S_1, MeasurableSet S_1 \u2227 (cylinder s S)\u1d9c = cylinder s_1 S_1", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\nt : Set ((i : \u03b9) \u2192 \u03b1 i)\ns : Finset \u03b9\nS : Set ((i : { x // x \u2208 s }) \u2192 \u03b1 \u2191i)\nhS : MeasurableSet S\n\u22a2 (cylinder s S)\u1d9c = cylinder s S\u1d9c"}, {"tactic": "rw [compl_cylinder]", "annotated_tactic": ["rw [<a>compl_cylinder</a>]", [{"full_name": "MeasureTheory.compl_cylinder", "def_path": "Mathlib/MeasureTheory/Constructions/Cylinders.lean", "def_pos": [209, 9], "def_end_pos": [209, 23]}]], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\nt : Set ((i : \u03b9) \u2192 \u03b1 i)\ns : Finset \u03b9\nS : Set ((i : { x // x \u2208 s }) \u2192 \u03b1 \u2191i)\nhS : MeasurableSet S\n\u22a2 (cylinder s S)\u1d9c = cylinder s S\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.image_closedBall", "start": [1072, 1], "end": [1073, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.le_inf_const_le", "start": [392, 1], "end": [393, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/LocalExtr/Basic.lean", "full_name": "IsLocalMaxOn.hasFDerivWithinAt_nonpos", "start": [114, 1], "end": [124, 62], "traced_tactics": [{"tactic": "rcases hy with \u27e8c, d, hd, hc, hcd\u27e9", "annotated_tactic": ["rcases hy with \u27e8c, d, hd, hc, hcd\u27e9", []], "state_before": "E : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y \u2208 posTangentConeAt s a\n\u22a2 f' y \u2264 0", "state_after": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 f' y \u2264 0"}, {"tactic": "suffices \u2200\u1da0 n in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0 from\n  le_of_tendsto (hf.lim atTop hd hc' hcd) this", "annotated_tactic": ["suffices \u2200\u1da0 n in <a>atTop</a>, c n \u2022 (f (a + d n) - f a) \u2264 0 from\n    <a>le_of_tendsto</a> (hf.lim <a>atTop</a> hd hc' hcd) this", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "le_of_tendsto", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [131, 9], "def_end_pos": [131, 22]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 f' y \u2264 0", "state_after": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0"}, {"tactic": "replace hd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0)) :=\n  tendsto_nhdsWithin_iff.2 \u27e8tendsto_const_nhds.add (tangentConeAt.lim_zero _ hc' hcd), hd\u27e9", "annotated_tactic": ["replace hd : <a>Tendsto</a> (fun n => a + d n) <a>atTop</a> (\ud835\udcdd[s] (a + 0)) :=\n    <a>tendsto_nhdsWithin_iff</a>.2 \u27e8tendsto_const_nhds.add (<a>tangentConeAt.lim_zero</a> _ hc' hcd), hd\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_nhdsWithin_iff", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [459, 9], "def_end_pos": [459, 31]}, {"full_name": "tangentConeAt.lim_zero", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [105, 9], "def_end_pos": [105, 31]}]], "state_before": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0", "state_after": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0"}, {"tactic": "rw [add_zero] at hd", "annotated_tactic": ["rw [<a>add_zero</a>] at hd", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0", "state_after": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0"}, {"tactic": "filter_upwards [hd.eventually h, hc.eventually_ge_atTop 0] with n hfn hcn", "annotated_tactic": ["filter_upwards [hd.eventually h, hc.eventually_ge_atTop 0] with n hfn hcn", []], "state_before": "case intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0", "state_after": "case h\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\nn : \u2115\nhfn : f (a + d n) \u2264 f a\nhcn : 0 \u2264 c n\n\u22a2 c n \u2022 (f (a + d n) - f a) \u2264 0"}, {"tactic": "exact mul_nonpos_of_nonneg_of_nonpos hcn (sub_nonpos.2 hfn)", "annotated_tactic": ["exact <a>mul_nonpos_of_nonneg_of_nonpos</a> hcn (<a>sub_nonpos</a>.2 hfn)", [{"full_name": "mul_nonpos_of_nonneg_of_nonpos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [440, 9], "def_end_pos": [440, 39]}, {"full_name": "sub_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [730, 15], "def_end_pos": [730, 25]}]], "state_before": "case h\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\nn : \u2115\nhfn : f (a + d n) \u2264 f a\nhcn : 0 \u2264 c n\n\u22a2 c n \u2022 (f (a + d n) - f a) \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.inv_bot", "start": [1487, 1], "end": [1488, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.isBounded_closure_iff", "start": [110, 1], "end": [111, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean", "full_name": "CategoryTheory.ProjectiveResolution.\u03c0_f_succ", "start": [89, 1], "end": [90, 56], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nZ : C\nP : ProjectiveResolution Z\nn : \u2115\n\u22a2 n + 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/FourierTransform.lean", "full_name": "VectorFourier.fourierIntegral_comp_add_right", "start": [104, 1], "end": [114, 73], "traced_tactics": [{"tactic": "ext1 w", "annotated_tactic": ["ext1 w", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\n\u22a2 fourierIntegral e \u03bc L (f \u2218 fun v => v + v\u2080) = fun w => e ((L v\u2080) w) \u2022 fourierIntegral e \u03bc L f w", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 fourierIntegral e \u03bc L (f \u2218 fun v => v + v\u2080) w = e ((L v\u2080) w) \u2022 fourierIntegral e \u03bc L f w"}, {"tactic": "dsimp only [fourierIntegral, Function.comp_apply, Submonoid.smul_def]", "annotated_tactic": ["dsimp only [<a>fourierIntegral</a>, <a>Function.comp_apply</a>, <a>Submonoid.smul_def</a>]", [{"full_name": "VectorFourier.fourierIntegral", "def_path": "Mathlib/Analysis/Fourier/FourierTransform.lean", "def_pos": [79, 5], "def_end_pos": [79, 20]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Submonoid.smul_def", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1461, 22], "def_end_pos": [1461, 30]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 fourierIntegral e \u03bc L (f \u2218 fun v => v + v\u2080) w = e ((L v\u2080) w) \u2022 fourierIntegral e \u03bc L f w", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f (v + v\u2080) \u2202\u03bc = \u2191(e ((L v\u2080) w)) \u2022 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f v \u2202\u03bc"}, {"tactic": "conv in L _ => rw [\u2190 add_sub_cancel_right v v\u2080]", "annotated_tactic": ["conv in L _ => rw [\u2190 <a>add_sub_cancel_right</a> v v\u2080]", [{"full_name": "add_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1008, 3], "def_end_pos": [1008, 14]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f (v + v\u2080) \u2202\u03bc = \u2191(e ((L v\u2080) w)) \u2022 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f v \u2202\u03bc", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(e (-(L (v + v\u2080 - v\u2080)) w)) \u2022 f (v + v\u2080) \u2202\u03bc = \u2191(e ((L v\u2080) w)) \u2022 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f v \u2202\u03bc"}, {"tactic": "rw [integral_add_right_eq_self fun v : V \u21a6 (e (-L (v - v\u2080) w) : \u2102) \u2022 f v, \u2190 integral_smul]", "annotated_tactic": ["rw [<a>integral_add_right_eq_self</a> fun v : V \u21a6 (e (-L (v - v\u2080) w) : \u2102) \u2022 f v, \u2190 <a>integral_smul</a>]", [{"full_name": "MeasureTheory.integral_add_right_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Integral.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MeasureTheory.integral_smul", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [915, 9], "def_end_pos": [915, 22]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(e (-(L (v + v\u2080 - v\u2080)) w)) \u2022 f (v + v\u2080) \u2202\u03bc = \u2191(e ((L v\u2080) w)) \u2022 \u222b (v : V), \u2191(e (-(L v) w)) \u2022 f v \u2202\u03bc", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (x : V), \u2191(e (-(L (x - v\u2080)) w)) \u2022 f x \u2202\u03bc = \u222b (a : V), \u2191(e ((L v\u2080) w)) \u2022 \u2191(e (-(L a) w)) \u2022 f a \u2202\u03bc"}, {"tactic": "congr 1 with v", "annotated_tactic": ["congr 1 with v", []], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (x : V), \u2191(e (-(L (x - v\u2080)) w)) \u2022 f x \u2202\u03bc = \u222b (a : V), \u2191(e ((L v\u2080) w)) \u2022 \u2191(e (-(L a) w)) \u2022 f a \u2202\u03bc", "state_after": "case h.e_f.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n\u22a2 \u2191(e (-(L (v - v\u2080)) w)) \u2022 f v = \u2191(e ((L v\u2080) w)) \u2022 \u2191(e (-(L v) w)) \u2022 f v"}, {"tactic": "rw [\u2190 smul_assoc, smul_eq_mul, \u2190 Submonoid.coe_mul, \u2190 e.map_add_eq_mul, \u2190 LinearMap.neg_apply,\n  \u2190 sub_eq_add_neg, \u2190 LinearMap.sub_apply, LinearMap.map_sub, neg_sub]", "annotated_tactic": ["rw [\u2190 <a>smul_assoc</a>, <a>smul_eq_mul</a>, \u2190 <a>Submonoid.coe_mul</a>, \u2190 e.map_add_eq_mul, \u2190 <a>LinearMap.neg_apply</a>,\n    \u2190 <a>sub_eq_add_neg</a>, \u2190 <a>LinearMap.sub_apply</a>, <a>LinearMap.map_sub</a>, <a>neg_sub</a>]", [{"full_name": "smul_assoc", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [255, 7], "def_end_pos": [255, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Submonoid.coe_mul", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [634, 9], "def_end_pos": [634, 16]}, {"full_name": "LinearMap.neg_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [936, 9], "def_end_pos": [936, 18]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "LinearMap.sub_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [958, 9], "def_end_pos": [958, 18]}, {"full_name": "LinearMap.map_sub", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [632, 19], "def_end_pos": [632, 26]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}]], "state_before": "case h.e_f.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u2102 F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u2102 G\ninst\u271d\u00b9 : MeasurableAdd V\ne : AddChar \ud835\udd5c \u21a5\ud835\udd4a\n\u03bc : Measure V\ninst\u271d : \u03bc.IsAddRightInvariant\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n\u22a2 \u2191(e (-(L (v - v\u2080)) w)) \u2022 f v = \u2191(e ((L v\u2080) w)) \u2022 \u2191(e (-(L v) w)) \u2022 f v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "ArithmeticFunction.pmul_assoc", "start": [509, 1], "end": [512, 36], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nf\u2081 f\u2082 f\u2083 : ArithmeticFunction R\n\u22a2 (f\u2081.pmul f\u2082).pmul f\u2083 = f\u2081.pmul (f\u2082.pmul f\u2083)", "state_after": "case h\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nf\u2081 f\u2082 f\u2083 : ArithmeticFunction R\nx\u271d : \u2115\n\u22a2 ((f\u2081.pmul f\u2082).pmul f\u2083) x\u271d = (f\u2081.pmul (f\u2082.pmul f\u2083)) x\u271d"}, {"tactic": "simp only [pmul_apply, mul_assoc]", "annotated_tactic": ["simp only [<a>pmul_apply</a>, <a>mul_assoc</a>]", [{"full_name": "ArithmeticFunction.pmul_apply", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [500, 9], "def_end_pos": [500, 19]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case h\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nf\u2081 f\u2082 f\u2083 : ArithmeticFunction R\nx\u271d : \u2115\n\u22a2 ((f\u2081.pmul f\u2082).pmul f\u2083) x\u271d = (f\u2081.pmul (f\u2082.pmul f\u2083)) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Meromorphic.lean", "full_name": "MeromorphicOn.mul", "start": [254, 1], "end": [254, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.ediam_univ_of_noncompact", "start": [483, 1], "end": [485, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isLittleO_zero_right_iff", "start": [1270, 1], "end": [1272, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomologySequence.lean", "full_name": "HomologicalComplex.opcyclesToCycles_iCycles", "start": [47, 1], "end": [50, 7], "traced_tactics": [{"tactic": "dsimp only [opcyclesToCycles]", "annotated_tactic": ["dsimp only [<a>opcyclesToCycles</a>]", [{"full_name": "HomologicalComplex.opcyclesToCycles", "def_path": "Mathlib/Algebra/Homology/HomologySequence.lean", "def_pos": [43, 19], "def_end_pos": [43, 35]}]], "state_before": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : HasZeroMorphisms C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\n\u03c6 : K \u27f6 L\ni j : \u03b9\ninst\u271d\u00b3 : K.HasHomology i\ninst\u271d\u00b2 : K.HasHomology j\ninst\u271d\u00b9 : L.HasHomology i\ninst\u271d : L.HasHomology j\n\u22a2 K.opcyclesToCycles i j \u226b K.iCycles j = K.fromOpcycles i j", "state_after": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : HasZeroMorphisms C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\n\u03c6 : K \u27f6 L\ni j : \u03b9\ninst\u271d\u00b3 : K.HasHomology i\ninst\u271d\u00b2 : K.HasHomology j\ninst\u271d\u00b9 : L.HasHomology i\ninst\u271d : L.HasHomology j\n\u22a2 K.liftCycles (K.fromOpcycles i j) (c.next j) \u22ef \u22ef \u226b K.iCycles j = K.fromOpcycles i j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : HasZeroMorphisms C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\n\u03c6 : K \u27f6 L\ni j : \u03b9\ninst\u271d\u00b3 : K.HasHomology i\ninst\u271d\u00b2 : K.HasHomology j\ninst\u271d\u00b9 : L.HasHomology i\ninst\u271d : L.HasHomology j\n\u22a2 K.liftCycles (K.fromOpcycles i j) (c.next j) \u22ef \u22ef \u226b K.iCycles j = K.fromOpcycles i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "full_name": "UniformInducing.cauchy_map_iff", "start": [88, 1], "end": [90, 97], "traced_tactics": [{"tactic": "simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, \u2190 hf.comap_uniformity]", "annotated_tactic": ["simp only [<a>Cauchy</a>, <a>map_neBot_iff</a>, <a>prod_map_map_eq</a>, <a>map_le_iff_le_comap</a>, \u2190 hf.comap_uniformity]", [{"full_name": "Cauchy", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [31, 5], "def_end_pos": [31, 11]}, {"full_name": "Filter.map_neBot_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2660, 9], "def_end_pos": [2660, 22]}, {"full_name": "Filter.prod_map_map_eq", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [356, 9], "def_end_pos": [356, 24]}, {"full_name": "Filter.map_le_iff_le_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2342, 9], "def_end_pos": [2342, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : UniformInducing f\nF : Filter \u03b1\n\u22a2 Cauchy (map f F) \u2194 Cauchy F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.bit1_of_bit1", "start": [1095, 1], "end": [1098, 97], "traced_tactics": [{"tactic": "rw [PosNum.one_sub', a.bit0_of_bit0]", "annotated_tactic": ["rw [<a>PosNum.one_sub'</a>, a.bit0_of_bit0]", [{"full_name": "PosNum.one_sub'", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [815, 9], "def_end_pos": [815, 17]}]], "state_before": "\u03b1 : Type u_1\na : PosNum\n\u22a2 sub' 1 (_root_.bit0 a) = (neg a).bit1", "state_after": "\u03b1 : Type u_1\na : PosNum\n\u22a2 a.bit0.pred'.toZNumNeg = (neg a).bit1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na : PosNum\n\u22a2 a.bit0.pred'.toZNumNeg = (neg a).bit1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "full_name": "Profinite.NobelingProof.GoodProducts.range_equiv_factorization", "start": [1081, 1], "end": [1083, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.mod_self", "start": [59, 1], "end": [60, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "GradedMonoid.GMonoid.gnpowRec_zero", "start": [205, 1], "end": [206, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.tan_sub_pi", "start": [809, 1], "end": [810, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.coe_toList", "start": [479, 1], "end": [480, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/PartitionOfUnity.lean", "full_name": "SmoothBumpCovering.mem_extChartAt_ind_source", "start": [474, 1], "end": [476, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "Sbtw.ne_right", "start": [244, 1], "end": [245, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.Prepartition.mem_biUnionTagged", "start": [139, 1], "end": [141, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "Dense.exists_countable_dense_subset", "start": [639, 1], "end": [643, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.setOf_app_iff", "start": [254, 1], "end": [255, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Linear.lean", "full_name": "LinearMap.deriv", "start": [95, 11], "end": [96, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Sym.lean", "full_name": "Multiset.sym2_coe", "start": [44, 9], "end": [44, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Sqrt.lean", "full_name": "NNReal.le_sqrt_iff_sq_le", "start": [74, 1], "end": [74, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Pochhammer.lean", "full_name": "monic_descPochhammer", "start": [262, 1], "end": [269, 38], "traced_tactics": [{"tactic": "induction' n with n hn", "annotated_tactic": ["induction' n with n hn", []], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\nn : \u2115\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\n\u22a2 (descPochhammer R n).Monic", "state_after": "case zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\n\u22a2 (descPochhammer R 0).Monic\n\ncase succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\n\u22a2 (descPochhammer R (n + 1)).Monic"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\n\u22a2 (descPochhammer R 0).Monic", "state_after": "no goals"}, {"tactic": "have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1", "annotated_tactic": ["have h : <a>leadingCoeff</a> (<a>X</a> - 1 : R[X]) = 1 := <a>leadingCoeff_X_sub_C</a> 1", [{"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [72, 5], "def_end_pos": [72, 17]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 6]}, {"full_name": "Polynomial.leadingCoeff_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [1640, 9], "def_end_pos": [1640, 29]}]], "state_before": "case succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\n\u22a2 (descPochhammer R (n + 1)).Monic", "state_after": "case succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\nh : (X - 1).leadingCoeff = 1\n\u22a2 (descPochhammer R (n + 1)).Monic"}, {"tactic": "have : natDegree (X - (1 : R[X])) \u2260 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R)", "annotated_tactic": ["have : <a>natDegree</a> (<a>X</a> - (1 : R[X])) \u2260 0 := <a>ne_zero_of_eq_one</a> <| <a>natDegree_X_sub_C</a> (1 : R)", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [67, 5], "def_end_pos": [67, 14]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 6]}, {"full_name": "ne_zero_of_eq_one", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [61, 7], "def_end_pos": [61, 24]}, {"full_name": "Polynomial.natDegree_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [1607, 9], "def_end_pos": [1607, 26]}]], "state_before": "case succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\nh : (X - 1).leadingCoeff = 1\n\u22a2 (descPochhammer R (n + 1)).Monic", "state_after": "case succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\nh : (X - 1).leadingCoeff = 1\nthis : (X - 1).natDegree \u2260 0\n\u22a2 (descPochhammer R (n + 1)).Monic"}, {"tactic": "rw [descPochhammer_succ_left, Monic.def, leadingCoeff_mul, leadingCoeff_comp this, hn, monic_X,\n    one_mul, one_mul, h, one_pow]", "annotated_tactic": ["rw [<a>descPochhammer_succ_left</a>, <a>Monic.def</a>, <a>leadingCoeff_mul</a>, <a>leadingCoeff_comp</a> this, hn, <a>monic_X</a>,\n        <a>one_mul</a>, <a>one_mul</a>, h, <a>one_pow</a>]", [{"full_name": "descPochhammer_succ_left", "def_path": "Mathlib/RingTheory/Polynomial/Pochhammer.lean", "def_pos": [258, 9], "def_end_pos": [258, 33]}, {"full_name": "Polynomial.Monic.def", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"full_name": "Polynomial.leadingCoeff_mul", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 25]}, {"full_name": "Polynomial.leadingCoeff_comp", "def_path": "Mathlib/Algebra/Polynomial/Degree/Lemmas.lean", "def_pos": [413, 9], "def_end_pos": [413, 26]}, {"full_name": "Polynomial.monic_X", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [895, 9], "def_end_pos": [895, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}]], "state_before": "case succ\nR : Type u\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nn : \u2115\nhn : (descPochhammer R n).Monic\nh : (X - 1).leadingCoeff = 1\nthis : (X - 1).natDegree \u2260 0\n\u22a2 (descPochhammer R (n + 1)).Monic", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean", "full_name": "HasStrictFDerivAt.approximates_deriv_on_nhds", "start": [74, 1], "end": [83, 58], "traced_tactics": [{"tactic": "cases' hc with hE hc", "annotated_tactic": ["cases' hc with hE hc", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : Subsingleton E \u2228 0 < c\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "case inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhE : Subsingleton E\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c\n\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c"}, {"tactic": "have := hf.def hc", "annotated_tactic": ["have := hf.def hc", []], "state_before": "case inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "case inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2200\u1da0 (x : E \u00d7 E) in \ud835\udcdd (a, a), \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c"}, {"tactic": "rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this", "annotated_tactic": ["rw [<a>nhds_prod_eq</a>, <a>Filter.Eventually</a>, <a>mem_prod_same_iff</a>] at this", [{"full_name": "nhds_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [549, 9], "def_end_pos": [549, 21]}, {"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1090, 15], "def_end_pos": [1090, 25]}, {"full_name": "Filter.mem_prod_same_iff", "def_path": "Mathlib/Order/Filter/Lift.lean", "def_pos": [428, 7], "def_end_pos": [428, 24]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2200\u1da0 (x : E \u00d7 E) in \ud835\udcdd (a, a), \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "case inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 t \u2208 \ud835\udcdd a, t \u00d7\u02e2 t \u2286 {x | \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016}\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c"}, {"tactic": "rcases this with \u27e8s, has, hs\u27e9", "annotated_tactic": ["rcases this with \u27e8s, has, hs\u27e9", []], "state_before": "case inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\nthis : \u2203 t \u2208 \ud835\udcdd a, t \u00d7\u02e2 t \u2286 {x | \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016}\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "case inr.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\ns : Set E\nhas : s \u2208 \ud835\udcdd a\nhs : s \u00d7\u02e2 s \u2286 {x | \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016}\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c"}, {"tactic": "exact \u27e8s, has, fun x hx y hy => hs (mk_mem_prod hx hy)\u27e9", "annotated_tactic": ["exact \u27e8s, has, fun x hx y hy => hs (<a>mk_mem_prod</a> hx hy)\u27e9", [{"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [214, 9], "def_end_pos": [214, 20]}]], "state_before": "case inr.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhc : 0 < c\ns : Set E\nhas : s \u2208 \ud835\udcdd a\nhs : s \u00d7\u02e2 s \u2286 {x | \u2016f x.1 - f x.2 - f' (x.1 - x.2)\u2016 \u2264 (fun a => \u2191a) c * \u2016x.1 - x.2\u2016}\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "no goals"}, {"tactic": "refine \u27e8univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>univ</a>, <a>IsOpen.mem_nhds</a> <a>isOpen_univ</a> <a>trivial</a>, fun x _ y _ => ?_\u27e9", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [88, 17], "def_end_pos": [88, 28]}, {"full_name": "trivial", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [645, 35], "def_end_pos": [645, 42]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhE : Subsingleton E\n\u22a2 \u2203 s \u2208 \ud835\udcdd a, ApproximatesLinearOn f f' s c", "state_after": "case inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhE : Subsingleton E\nx : E\nx\u271d\u00b9 : x \u2208 univ\ny : E\nx\u271d : y \u2208 univ\n\u22a2 \u2016f x - f y - f' (x - y)\u2016 \u2264 \u2191c * \u2016x - y\u2016"}, {"tactic": "simp [@Subsingleton.elim E hE x y]", "annotated_tactic": ["simp [@<a>Subsingleton.elim</a> E hE x y]", [{"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f f' a\nc : \u211d\u22650\nhE : Subsingleton E\nx : E\nx\u271d\u00b9 : x \u2208 univ\ny : E\nx\u271d : y \u2208 univ\n\u22a2 \u2016f x - f y - f' (x - y)\u2016 \u2264 \u2191c * \u2016x - y\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean", "full_name": "isLocalStructomorphOn_contDiffGroupoid_iff", "start": [181, 1], "end": [270, 41], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\n\u22a2 LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source \u2194\n    SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\n\u22a2 LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source \u2192\n    SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target\n\ncase mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\n\u22a2 SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target \u2192\n    LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\n\u22a2 LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source \u2192\n    SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\n\u22a2 SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target"}, {"tactic": "refine \u27e8isLocalStructomorphOn_contDiffGroupoid_iff_aux h,\n  isLocalStructomorphOn_contDiffGroupoid_iff_aux ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>isLocalStructomorphOn_contDiffGroupoid_iff_aux</a> h,\n      <a>isLocalStructomorphOn_contDiffGroupoid_iff_aux</a> ?_\u27e9", [{"full_name": "isLocalStructomorphOn_contDiffGroupoid_iff_aux", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean", "def_pos": [134, 9], "def_end_pos": [134, 55]}, {"full_name": "isLocalStructomorphOn_contDiffGroupoid_iff_aux", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean", "def_pos": [134, 9], "def_end_pos": [134, 55]}]], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\n\u22a2 SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\n\u22a2 LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source"}, {"tactic": "intro X hX", "annotated_tactic": ["intro X hX", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\n\u22a2 LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "let x := f.symm X", "annotated_tactic": ["let x := f.symm X", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "have hx : x \u2208 f.source := f.symm.mapsTo hX", "annotated_tactic": ["have hx : x \u2208 f.source := f.symm.mapsTo hX", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "let c := chartAt H x", "annotated_tactic": ["let c := <a>chartAt</a> H x", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}]], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "let c' := chartAt H X", "annotated_tactic": ["let c' := <a>chartAt</a> H X", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}]], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "obtain \u27e8-, hxf\u27e9 := h x hx", "annotated_tactic": ["obtain \u27e8-, hxf\u27e9 := h x hx", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X"}, {"tactic": "refine \u27e8(f.symm.continuousAt hX).continuousWithinAt, fun h2x => ?_\u27e9", "annotated_tactic": ["refine \u27e8(f.symm.continuousAt hX).<a>continuousWithinAt</a>, fun h2x => ?_\u27e9", [{"full_name": "ContinuousAt.continuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [897, 9], "def_end_pos": [897, 40]}]], "state_before": "case mp.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f.symm) f.symm.source X", "state_after": "case mp.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f.symm X)) \u2218 \u2191f.symm \u2218 \u2191(chartAt H X).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H X).symm \u207b\u00b9' f.symm.source \u2229 e.source) \u2227\n      \u2191(chartAt H X) X \u2208 e.source"}, {"tactic": "have h2X : c' X = e (c (f.symm X)) := by\n  rw [\u2190 hef hex]\n  dsimp only [Function.comp_def]\n  have hfX : f.symm X \u2208 c.source := by simp only [c, hX, mfld_simps]\n  rw [c.left_inv hfX, f.right_inv hX]", "annotated_tactic": ["have h2X : c' X = e (c (f.symm X)) := by\n      rw [\u2190 hef hex]\n      dsimp only [<a>Function.comp_def</a>]\n      have hfX : f.symm X \u2208 c.source := by simp only [c, hX, mfld_simps]\n      rw [c.left_inv hfX, f.right_inv hX]", [{"full_name": "Function.comp_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [37, 9], "def_end_pos": [37, 26]}]], "state_before": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f.symm X)) \u2218 \u2191f.symm \u2218 \u2191(chartAt H X).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H X).symm \u207b\u00b9' f.symm.source \u2229 e.source) \u2227\n      \u2191(chartAt H X) X \u2208 e.source", "state_after": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f.symm X)) \u2218 \u2191f.symm \u2218 \u2191(chartAt H X).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H X).symm \u207b\u00b9' f.symm.source \u2229 e.source) \u2227\n      \u2191(chartAt H X) X \u2208 e.source"}, {"tactic": "refine \u27e8e.symm, StructureGroupoid.symm _ he, h3e, ?_\u27e9", "annotated_tactic": ["refine \u27e8e.symm, <a>StructureGroupoid.symm</a> _ he, h3e, ?_\u27e9", [{"full_name": "StructureGroupoid.symm", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [243, 9], "def_end_pos": [243, 31]}]], "state_before": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh3e : EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f.symm X)) \u2218 \u2191f.symm \u2218 \u2191(chartAt H X).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H X).symm \u207b\u00b9' f.symm.source \u2229 e.source) \u2227\n      \u2191(chartAt H X) X \u2208 e.source", "state_after": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh3e : EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)\n\u22a2 \u2191(chartAt H X) X \u2208 e.symm.source"}, {"tactic": "rw [h2X]", "annotated_tactic": ["rw [h2X]", []], "state_before": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh3e : EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)\n\u22a2 \u2191(chartAt H X) X \u2208 e.symm.source", "state_after": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh3e : EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)\n\u22a2 \u2191e (\u2191c (\u2191f.symm X)) \u2208 e.symm.source"}, {"tactic": "exact e.mapsTo hex", "annotated_tactic": ["exact e.mapsTo hex", []], "state_before": "case mp.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh3e : EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)\n\u22a2 \u2191e (\u2191c (\u2191f.symm X)) \u2208 e.symm.source", "state_after": "no goals"}, {"tactic": "have h1 : c' = chartAt H (f x) := by simp only [f.right_inv hX]", "annotated_tactic": ["have h1 : c' = <a>chartAt</a> H (f x) := by simp only [f.right_inv hX]", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source"}, {"tactic": "have h2 : c' \u2218 f \u2218 c.symm = \u21d1(c.symm \u226b\u2095 f \u226b\u2095 c') := rfl", "annotated_tactic": ["have h2 : c' \u2218 f \u2218 c.symm = \u21d1(c.symm \u226b\u2095 f \u226b\u2095 c') := <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source"}, {"tactic": "have hcx : c x \u2208 c.symm \u207b\u00b9' f.source := by simp only [c, hx, mfld_simps]", "annotated_tactic": ["have hcx : c x \u2208 c.symm \u207b\u00b9' f.source := by simp only [c, hx, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source"}, {"tactic": "rw [h2]", "annotated_tactic": ["rw [h2]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source"}, {"tactic": "rw [\u2190 h1, h2, PartialHomeomorph.isLocalStructomorphWithinAt_iff'] at hxf", "annotated_tactic": ["rw [\u2190 h1, h2, <a>PartialHomeomorph.isLocalStructomorphWithinAt_iff'</a>] at hxf", [{"full_name": "PartialHomeomorph.isLocalStructomorphWithinAt_iff'", "def_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "def_pos": [671, 9], "def_end_pos": [671, 66]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  \u2191(chartAt H x) x \u2208 \u2191(chartAt H x).symm \u207b\u00b9' f.source \u2192\n    \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n      e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191(chartAt H x) x \u2208 e.source\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source\n\ncase hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2286 \u2191(chartAt H x).symm \u207b\u00b9' f.source\n\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u222a (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\u1d9c"}, {"tactic": "simp only [f.right_inv hX]", "annotated_tactic": ["simp only [f.right_inv hX]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\n\u22a2 c' = chartAt H (\u2191f x)", "state_after": "no goals"}, {"tactic": "simp only [c, hx, mfld_simps]", "annotated_tactic": ["simp only [c, hx, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\n\u22a2 \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source", "state_after": "no goals"}, {"tactic": "exact hxf hcx", "annotated_tactic": ["exact hxf hcx", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  \u2191(chartAt H x) x \u2208 \u2191(chartAt H x).symm \u207b\u00b9' f.source \u2192\n    \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n      e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191(chartAt H x) x \u2208 e.source\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2227 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191e) e.source \u2227 \u2191c x \u2208 e.source", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "case hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2286 \u2191(chartAt H x).symm \u207b\u00b9' f.source", "state_after": "no goals"}, {"tactic": "apply Or.inl", "annotated_tactic": ["apply <a>Or.inl</a>", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').source \u222a (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\u1d9c", "state_after": "case hx.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').source"}, {"tactic": "simp only [c, hx, h1, mfld_simps]", "annotated_tactic": ["simp only [c, hx, h1, mfld_simps]", []], "state_before": "case hx.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (\u2191(chartAt H x).symm \u207b\u00b9' f.source)\n    (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\nh1 : c' = chartAt H (\u2191f x)\nh2 : \u2191c' \u2218 \u2191f \u2218 \u2191c.symm = \u2191(c.symm \u226b\u2095 f \u226b\u2095 c')\nhcx : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').source", "state_after": "no goals"}, {"tactic": "rw [\u2190 hef hex]", "annotated_tactic": ["rw [\u2190 hef hex]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191c' X = (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191c x)"}, {"tactic": "dsimp only [Function.comp_def]", "annotated_tactic": ["dsimp only [<a>Function.comp_def</a>]", [{"full_name": "Function.comp_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [37, 9], "def_end_pos": [37, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191c' X = (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191c x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191c' X = \u2191c' (\u2191f (\u2191c.symm (\u2191c x)))"}, {"tactic": "have hfX : f.symm X \u2208 c.source := by simp only [c, hX, mfld_simps]", "annotated_tactic": ["have hfX : f.symm X \u2208 c.source := by simp only [c, hX, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191c' X = \u2191c' (\u2191f (\u2191c.symm (\u2191c x)))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nhfX : \u2191f.symm X \u2208 c.source\n\u22a2 \u2191c' X = \u2191c' (\u2191f (\u2191c.symm (\u2191c x)))"}, {"tactic": "rw [c.left_inv hfX, f.right_inv hX]", "annotated_tactic": ["rw [c.left_inv hfX, f.right_inv hX]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nhfX : \u2191f.symm X \u2208 c.source\n\u22a2 \u2191c' X = \u2191c' (\u2191f (\u2191c.symm (\u2191c x)))", "state_after": "no goals"}, {"tactic": "simp only [c, hX, mfld_simps]", "annotated_tactic": ["simp only [c, hX, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\n\u22a2 \u2191f.symm X \u2208 c.source", "state_after": "no goals"}, {"tactic": "have h2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target := by\n  intro x hx; rw [\u2190 e.right_inv hx, \u2190 hef (e.symm.mapsTo hx)]\n  exact PartialHomeomorph.mapsTo _ (h2e <| e.symm.mapsTo hx)", "annotated_tactic": ["have h2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').<a>target</a> := by\n        intro x hx; rw [\u2190 e.right_inv hx, \u2190 hef (e.symm.mapsTo hx)]\n        exact <a>PartialHomeomorph.mapsTo</a> _ (h2e <| e.symm.mapsTo hx)", [{"full_name": "PartialEquiv.target", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [129, 3], "def_end_pos": [129, 9]}, {"full_name": "PartialHomeomorph.mapsTo", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [175, 19], "def_end_pos": [175, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\n\u22a2 EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)"}, {"tactic": "rw [inter_self] at h1", "annotated_tactic": ["rw [<a>inter_self</a>] at h1", [{"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)"}, {"tactic": "rwa [inter_eq_right.mpr]", "annotated_tactic": ["rwa [inter_eq_right.mpr]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 EqOn (\u2191c \u2218 \u2191f.symm \u2218 \u2191c'.symm) (\u2191e.symm) (\u2191c'.symm \u207b\u00b9' f.target \u2229 e.target)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 e.target \u2286 \u2191c'.symm \u207b\u00b9' f.target"}, {"tactic": "refine h2.trans ?_", "annotated_tactic": ["refine h2.trans ?_", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 e.target \u2286 \u2191c'.symm \u207b\u00b9' f.target", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 (c.symm \u226b\u2095 f \u226b\u2095 c').target \u2286 \u2191c'.symm \u207b\u00b9' f.target"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) e.target\nh2 : e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target\n\u22a2 (c.symm \u226b\u2095 f \u226b\u2095 c').target \u2286 \u2191c'.symm \u207b\u00b9' f.target", "state_after": "no goals"}, {"tactic": "apply EqOn.symm", "annotated_tactic": ["apply <a>EqOn.symm</a>", [{"full_name": "Set.EqOn.symm", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e.symm) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (e.target \u2229 e.target)"}, {"tactic": "refine e.isImage_source_target.symm_eqOn_of_inter_eq_of_eqOn ?_ ?_", "annotated_tactic": ["refine e.isImage_source_target.symm_eqOn_of_inter_eq_of_eqOn ?_ ?_", []], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e.symm) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (e.target \u2229 e.target)", "state_after": "case h.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 e.source \u2229 e.source = (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2229 e.source\n\ncase h.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (e.source \u2229 e.source)"}, {"tactic": "rw [inter_self, inter_eq_right.mpr h2e]", "annotated_tactic": ["rw [<a>inter_self</a>, inter_eq_right.mpr h2e]", [{"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 19]}]], "state_before": "case h.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 e.source \u2229 e.source = (c.symm \u226b\u2095 f \u226b\u2095 c').source \u2229 e.source", "state_after": "no goals"}, {"tactic": "rw [inter_self]", "annotated_tactic": ["rw [<a>inter_self</a>]", [{"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 19]}]], "state_before": "case h.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) (e.source \u2229 e.source)", "state_after": "case h.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) e.source"}, {"tactic": "exact hef.symm", "annotated_tactic": ["exact hef.symm", []], "state_before": "case h.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\n\u22a2 EqOn (\u2191e) (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c')) e.source", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx : M := \u2191f.symm X\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\n\u22a2 e.target \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').target", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d\u00b9 : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx\u271d : M := \u2191f.symm X\nhx\u271d : x\u271d \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\u271d\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x\u271d)) \u2218 \u2191f \u2218 \u2191(chartAt H x\u271d).symm)\n    (\u2191(chartAt H x\u271d).symm \u207b\u00b9' f.source) (\u2191(chartAt H x\u271d) x\u271d)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x\u271d \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nx : H\nhx : x \u2208 e.target\n\u22a2 x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').target"}, {"tactic": "rw [\u2190 e.right_inv hx, \u2190 hef (e.symm.mapsTo hx)]", "annotated_tactic": ["rw [\u2190 e.right_inv hx, \u2190 hef (e.symm.mapsTo hx)]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d\u00b9 : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx\u271d : M := \u2191f.symm X\nhx\u271d : x\u271d \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\u271d\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x\u271d)) \u2218 \u2191f \u2218 \u2191(chartAt H x\u271d).symm)\n    (\u2191(chartAt H x\u271d).symm \u207b\u00b9' f.source) (\u2191(chartAt H x\u271d) x\u271d)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x\u271d \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nx : H\nhx : x \u2208 e.target\n\u22a2 x \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').target", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d\u00b9 : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx\u271d : M := \u2191f.symm X\nhx\u271d : x\u271d \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\u271d\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x\u271d)) \u2218 \u2191f \u2218 \u2191(chartAt H x\u271d).symm)\n    (\u2191(chartAt H x\u271d).symm \u207b\u00b9' f.source) (\u2191(chartAt H x\u271d) x\u271d)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x\u271d \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nx : H\nhx : x \u2208 e.target\n\u22a2 (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e.symm x) \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').target"}, {"tactic": "exact PartialHomeomorph.mapsTo _ (h2e <| e.symm.mapsTo hx)", "annotated_tactic": ["exact <a>PartialHomeomorph.mapsTo</a> _ (h2e <| e.symm.mapsTo hx)", [{"full_name": "PartialHomeomorph.mapsTo", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [175, 19], "def_end_pos": [175, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne\u271d : PartialHomeomorph M H\nx\u271d\u00b9 : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh : LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source\nX : M'\nhX : X \u2208 f.symm.source\nx\u271d : M := \u2191f.symm X\nhx\u271d : x\u271d \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\u271d\nc' : PartialHomeomorph M' H := chartAt H X\nhxf :\n  (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x\u271d)) \u2218 \u2191f \u2218 \u2191(chartAt H x\u271d).symm)\n    (\u2191(chartAt H x\u271d).symm \u207b\u00b9' f.source) (\u2191(chartAt H x\u271d) x\u271d)\nh2x : \u2191(chartAt H X) X \u2208 \u2191(chartAt H X).symm \u207b\u00b9' f.symm.source\ne : PartialHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nh2e : e.source \u2286 (c.symm \u226b\u2095 f \u226b\u2095 c').source\nhef : EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e) e.source\nhex : \u2191c x\u271d \u2208 e.source\nh2X : \u2191c' X = \u2191e (\u2191c (\u2191f.symm X))\nh1 : EqOn (\u2191(c.symm \u226b\u2095 f \u226b\u2095 c').symm) (\u2191e.symm) (e.target \u2229 e.target)\nx : H\nhx : x \u2208 e.target\n\u22a2 (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191e.symm x) \u2208 (c.symm \u226b\u2095 f \u226b\u2095 c').target", "state_after": "no goals"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9 x hx", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9 x hx", []], "state_before": "case mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\n\u22a2 SmoothOn I I (\u2191f) f.source \u2227 SmoothOn I I (\u2191f.symm) f.target \u2192\n    LiftPropOn (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source", "state_after": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source x"}, {"tactic": "refine \u27e8(h\u2081 x hx).continuousWithinAt, ?_\u27e9", "annotated_tactic": ["refine \u27e8(h\u2081 x hx).<a>continuousWithinAt</a>, ?_\u27e9", [{"full_name": "ContMDiffWithinAt.continuousWithinAt", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [669, 9], "def_end_pos": [669, 45]}]], "state_before": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\n\u22a2 LiftPropWithinAt (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191f) f.source x", "state_after": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)"}, {"tactic": "let c := chartAt H x", "annotated_tactic": ["let c := <a>chartAt</a> H x", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}]], "state_before": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)", "state_after": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)"}, {"tactic": "let c' := chartAt H (f x)", "annotated_tactic": ["let c' := <a>chartAt</a> H (f x)", [{"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}]], "state_before": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)", "state_after": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)"}, {"tactic": "rintro (hx' : c x \u2208 c.symm \u207b\u00b9' f.source)", "annotated_tactic": ["rintro (hx' : c x \u2208 c.symm \u207b\u00b9' f.source)", []], "state_before": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\n\u22a2 (contDiffGroupoid \u22a4 I).IsLocalStructomorphWithinAt (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source) (\u2191(chartAt H x) x)", "state_after": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 e.source) \u2227\n      \u2191(chartAt H x) x \u2208 e.source"}, {"tactic": "refine \u27e8(c.symm.trans f).trans c', \u27e8?_, ?_\u27e9, (?_ : EqOn (c' \u2218 f \u2218 c.symm) _ _), ?_\u27e9", "annotated_tactic": ["refine \u27e8(c.symm.trans f).<a>trans</a> c', \u27e8?_, ?_\u27e9, (?_ : <a>EqOn</a> (c' \u2218 f \u2218 c.symm) _ _), ?_\u27e9", [{"full_name": "PartialHomeomorph.trans", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [842, 15], "def_end_pos": [842, 20]}, {"full_name": "Set.EqOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [263, 5], "def_end_pos": [263, 9]}]], "state_before": "case mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2203 e \u2208 contDiffGroupoid \u22a4 I,\n    EqOn (\u2191(chartAt H (\u2191f x)) \u2218 \u2191f \u2218 \u2191(chartAt H x).symm) (\u2191e.toPartialEquiv)\n        (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 e.source) \u2227\n      \u2191(chartAt H x) x \u2208 e.source", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (contDiffPregroupoid \u22a4 I).property (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c')) ((c.symm \u226b\u2095 f) \u226b\u2095 c').source\n\ncase mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (contDiffPregroupoid \u22a4 I).property (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm) ((c.symm \u226b\u2095 f) \u226b\u2095 c').target\n\ncase mpr.intro.refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').toPartialEquiv)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 ((c.symm \u226b\u2095 f) \u226b\u2095 c').source)\n\ncase mpr.intro.refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 ((c.symm \u226b\u2095 f) \u226b\u2095 c').source"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (contDiffPregroupoid \u22a4 I).property (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c')) ((c.symm \u226b\u2095 f) \u226b\u2095 c').source", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : y \u2208 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y"}, {"tactic": "simp only [mfld_simps] at hy", "annotated_tactic": ["simp only [mfld_simps] at hy", []], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : y \u2208 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y"}, {"tactic": "have hy' : (extChartAt I x).symm y \u2208 c.source := by simp only [hy, mfld_simps]", "annotated_tactic": ["have hy' : (<a>extChartAt</a> I x).<a>symm</a> y \u2208 c.source := by simp only [hy, mfld_simps]", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}]], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y"}, {"tactic": "have hy'' : f ((extChartAt I x).symm y) \u2208 c'.source := by simp only [hy, mfld_simps]", "annotated_tactic": ["have hy'' : f ((<a>extChartAt</a> I x).<a>symm</a> y) \u2208 c'.source := by simp only [hy, mfld_simps]", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}]], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y"}, {"tactic": "rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H", "annotated_tactic": ["rw [<a>contMDiffWithinAt_iff_of_mem_source</a> hy' hy''] at H", [{"full_name": "contMDiffWithinAt_iff_of_mem_source", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [412, 9], "def_end_pos": [412, 44]}]], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y"}, {"tactic": "convert H.2.mono _", "annotated_tactic": ["convert H.2.<a>mono</a> _", [{"full_name": "ContDiffWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [497, 9], "def_end_pos": [497, 30]}]], "state_before": "case mpr.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c') \u2218 \u2191I.symm) (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I) y", "state_after": "case h.e'_12\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 y = \u2191(extChartAt I x) (\u2191(extChartAt I x).symm y)\n\ncase mpr.intro.refine_1.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I \u2286 \u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I"}, {"tactic": "refine (h\u2081 ((extChartAt I x).symm y) ?_).mono ?_", "annotated_tactic": ["refine (h\u2081 ((<a>extChartAt</a> I x).<a>symm</a> y) ?_).<a>mono</a> ?_", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}, {"full_name": "ContMDiffWithinAt.mono", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [722, 9], "def_end_pos": [722, 31]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)", "state_after": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 \u2191(extChartAt I x).symm y \u2208 f.source\n\ncase refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 (f \u226b\u2095 c').source \u2286 f.source"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 \u2191(extChartAt I x).symm y \u2208 f.source", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "case refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\n\u22a2 (f \u226b\u2095 c').source \u2286 f.source", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\n\u22a2 \u2191(extChartAt I x).symm y \u2208 c.source", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y)\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\n\u22a2 \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "case h.e'_12\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 y = \u2191(extChartAt I x) (\u2191(extChartAt I x).symm y)", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "case mpr.intro.refine_1.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : ((\u2191I.symm y \u2208 c.target \u2227 \u2191c.symm (\u2191I.symm y) \u2208 f.source) \u2227 \u2191f (\u2191c.symm (\u2191I.symm y)) \u2208 c'.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f) (f \u226b\u2095 c').source (\u2191(extChartAt I x).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I (\u2191f x)) \u2218 \u2191f \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I) (\u2191(extChartAt I x) (\u2191(extChartAt I x).symm y))\nhy' : \u2191(extChartAt I x).symm y \u2208 c.source\nhy'' : \u2191f (\u2191(extChartAt I x).symm y) \u2208 c'.source\n\u22a2 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').source \u2229 range \u2191I \u2286 \u2191(extChartAt I x).symm \u207b\u00b9' (f \u226b\u2095 c').source \u2229 range \u2191I", "state_after": "no goals"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 (contDiffPregroupoid \u22a4 I).property (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm) ((c.symm \u226b\u2095 f) \u226b\u2095 c').target", "state_after": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : y \u2208 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y"}, {"tactic": "simp only [mfld_simps] at hy", "annotated_tactic": ["simp only [mfld_simps] at hy", []], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy : y \u2208 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y"}, {"tactic": "have hy' : (extChartAt I (f x)).symm y \u2208 c'.source := by simp only [hy, mfld_simps]", "annotated_tactic": ["have hy' : (<a>extChartAt</a> I (f x)).<a>symm</a> y \u2208 c'.source := by simp only [hy, mfld_simps]", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}]], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y"}, {"tactic": "have hy'' : f.symm ((extChartAt I (f x)).symm y) \u2208 c.source := by simp only [hy, mfld_simps]", "annotated_tactic": ["have hy'' : f.symm ((<a>extChartAt</a> I (f x)).<a>symm</a> y) \u2208 c.source := by simp only [hy, mfld_simps]", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}]], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y"}, {"tactic": "rw [contMDiffWithinAt_iff_of_mem_source hy' hy''] at H", "annotated_tactic": ["rw [<a>contMDiffWithinAt_iff_of_mem_source</a> hy' hy''] at H", [{"full_name": "contMDiffWithinAt_iff_of_mem_source", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [412, 9], "def_end_pos": [412, 44]}]], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y", "state_after": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y"}, {"tactic": "convert H.2.mono _", "annotated_tactic": ["convert H.2.<a>mono</a> _", [{"full_name": "ContDiffWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [497, 9], "def_end_pos": [497, 30]}]], "state_before": "case mpr.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').symm \u2218 \u2191I.symm)\n    (\u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I) y", "state_after": "case h.e'_12\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 y = \u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y)\n\ncase mpr.intro.refine_2.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I \u2286 \u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I"}, {"tactic": "refine (h\u2082 ((extChartAt I (f x)).symm y) ?_).mono ?_", "annotated_tactic": ["refine (h\u2082 ((<a>extChartAt</a> I (f x)).<a>symm</a> y) ?_).<a>mono</a> ?_", [{"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1124, 5], "def_end_pos": [1124, 15]}, {"full_name": "PartialEquiv.symm", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [152, 15], "def_end_pos": [152, 19]}, {"full_name": "ContMDiffWithinAt.mono", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [722, 9], "def_end_pos": [722, 31]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)", "state_after": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 \u2191(extChartAt I (\u2191f x)).symm y \u2208 f.target\n\ncase refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 (f.symm \u226b\u2095 c).source \u2286 f.target"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 \u2191(extChartAt I (\u2191f x)).symm y \u2208 f.target", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "case refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\n\u22a2 (f.symm \u226b\u2095 c).source \u2286 f.target", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\n\u22a2 \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH : ContMDiffWithinAt I I \u22a4 (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y)\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\n\u22a2 \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source", "state_after": "no goals"}, {"tactic": "simp only [hy, mfld_simps]", "annotated_tactic": ["simp only [hy, mfld_simps]", []], "state_before": "case h.e'_12\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 y = \u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y)", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "annotated_tactic": ["mfld_set_tac", []], "state_before": "case mpr.intro.refine_2.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH\u271d : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\u271d\nI : ModelWithCorners \ud835\udd5c E H\u271d\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H\u271d M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\u271d\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H\u271d M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H\u271d := chartAt H\u271d x\nc' : PartialHomeomorph M' H\u271d := chartAt H\u271d (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\ny : E\nhy :\n  (\u2191I.symm y \u2208 c'.target \u2227 \u2191c'.symm (\u2191I.symm y) \u2208 f.target \u2227 \u2191f.symm (\u2191c'.symm (\u2191I.symm y)) \u2208 c.source) \u2227 y \u2208 range \u2191I\nH :\n  ContinuousWithinAt (\u2191f.symm) (f.symm \u226b\u2095 c).source (\u2191(extChartAt I (\u2191f x)).symm y) \u2227\n    ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(extChartAt I x) \u2218 \u2191f.symm \u2218 \u2191(extChartAt I (\u2191f x)).symm)\n      (\u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I)\n      (\u2191(extChartAt I (\u2191f x)) (\u2191(extChartAt I (\u2191f x)).symm y))\nhy' : \u2191(extChartAt I (\u2191f x)).symm y \u2208 c'.source\nhy'' : \u2191f.symm (\u2191(extChartAt I (\u2191f x)).symm y) \u2208 c.source\n\u22a2 \u2191I.symm \u207b\u00b9' ((c.symm \u226b\u2095 f) \u226b\u2095 c').target \u2229 range \u2191I \u2286 \u2191(extChartAt I (\u2191f x)).symm \u207b\u00b9' (f.symm \u226b\u2095 c).source \u2229 range \u2191I", "state_after": "no goals"}, {"tactic": "simp only [mfld_simps]", "annotated_tactic": ["simp only [mfld_simps]", []], "state_before": "case mpr.intro.refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191((c.symm \u226b\u2095 f) \u226b\u2095 c').toPartialEquiv)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 ((c.symm \u226b\u2095 f) \u226b\u2095 c').source)", "state_after": "case mpr.intro.refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 (c.target \u2229 \u2191c.symm \u207b\u00b9' f.source \u2229 \u2191f \u2218 \u2191c.symm \u207b\u00b9' c'.source))"}, {"tactic": "apply eqOn_refl", "annotated_tactic": ["apply <a>eqOn_refl</a>", [{"full_name": "Set.eqOn_refl", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [208, 9], "def_end_pos": [208, 18]}]], "state_before": "case mpr.intro.refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 EqOn (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm) (\u2191c' \u2218 \u2191f \u2218 \u2191c.symm)\n    (\u2191(chartAt H x).symm \u207b\u00b9' f.source \u2229 (c.target \u2229 \u2191c.symm \u207b\u00b9' f.source \u2229 \u2191f \u2218 \u2191c.symm \u207b\u00b9' c'.source))", "state_after": "no goals"}, {"tactic": "simp only [c, c', hx', mfld_simps]", "annotated_tactic": ["simp only [c, c', hx', mfld_simps]", []], "state_before": "case mpr.intro.refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2070 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : ChartedSpace H M\ninst\u271d\u2077 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2074 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ne : PartialHomeomorph M H\nx\u271d : M\nm n : \u2115\u221e\ninst\u271d : ChartedSpace H M'\nIsM' : SmoothManifoldWithCorners I M'\nf : PartialHomeomorph M M'\nh\u2081 : SmoothOn I I (\u2191f) f.source\nh\u2082 : SmoothOn I I (\u2191f.symm) f.target\nx : M\nhx : x \u2208 f.source\nc : PartialHomeomorph M H := chartAt H x\nc' : PartialHomeomorph M' H := chartAt H (\u2191f x)\nhx' : \u2191c x \u2208 \u2191c.symm \u207b\u00b9' f.source\n\u22a2 \u2191(chartAt H x) x \u2208 ((c.symm \u226b\u2095 f) \u226b\u2095 c').source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Irreducible.lean", "full_name": "mem_irreducibleComponent", "start": [141, 1], "end": [142, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.ceil_eq_zero", "start": [347, 1], "end": [347, 89], "traced_tactics": [{"tactic": "rw [\u2190 Nat.le_zero, ceil_le, Nat.cast_zero]", "annotated_tactic": ["rw [\u2190 <a>Nat.le_zero</a>, <a>ceil_le</a>, <a>Nat.cast_zero</a>]", [{"full_name": "Nat.le_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [592, 9], "def_end_pos": [592, 16]}, {"full_name": "Nat.ceil_le", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [286, 9], "def_end_pos": [286, 16]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2308a\u2309\u208a = 0 \u2194 a \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.map_coe", "start": [202, 1], "end": [203, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "pow_two", "start": [678, 1], "end": [679, 92], "traced_tactics": [{"tactic": "rw [pow_succ, pow_one]", "annotated_tactic": ["rw [<a>pow_succ</a>, <a>pow_one</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "G : Type u_1\nM : Type u_2\ninst\u271d : Monoid M\na\u271d b c : M\nm n : \u2115\na : M\n\u22a2 a ^ 2 = a * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor/Div.lean", "full_name": "smul_ceilDiv", "start": [151, 1], "end": [153, 67], "traced_tactics": [{"tactic": "simp [smul_le_smul_iff_of_pos_left, ha]", "annotated_tactic": ["simp [<a>smul_le_smul_iff_of_pos_left</a>, ha]", [{"full_name": "smul_le_smul_iff_of_pos_left", "def_path": "Mathlib/Algebra/Order/Module/Defs.lean", "def_pos": [292, 7], "def_end_pos": [292, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2075 : OrderedSemiring \u03b1\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b3 : MulActionWithZero \u03b1 \u03b2\ninst\u271d\u00b2 : CeilDiv \u03b1 \u03b2\na : \u03b1\ninst\u271d\u00b9 : PosSMulMono \u03b1 \u03b2\ninst\u271d : PosSMulReflectLE \u03b1 \u03b2\nha : 0 < a\nb : \u03b2\n\u22a2 \u2200 (c : \u03b2), a \u2022 b \u2308/\u2309 a \u2264 c \u2194 b \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Irreducible.lean", "full_name": "InfPrime.inf_le", "start": [185, 1], "end": [186, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "full_name": "Ideal.IsHomogeneous.sSup", "start": [308, 1], "end": [311, 16], "traced_tactics": [{"tactic": "rw [sSup_eq_iSup]", "annotated_tactic": ["rw [<a>sSup_eq_iSup</a>]", [{"full_name": "sSup_eq_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [874, 9], "def_end_pos": [874, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 I \u2208 \u2110, IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (SupSet.sSup \u2110)", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 I \u2208 \u2110, IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a06 a \u2208 \u2110, a)"}, {"tactic": "exact iSup\u2082 h", "annotated_tactic": ["exact <a>iSup\u2082</a> h", [{"full_name": "Ideal.IsHomogeneous.iSup\u2082", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [298, 9], "def_end_pos": [298, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 I \u2208 \u2110, IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a06 a \u2208 \u2110, a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "full_name": "LieAlgebra.IsKilling.eq_neg_or_eq_of_eq_smul", "start": [337, 1], "end": [345, 42], "traced_tactics": [{"tactic": "by_cases h\u03b1 : \u03b1.IsZero", "annotated_tactic": ["by_cases h\u03b1 : \u03b1.IsZero", []], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1 : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = k \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = k \u2022 \u21d1\u03b1\nh\u03b1 : \u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1\n\ncase neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = k \u2022 \u21d1\u03b1\nh\u03b1 : \u00ac\u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1"}, {"tactic": "rcases eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul \u03b1 \u03b2 h\u03b1 k h with (rfl | rfl | rfl)", "annotated_tactic": ["rcases <a>eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul</a> \u03b1 \u03b2 h\u03b1 k h with (rfl | rfl | rfl)", [{"full_name": "LieAlgebra.IsKilling.eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul", "def_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "def_pos": [301, 7], "def_end_pos": [301, 49]}]], "state_before": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = k \u2022 \u21d1\u03b1\nh\u03b1 : \u00ac\u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "case neg.inl\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1\n\ncase neg.inr.inl\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 0 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1\n\ncase neg.inr.inr\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1"}, {"tactic": "rw [h\u03b1, smul_zero] at h", "annotated_tactic": ["rw [h\u03b1, <a>smul_zero</a>] at h", [{"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}]], "state_before": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = k \u2022 \u21d1\u03b1\nh\u03b1 : \u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = 0\nh\u03b1 : \u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1"}, {"tactic": "cases h\u03b2 h", "annotated_tactic": ["cases h\u03b2 h", []], "state_before": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nk : K\nh : \u21d1\u03b2 = 0\nh\u03b1 : \u03b1.IsZero\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "no goals"}, {"tactic": "exact .inl (by ext; rw [h, neg_one_smul]; rfl)", "annotated_tactic": ["exact .inl (by ext; rw [h, <a>neg_one_smul</a>]; rfl)", [{"full_name": "neg_one_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [269, 9], "def_end_pos": [269, 21]}]], "state_before": "case neg.inl\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1", "state_after": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 \u03b2 x\u271d = (-\u03b1) x\u271d"}, {"tactic": "rw [h, neg_one_smul]", "annotated_tactic": ["rw [h, <a>neg_one_smul</a>]", [{"full_name": "neg_one_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [269, 9], "def_end_pos": [269, 21]}]], "state_before": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 \u03b2 x\u271d = (-\u03b1) x\u271d", "state_after": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 (-\u21d1\u03b1) x\u271d = (-\u03b1) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = -1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 (-\u21d1\u03b1) x\u271d = (-\u03b1) x\u271d", "state_after": "no goals"}, {"tactic": "cases h\u03b2 (by rwa [zero_smul] at h)", "annotated_tactic": ["cases h\u03b2 (by rwa [<a>zero_smul</a>] at h)", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case neg.inr.inl\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 0 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "no goals"}, {"tactic": "rwa [zero_smul] at h", "annotated_tactic": ["rwa [<a>zero_smul</a>] at h", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 0 \u2022 \u21d1\u03b1\n\u22a2 \u03b2.IsZero", "state_after": "no goals"}, {"tactic": "exact .inr (by ext; rw [h, one_smul])", "annotated_tactic": ["exact .inr (by ext; rw [h, <a>one_smul</a>])", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case neg.inr.inr\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = -\u03b1 \u2228 \u03b2 = \u03b1", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 1 \u2022 \u21d1\u03b1\n\u22a2 \u03b2 = \u03b1", "state_after": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 \u03b2 x\u271d = \u03b1 x\u271d"}, {"tactic": "rw [h, one_smul]", "annotated_tactic": ["rw [h, <a>one_smul</a>]", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case h\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b2 : \u03b2.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nh : \u21d1\u03b2 = 1 \u2022 \u21d1\u03b1\nx\u271d : \u21a5H\n\u22a2 \u03b2 x\u271d = \u03b1 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.measure_union_lt_top_iff", "start": [254, 1], "end": [257, 59], "traced_tactics": [{"tactic": "refine \u27e8fun h => \u27e8?_, ?_\u27e9, fun h => measure_union_lt_top h.1 h.2\u27e9", "annotated_tactic": ["refine \u27e8fun h => \u27e8?_, ?_\u27e9, fun h => <a>measure_union_lt_top</a> h.1 h.2\u27e9", [{"full_name": "MeasureTheory.measure_union_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [249, 9], "def_end_pos": [249, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 \u03bc (s \u222a t) < \u22a4 \u2194 \u03bc s < \u22a4 \u2227 \u03bc t < \u22a4", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u03bc (s \u222a t) < \u22a4\n\u22a2 \u03bc s < \u22a4\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u03bc (s \u222a t) < \u22a4\n\u22a2 \u03bc t < \u22a4"}, {"tactic": "exact (measure_mono Set.subset_union_left).trans_lt h", "annotated_tactic": ["exact (<a>measure_mono</a> <a>Set.subset_union_left</a>).<a>trans_lt</a> h", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 26]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u03bc (s \u222a t) < \u22a4\n\u22a2 \u03bc s < \u22a4", "state_after": "no goals"}, {"tactic": "exact (measure_mono Set.subset_union_right).trans_lt h", "annotated_tactic": ["exact (<a>measure_mono</a> <a>Set.subset_union_right</a>).<a>trans_lt</a> h", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.subset_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 27]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u03bc (s \u222a t) < \u22a4\n\u22a2 \u03bc t < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.le_maximum_of_mem", "start": [360, 1], "end": [361, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.floor_int", "start": [656, 1], "end": [657, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean", "full_name": "TangentBundle.trivializationAt_eq_localTriv", "start": [241, 1], "end": [244, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/IntermediateValue.lean", "full_name": "IsPreconnected.intermediate_value\u2082_eventually\u2082", "start": [115, 1], "end": [124, 23], "traced_tactics": [{"tactic": "rw [continuousOn_iff_continuous_restrict] at hf hg", "annotated_tactic": ["rw [<a>continuousOn_iff_continuous_restrict</a>] at hf hg", [{"full_name": "continuousOn_iff_continuous_restrict", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [656, 9], "def_end_pos": [656, 45]}]], "state_before": "X : Type u\n\u03b1 : Type v\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ns : Set X\nhs : IsPreconnected s\nl\u2081 l\u2082 : Filter X\ninst\u271d\u00b9 : l\u2081.NeBot\ninst\u271d : l\u2082.NeBot\nhl\u2081 : l\u2081 \u2264 \ud835\udcdf s\nhl\u2082 : l\u2082 \u2264 \ud835\udcdf s\nf g : X \u2192 \u03b1\nhf : ContinuousOn f s\nhg : ContinuousOn g s\nhe\u2081 : f \u2264\u1da0[l\u2081] g\nhe\u2082 : g \u2264\u1da0[l\u2082] f\n\u22a2 \u2203 x \u2208 s, f x = g x", "state_after": "X : Type u\n\u03b1 : Type v\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ns : Set X\nhs : IsPreconnected s\nl\u2081 l\u2082 : Filter X\ninst\u271d\u00b9 : l\u2081.NeBot\ninst\u271d : l\u2082.NeBot\nhl\u2081 : l\u2081 \u2264 \ud835\udcdf s\nhl\u2082 : l\u2082 \u2264 \ud835\udcdf s\nf g : X \u2192 \u03b1\nhf : Continuous (s.restrict f)\nhg : Continuous (s.restrict g)\nhe\u2081 : f \u2264\u1da0[l\u2081] g\nhe\u2082 : g \u2264\u1da0[l\u2082] f\n\u22a2 \u2203 x \u2208 s, f x = g x"}, {"tactic": "obtain \u27e8b, h\u27e9 :=\n  @intermediate_value_univ\u2082_eventually\u2082 _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _\n    (comap_coe_neBot_of_le_principal hl\u2081) (comap_coe_neBot_of_le_principal hl\u2082) _ _ hf hg\n    (he\u2081.comap _) (he\u2082.comap _)", "annotated_tactic": ["obtain \u27e8b, h\u27e9 :=\n    @<a>intermediate_value_univ\u2082_eventually\u2082</a> _ _ _ _ _ _ (<a>Subtype.preconnectedSpace</a> hs) _ _\n      (<a>comap_coe_neBot_of_le_principal</a> hl\u2081) (<a>comap_coe_neBot_of_le_principal</a> hl\u2082) _ _ hf hg\n      (he\u2081.comap _) (he\u2082.comap _)", [{"full_name": "intermediate_value_univ\u2082_eventually\u2082", "def_path": "Mathlib/Topology/Order/IntermediateValue.lean", "def_pos": [84, 9], "def_end_pos": [84, 45]}, {"full_name": "Subtype.preconnectedSpace", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [946, 9], "def_end_pos": [946, 34]}, {"full_name": "Filter.comap_coe_neBot_of_le_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2638, 9], "def_end_pos": [2638, 40]}, {"full_name": "Filter.comap_coe_neBot_of_le_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2638, 9], "def_end_pos": [2638, 40]}]], "state_before": "X : Type u\n\u03b1 : Type v\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ns : Set X\nhs : IsPreconnected s\nl\u2081 l\u2082 : Filter X\ninst\u271d\u00b9 : l\u2081.NeBot\ninst\u271d : l\u2082.NeBot\nhl\u2081 : l\u2081 \u2264 \ud835\udcdf s\nhl\u2082 : l\u2082 \u2264 \ud835\udcdf s\nf g : X \u2192 \u03b1\nhf : Continuous (s.restrict f)\nhg : Continuous (s.restrict g)\nhe\u2081 : f \u2264\u1da0[l\u2081] g\nhe\u2082 : g \u2264\u1da0[l\u2082] f\n\u22a2 \u2203 x \u2208 s, f x = g x", "state_after": "case intro\nX : Type u\n\u03b1 : Type v\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ns : Set X\nhs : IsPreconnected s\nl\u2081 l\u2082 : Filter X\ninst\u271d\u00b9 : l\u2081.NeBot\ninst\u271d : l\u2082.NeBot\nhl\u2081 : l\u2081 \u2264 \ud835\udcdf s\nhl\u2082 : l\u2082 \u2264 \ud835\udcdf s\nf g : X \u2192 \u03b1\nhf : Continuous (s.restrict f)\nhg : Continuous (s.restrict g)\nhe\u2081 : f \u2264\u1da0[l\u2081] g\nhe\u2082 : g \u2264\u1da0[l\u2082] f\nb : \u2191s\nh : s.restrict f b = s.restrict g b\n\u22a2 \u2203 x \u2208 s, f x = g x"}, {"tactic": "exact \u27e8b, b.prop, h\u27e9", "annotated_tactic": ["exact \u27e8b, b.prop, h\u27e9", []], "state_before": "case intro\nX : Type u\n\u03b1 : Type v\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ns : Set X\nhs : IsPreconnected s\nl\u2081 l\u2082 : Filter X\ninst\u271d\u00b9 : l\u2081.NeBot\ninst\u271d : l\u2082.NeBot\nhl\u2081 : l\u2081 \u2264 \ud835\udcdf s\nhl\u2082 : l\u2082 \u2264 \ud835\udcdf s\nf g : X \u2192 \u03b1\nhf : Continuous (s.restrict f)\nhg : Continuous (s.restrict g)\nhe\u2081 : f \u2264\u1da0[l\u2081] g\nhe\u2082 : g \u2264\u1da0[l\u2082] f\nb : \u2191s\nh : s.restrict f b = s.restrict g b\n\u22a2 \u2203 x \u2208 s, f x = g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.gcd_add_self_left", "start": [80, 1], "end": [81, 46], "traced_tactics": [{"tactic": "rw [gcd_comm, gcd_add_self_right, gcd_comm]", "annotated_tactic": ["rw [<a>gcd_comm</a>, <a>gcd_add_self_right</a>, <a>gcd_comm</a>]", [{"full_name": "Nat.gcd_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}, {"full_name": "Nat.gcd_add_self_right", "def_path": "Mathlib/Data/Nat/GCD/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 27]}, {"full_name": "Nat.gcd_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}]], "state_before": "m n : \u2115\n\u22a2 (m + n).gcd n = m.gcd n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Circular.lean", "full_name": "sbtw_cyclic_right", "start": [221, 1], "end": [222, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/CC/Lemmas.lean", "full_name": "Mathlib.Tactic.CC.not_eq_of_eq_true", "start": [73, 1], "end": [74, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Zip.lean", "full_name": "List.mem_zip_inits_tails", "start": [349, 1], "end": [363, 17], "traced_tactics": [{"tactic": "induction' l with hd tl ih generalizing init tail <;> simp_rw [tails, inits, zip_cons_cons]", "annotated_tactic": ["induction' l with hd tl ih generalizing init tail <;> simp_rw [<a>tails</a>, <a>inits</a>, <a>zip_cons_cons</a>]", [{"full_name": "List.tails", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [497, 13], "def_end_pos": [497, 18]}, {"full_name": "List.inits", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [480, 13], "def_end_pos": [480, 18]}, {"full_name": "List.zip_cons_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1111, 17], "def_end_pos": [1111, 30]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl init tail : List \u03b1\n\u22a2 (init, tail) \u2208 l.inits.zip l.tails \u2194 init ++ tail = l", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], []) :: [].zip [] \u2194 init ++ tail = []\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], hd :: tl) :: (map (fun t => hd :: t) tl.inits).zip tl.tails \u2194 init ++ tail = hd :: tl"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], []) :: [].zip [] \u2194 init ++ tail = []", "state_after": "no goals"}, {"tactic": "constructor <;> rw [mem_cons, zip_map_left, mem_map, Prod.exists]", "annotated_tactic": ["constructor <;> rw [<a>mem_cons</a>, <a>zip_map_left</a>, <a>mem_map</a>, <a>Prod.exists</a>]", [{"full_name": "List.mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 25]}, {"full_name": "List.zip_map_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [2162, 9], "def_end_pos": [2162, 21]}, {"full_name": "List.mem_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [750, 17], "def_end_pos": [750, 24]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], hd :: tl) :: (map (fun t => hd :: t) tl.inits).zip tl.tails \u2194 init ++ tail = hd :: tl", "state_after": "case cons.mp\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 ((init, tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)) \u2192\n    init ++ tail = hd :: tl\n\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 init ++ tail = hd :: tl \u2192\n    (init, tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)"}, {"tactic": "rintro (\u27e8rfl, rfl\u27e9 | \u27e8_, _, h, rfl, rfl\u27e9)", "annotated_tactic": ["rintro (\u27e8rfl, rfl\u27e9 | \u27e8_, _, h, rfl, rfl\u27e9)", []], "state_before": "case cons.mp\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 ((init, tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)) \u2192\n    init ++ tail = hd :: tl", "state_after": "case cons.mp.inl.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\n\u22a2 [] ++ hd :: tl = hd :: tl\n\ncase cons.mp.inr.intro.intro.intro.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\nw\u271d\u00b9 w\u271d : List \u03b1\nh : (w\u271d\u00b9, w\u271d) \u2208 tl.inits.zip tl.tails\n\u22a2 (fun t => hd :: t) w\u271d\u00b9 ++ id w\u271d = hd :: tl"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons.mp.inl.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\n\u22a2 [] ++ hd :: tl = hd :: tl", "state_after": "no goals"}, {"tactic": "simp [ih.mp h]", "annotated_tactic": ["simp [ih.mp h]", []], "state_before": "case cons.mp.inr.intro.intro.intro.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\nw\u271d\u00b9 w\u271d : List \u03b1\nh : (w\u271d\u00b9, w\u271d) \u2208 tl.inits.zip tl.tails\n\u22a2 (fun t => hd :: t) w\u271d\u00b9 ++ id w\u271d = hd :: tl", "state_after": "no goals"}, {"tactic": "cases' init with hd' tl'", "annotated_tactic": ["cases' init with hd' tl'", []], "state_before": "case cons.mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 init ++ tail = hd :: tl \u2192\n    (init, tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)", "state_after": "case cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\n\u22a2 [] ++ tail = hd :: tl \u2192\n    ([], tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], tail)\n\ncase cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\n\u22a2 hd' :: tl' ++ tail = hd :: tl \u2192\n    (hd' :: tl', tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\n\u22a2 [] ++ tail = hd :: tl \u2192\n    ([], tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], tail)", "state_after": "case cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\n\u22a2 ([], hd :: tl) = ([], hd :: tl) \u2228\n    \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], hd :: tl)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\n\u22a2 ([], hd :: tl) = ([], hd :: tl) \u2228\n    \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], hd :: tl)", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\n\u22a2 hd' :: tl' ++ tail = hd :: tl \u2192\n    (hd' :: tl', tail) = ([], hd :: tl) \u2228\n      \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)", "state_after": "case cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (hd' :: tl', tail) = ([], hd :: tl) \u2228\n    \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (hd' :: tl', tail) = ([], hd :: tl) \u2228\n    \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)", "state_after": "case cons.mpr.cons.h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)"}, {"tactic": "use tl', tail", "annotated_tactic": ["use tl', tail", []], "state_before": "case cons.mpr.cons.h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 \u2203 a b, (a, b) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (tl', tail) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (tl', tail) = (hd' :: tl', tail)"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 tl.inits.zip tl.tails \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (tl', tail) \u2208 tl.inits.zip tl.tails \u2227 Prod.map (fun t => hd :: t) id (tl', tail) = (hd' :: tl', tail)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.supports_singleton", "start": [1916, 1], "end": [1916, 100], "traced_tactics": [{"tactic": "simp [Supports]", "annotated_tactic": ["simp [<a>Supports</a>]", [{"full_name": "Turing.PartrecToTM2.Supports", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1908, 5], "def_end_pos": [1908, 13]}]], "state_before": "S : Finset \u039b'\nq : \u039b'\n\u22a2 Supports {q} S \u2194 TM2.SupportsStmt S (tr q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Transfer.lean", "full_name": "MonoidHom.ker_transferSylow_disjoint", "start": [268, 1], "end": [273, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Instances.lean", "full_name": "MonoidHom.pow_apply", "start": [102, 1], "end": [105, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Liouville/LiouvilleWith.lean", "full_name": "LiouvilleWith.int_sub_iff", "start": [295, 1], "end": [295, 94], "traced_tactics": [{"tactic": "simp [sub_eq_add_neg]", "annotated_tactic": ["simp [<a>sub_eq_add_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}]], "state_before": "p q x y : \u211d\nr : \u211a\nm : \u2124\nn : \u2115\n\u22a2 LiouvilleWith p (\u2191m - x) \u2194 LiouvilleWith p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Norm.lean", "full_name": "Ideal.irreducible_of_irreducible_absNorm", "start": [414, 1], "end": [422, 58], "traced_tactics": [{"tactic": "simpa only [Ideal.isUnit_iff, Nat.isUnit_iff, absNorm_eq_one_iff] using h", "annotated_tactic": ["simpa only [<a>Ideal.isUnit_iff</a>, <a>Nat.isUnit_iff</a>, <a>absNorm_eq_one_iff</a>] using h", [{"full_name": "Ideal.isUnit_iff", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 19]}, {"full_name": "Nat.isUnit_iff", "def_path": "Mathlib/Algebra/Group/Nat.lean", "def_pos": [181, 25], "def_end_pos": [181, 35]}, {"full_name": "Ideal.absNorm_eq_one_iff", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [277, 9], "def_end_pos": [277, 27]}]], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Nontrivial S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : Irreducible (absNorm I)\nh : IsUnit I\n\u22a2 IsUnit (absNorm I)", "state_after": "no goals"}, {"tactic": "rintro a b rfl", "annotated_tactic": ["rintro a b rfl", []], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Nontrivial S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : Irreducible (absNorm I)\n\u22a2 \u2200 (a b : Ideal S), I = a * b \u2192 IsUnit a \u2228 IsUnit b", "state_after": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Nontrivial S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\na b : Ideal S\nhI : Irreducible (absNorm (a * b))\n\u22a2 IsUnit a \u2228 IsUnit b"}, {"tactic": "simpa only [Ideal.isUnit_iff, Nat.isUnit_iff, absNorm_eq_one_iff] using\n  hI.isUnit_or_isUnit (_root_.map_mul absNorm a b)", "annotated_tactic": ["simpa only [<a>Ideal.isUnit_iff</a>, <a>Nat.isUnit_iff</a>, <a>absNorm_eq_one_iff</a>] using\n        hI.isUnit_or_isUnit (<a>_root_.map_mul</a> <a>absNorm</a> a b)", [{"full_name": "Ideal.isUnit_iff", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1318, 9], "def_end_pos": [1318, 19]}, {"full_name": "Nat.isUnit_iff", "def_path": "Mathlib/Algebra/Group/Nat.lean", "def_pos": [181, 25], "def_end_pos": [181, 35]}, {"full_name": "Ideal.absNorm_eq_one_iff", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [277, 9], "def_end_pos": [277, 27]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}]], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Nontrivial S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\na b : Ideal S\nhI : Irreducible (absNorm (a * b))\n\u22a2 IsUnit a \u2228 IsUnit b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "vectorSpan_insert_eq_vectorSpan", "start": [1415, 1], "end": [1417, 72], "traced_tactics": [{"tactic": "simp_rw [\u2190 direction_affineSpan, affineSpan_insert_eq_affineSpan _ h]", "annotated_tactic": ["simp_rw [\u2190 <a>direction_affineSpan</a>, <a>affineSpan_insert_eq_affineSpan</a> _ h]", [{"full_name": "direction_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [559, 9], "def_end_pos": [559, 29]}, {"full_name": "affineSpan_insert_eq_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 40]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : P\nps : Set P\nh : p \u2208 affineSpan k ps\n\u22a2 vectorSpan k (insert p ps) = vectorSpan k ps", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "full_name": "LinearMap.injective_domRestrict_iff", "start": [214, 1], "end": [225, 21], "traced_tactics": [{"tactic": "rw [\u2190 LinearMap.ker_eq_bot]", "annotated_tactic": ["rw [\u2190 <a>LinearMap.ker_eq_bot</a>]", [{"full_name": "LinearMap.ker_eq_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [210, 9], "def_end_pos": [210, 19]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\n\u22a2 Injective \u21d1(f.domRestrict S) \u2194 S \u2293 ker f = \u22a5", "state_after": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\n\u22a2 ker (f.domRestrict S) = \u22a5 \u2194 S \u2293 ker f = \u22a5"}, {"tactic": "refine \u27e8fun h \u21a6 le_bot_iff.1 ?_, fun h \u21a6 le_bot_iff.1 ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 <a>le_bot_iff</a>.1 ?_, fun h \u21a6 <a>le_bot_iff</a>.1 ?_\u27e9", [{"full_name": "le_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [323, 9], "def_end_pos": [323, 19]}, {"full_name": "le_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [323, 9], "def_end_pos": [323, 19]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\n\u22a2 ker (f.domRestrict S) = \u22a5 \u2194 S \u2293 ker f = \u22a5", "state_after": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\n\u22a2 S \u2293 ker f \u2264 \u22a5\n\ncase refine_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\n\u22a2 ker (f.domRestrict S) \u2264 \u22a5"}, {"tactic": "intro x \u27e8hx, h'x\u27e9", "annotated_tactic": ["intro x \u27e8hx, h'x\u27e9", []], "state_before": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\n\u22a2 S \u2293 ker f \u2264 \u22a5", "state_after": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\n\u22a2 x \u2208 \u22a5"}, {"tactic": "have : \u27e8x, hx\u27e9 \u2208 LinearMap.ker (LinearMap.domRestrict f S) := by simpa using h'x", "annotated_tactic": ["have : \u27e8x, hx\u27e9 \u2208 <a>LinearMap.ker</a> (<a>LinearMap.domRestrict</a> f S) := by simpa using h'x", [{"full_name": "LinearMap.ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [60, 5], "def_end_pos": [60, 8]}, {"full_name": "LinearMap.domRestrict", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [135, 5], "def_end_pos": [135, 16]}]], "state_before": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\n\u22a2 x \u2208 \u22a5", "state_after": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\nthis : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\n\u22a2 x \u2208 \u22a5"}, {"tactic": "rw [h] at this", "annotated_tactic": ["rw [h] at this", []], "state_before": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\nthis : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\n\u22a2 x \u2208 \u22a5", "state_after": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\nthis : \u27e8x, hx\u27e9 \u2208 \u22a5\n\u22a2 x \u2208 \u22a5"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "case refine_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\nthis : \u27e8x, hx\u27e9 \u2208 \u22a5\n\u22a2 x \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "simpa using h'x", "annotated_tactic": ["simpa using h'x", []], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : ker (f.domRestrict S) = \u22a5\nx : M\nhx : x \u2208 \u2191S\nh'x : x \u2208 \u2191(ker f)\n\u22a2 \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, hx\u27e9 h'x", "annotated_tactic": ["rintro \u27e8x, hx\u27e9 h'x", []], "state_before": "case refine_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\n\u22a2 ker (f.domRestrict S) \u2264 \u22a5", "state_after": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5"}, {"tactic": "have : x \u2208 S \u2293 LinearMap.ker f := \u27e8hx, h'x\u27e9", "annotated_tactic": ["have : x \u2208 S \u2293 <a>LinearMap.ker</a> f := \u27e8hx, h'x\u27e9", [{"full_name": "LinearMap.ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [60, 5], "def_end_pos": [60, 8]}]], "state_before": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5", "state_after": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\nthis : x \u2208 S \u2293 ker f\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5"}, {"tactic": "rw [h] at this", "annotated_tactic": ["rw [h] at this", []], "state_before": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\nthis : x \u2208 S \u2293 ker f\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5", "state_after": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\nthis : x \u2208 \u22a5\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "case refine_2.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nK : Type u_5\nM : Type u_6\nM\u2081 : Type u_7\nM\u2082 : Type u_8\nM\u2083 : Type u_9\nV : Type u_10\nV\u2082 : Type u_11\ninst\u271d\u00b9\u00b9 : Ring R\ninst\u271d\u00b9\u2070 : Ring R\u2082\ninst\u271d\u2079 : Ring R\u2083\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_12\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf\u271d : F\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nS : Submodule R M\nh : S \u2293 ker f = \u22a5\nx : M\nhx : x \u2208 S\nh'x : \u27e8x, hx\u27e9 \u2208 ker (f.domRestrict S)\nthis : x \u2208 \u22a5\n\u22a2 \u27e8x, hx\u27e9 \u2208 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/StructurePolynomial.lean", "full_name": "C_p_pow_dvd_bind\u2081_rename_wittPolynomial_sub_sum", "start": [230, 1], "end": [268, 30], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH : \u2200 m < n, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\n\u22a2 C \u2191(p ^ n) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 n)) \u03a6 -\n      \u2211 i \u2208 Finset.range n, C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n - i)", "state_after": "case zero\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nIH : \u2200 m < 0, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\n\u22a2 C \u2191(p ^ 0) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 0)) \u03a6 -\n      \u2211 i \u2208 Finset.range 0, C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (0 - i)\n\ncase succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)"}, {"tactic": "have key := bind\u2081_rename_expand_wittPolynomial \u03a6 n IH", "annotated_tactic": ["have key := <a>bind\u2081_rename_expand_wittPolynomial</a> \u03a6 n IH", [{"full_name": "bind\u2081_rename_expand_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/StructurePolynomial.lean", "def_pos": [210, 9], "def_end_pos": [210, 43]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun b => (rename fun i => (b, i)) ((expand p) (W_ \u2124 n))) \u03a6 =\n    (bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)"}, {"tactic": "apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key", "annotated_tactic": ["apply_fun <a>map</a> (<a>Int.castRingHom</a> (<a>ZMod</a> (p ^ (n + 1)))) at key", [{"full_name": "MvPolynomial.map", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1288, 5], "def_end_pos": [1288, 8]}, {"full_name": "Int.castRingHom", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [98, 5], "def_end_pos": [98, 16]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun b => (rename fun i => (b, i)) ((expand p) (W_ \u2124 n))) \u03a6 =\n    (bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun b => (rename fun i => (b, i)) ((expand p) (W_ \u2124 n))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)"}, {"tactic": "conv_lhs at key => simp only [map_bind\u2081, map_rename, map_expand, map_wittPolynomial]", "annotated_tactic": ["conv_lhs at key => simp only [<a>map_bind\u2081</a>, <a>map_rename</a>, <a>map_expand</a>, <a>map_wittPolynomial</a>]", [{"full_name": "MvPolynomial.map_bind\u2081", "def_path": "Mathlib/Algebra/MvPolynomial/Monad.lean", "def_pos": [285, 9], "def_end_pos": [285, 18]}, {"full_name": "MvPolynomial.map_rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 19]}, {"full_name": "MvPolynomial.map_expand", "def_path": "Mathlib/Algebra/MvPolynomial/Expand.lean", "def_pos": [77, 9], "def_end_pos": [77, 19]}, {"full_name": "map_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [116, 9], "def_end_pos": [116, 27]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun b => (rename fun i => (b, i)) ((expand p) (W_ \u2124 n))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)"}, {"tactic": "rw [C_dvd_iff_zmod, RingHom.map_sub, sub_eq_zero, map_bind\u2081]", "annotated_tactic": ["rw [<a>C_dvd_iff_zmod</a>, <a>RingHom.map_sub</a>, <a>sub_eq_zero</a>, <a>map_bind\u2081</a>]", [{"full_name": "MvPolynomial.C_dvd_iff_zmod", "def_path": "Mathlib/FieldTheory/Finite/Polynomial.lean", "def_pos": [23, 9], "def_end_pos": [23, 23]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [612, 19], "def_end_pos": [612, 26]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}, {"full_name": "MvPolynomial.map_bind\u2081", "def_path": "Mathlib/Algebra/MvPolynomial/Monad.lean", "def_pos": [285, 9], "def_end_pos": [285, 18]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 C \u2191(p ^ (n + 1)) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 (n + 1))) \u03a6 -\n      \u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i)", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (bind\u2081 fun i => (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((rename fun i_1 => (i, i_1)) (W_ \u2124 (n + 1))))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))"}, {"tactic": "simp only [map_rename, map_wittPolynomial, wittPolynomial_zmod_self]", "annotated_tactic": ["simp only [<a>map_rename</a>, <a>map_wittPolynomial</a>, <a>wittPolynomial_zmod_self</a>]", [{"full_name": "MvPolynomial.map_rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 19]}, {"full_name": "map_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [116, 9], "def_end_pos": [116, 27]}, {"full_name": "wittPolynomial_zmod_self", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [154, 9], "def_end_pos": [154, 33]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (bind\u2081 fun i => (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((rename fun i_1 => (i, i_1)) (W_ \u2124 (n + 1))))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))"}, {"tactic": "rw [key]", "annotated_tactic": ["rw [key]", []], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))"}, {"tactic": "clear key IH", "annotated_tactic": ["clear key IH", []], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\nIH :\n  \u2200 m < n + 1, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\nkey :\n  (bind\u2081 fun i => (rename fun i_1 => (i, i_1)) ((expand p) (W_ (ZMod (p ^ (n + 1))) n)))\n      ((map (Int.castRingHom (ZMod (p ^ (n + 1))))) \u03a6) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n))\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))"}, {"tactic": "rw [bind\u2081, aeval_wittPolynomial, map_sum, map_sum, Finset.sum_congr rfl]", "annotated_tactic": ["rw [<a>bind\u2081</a>, <a>aeval_wittPolynomial</a>, <a>map_sum</a>, <a>map_sum</a>, <a>Finset.sum_congr</a> <a>rfl</a>]", [{"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Algebra/MvPolynomial/Monad.lean", "def_pos": [66, 5], "def_end_pos": [66, 10]}, {"full_name": "aeval_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [146, 9], "def_end_pos": [146, 29]}, {"full_name": "map_sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [295, 3], "def_end_pos": [295, 14]}, {"full_name": "map_sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [295, 3], "def_end_pos": [295, 14]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) ((bind\u2081 fun i => (expand p) (wittStructureInt p \u03a6 i)) (W_ \u2124 n)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1)))))\n      (\u2211 i \u2208 Finset.range (n + 1), C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (n + 1 - i))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\n\u22a2 \u2200 x \u2208 Finset.range (n + 1),\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ x * (expand p) (wittStructureInt p \u03a6 x) ^ p ^ (n - x)) =\n      (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ x) * wittStructureInt p \u03a6 x ^ p ^ (n + 1 - x))"}, {"tactic": "intro k hk", "annotated_tactic": ["intro k hk", []], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn : \u2115\n\u22a2 \u2200 x \u2208 Finset.range (n + 1),\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ x * (expand p) (wittStructureInt p \u03a6 x) ^ p ^ (n - x)) =\n      (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ x) * wittStructureInt p \u03a6 x ^ p ^ (n + 1 - x))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2208 Finset.range (n + 1)\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ k * (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ k) * wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "rw [Finset.mem_range, Nat.lt_succ_iff] at hk", "annotated_tactic": ["rw [<a>Finset.mem_range</a>, <a>Nat.lt_succ_iff</a>] at hk", [{"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 18]}, {"full_name": "Nat.lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [573, 19], "def_end_pos": [573, 30]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2208 Finset.range (n + 1)\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ k * (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ k) * wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ k * (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ k) * wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "rw [\u2190 sub_eq_zero, \u2190 RingHom.map_sub, \u2190 C_dvd_iff_zmod, C_eq_coe_nat, \u2190 Nat.cast_pow,\n  \u2190 Nat.cast_pow, C_eq_coe_nat, \u2190 mul_sub]", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>, \u2190 <a>RingHom.map_sub</a>, \u2190 <a>C_dvd_iff_zmod</a>, <a>C_eq_coe_nat</a>, \u2190 <a>Nat.cast_pow</a>,\n    \u2190 <a>Nat.cast_pow</a>, <a>C_eq_coe_nat</a>, \u2190 <a>mul_sub</a>]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [612, 19], "def_end_pos": [612, 26]}, {"full_name": "MvPolynomial.C_dvd_iff_zmod", "def_path": "Mathlib/FieldTheory/Finite/Polynomial.lean", "def_pos": [23, 9], "def_end_pos": [23, 23]}, {"full_name": "MvPolynomial.C_eq_coe_nat", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [272, 9], "def_end_pos": [272, 21]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "MvPolynomial.C_eq_coe_nat", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [272, 9], "def_end_pos": [272, 21]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (\u2191p ^ k * (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k)) =\n    (map (Int.castRingHom (ZMod (p ^ (n + 1))))) (C (\u2191p ^ k) * wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 \u2191(p ^ (n + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "have : p ^ (n + 1) = p ^ k * p ^ (n - k + 1) := by\n  rw [\u2190 pow_add, \u2190 add_assoc]; congr 2; rw [add_comm, \u2190 tsub_eq_iff_eq_add_of_le hk]", "annotated_tactic": ["have : p ^ (n + 1) = p ^ k * p ^ (n - k + 1) := by\n    rw [\u2190 <a>pow_add</a>, \u2190 <a>add_assoc</a>]; congr 2; rw [<a>add_comm</a>, \u2190 <a>tsub_eq_iff_eq_add_of_le</a> hk]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "tsub_eq_iff_eq_add_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [210, 9], "def_end_pos": [210, 33]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 \u2191(p ^ (n + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191(p ^ (n + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191(p ^ (n + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191(p ^ k * p ^ (n - k + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "rw [Nat.cast_mul, Nat.cast_pow, Nat.cast_pow]", "annotated_tactic": ["rw [<a>Nat.cast_mul</a>, <a>Nat.cast_pow</a>, <a>Nat.cast_pow</a>]", [{"full_name": "Nat.cast_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [62, 26], "def_end_pos": [62, 34]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191(p ^ k * p ^ (n - k + 1)) \u2223\n    \u2191(p ^ k) * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ k * \u2191p ^ (n - k + 1) \u2223\n    \u2191p ^ k * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))"}, {"tactic": "apply mul_dvd_mul_left ((p : MvPolynomial (idx \u00d7 \u2115) \u2124) ^ k)", "annotated_tactic": ["apply <a>mul_dvd_mul_left</a> ((p : <a>MvPolynomial</a> (idx \u00d7 \u2115) \u2124) ^ k)", [{"full_name": "mul_dvd_mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 25]}, {"full_name": "MvPolynomial", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [84, 5], "def_end_pos": [84, 17]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ k * \u2191p ^ (n - k + 1) \u2223\n    \u2191p ^ k * ((expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k)"}, {"tactic": "rw [show p ^ (n + 1 - k) = p * p ^ (n - k) by rw [\u2190 pow_succ', \u2190 tsub_add_eq_add_tsub hk]]", "annotated_tactic": ["rw [show p ^ (n + 1 - k) = p * p ^ (n - k) by rw [\u2190 <a>pow_succ'</a>, \u2190 <a>tsub_add_eq_add_tsub</a> hk]]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "tsub_add_eq_add_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [219, 9], "def_end_pos": [219, 29]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ p ^ (n + 1 - k)", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ (p * p ^ (n - k))"}, {"tactic": "rw [pow_mul]", "annotated_tactic": ["rw [<a>pow_mul</a>]", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - wittStructureInt p \u03a6 k ^ (p * p ^ (n - k))", "state_after": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - (wittStructureInt p \u03a6 k ^ p) ^ p ^ (n - k)"}, {"tactic": "apply dvd_sub_pow_of_dvd_sub", "annotated_tactic": ["apply <a>dvd_sub_pow_of_dvd_sub</a>", [{"full_name": "dvd_sub_pow_of_dvd_sub", "def_path": "Mathlib/NumberTheory/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 31]}]], "state_before": "case succ\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p ^ (n - k + 1) \u2223 (expand p) (wittStructureInt p \u03a6 k) ^ p ^ (n - k) - (wittStructureInt p \u03a6 k ^ p) ^ p ^ (n - k)", "state_after": "case succ.h\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p \u2223 (expand p) (wittStructureInt p \u03a6 k) - wittStructureInt p \u03a6 k ^ p"}, {"tactic": "rw [\u2190 C_eq_coe_nat, C_dvd_iff_zmod, RingHom.map_sub, sub_eq_zero, map_expand, RingHom.map_pow,\n  MvPolynomial.expand_zmod]", "annotated_tactic": ["rw [\u2190 <a>C_eq_coe_nat</a>, <a>C_dvd_iff_zmod</a>, <a>RingHom.map_sub</a>, <a>sub_eq_zero</a>, <a>map_expand</a>, <a>RingHom.map_pow</a>,\n    <a>MvPolynomial.expand_zmod</a>]", [{"full_name": "MvPolynomial.C_eq_coe_nat", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [272, 9], "def_end_pos": [272, 21]}, {"full_name": "MvPolynomial.C_dvd_iff_zmod", "def_path": "Mathlib/FieldTheory/Finite/Polynomial.lean", "def_pos": [23, 9], "def_end_pos": [23, 23]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [612, 19], "def_end_pos": [612, 26]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}, {"full_name": "MvPolynomial.map_expand", "def_path": "Mathlib/Algebra/MvPolynomial/Expand.lean", "def_pos": [77, 9], "def_end_pos": [77, 19]}, {"full_name": "RingHom.map_pow", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [727, 17], "def_end_pos": [727, 32]}, {"full_name": "MvPolynomial.expand_zmod", "def_path": "Mathlib/FieldTheory/Finite/Polynomial.lean", "def_pos": [41, 9], "def_end_pos": [41, 20]}]], "state_before": "case succ.h\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 \u2191p \u2223 (expand p) (wittStructureInt p \u03a6 k) - wittStructureInt p \u03a6 k ^ p", "state_after": "no goals"}, {"tactic": "simp only [isUnit_one, Int.ofNat_zero, Int.ofNat_succ, zero_add, pow_zero, C_1, IsUnit.dvd,\n  Nat.cast_one, Nat.zero_eq]", "annotated_tactic": ["simp only [<a>isUnit_one</a>, <a>Int.ofNat_zero</a>, <a>Int.ofNat_succ</a>, <a>zero_add</a>, <a>pow_zero</a>, <a>C_1</a>, <a>IsUnit.dvd</a>,\n      <a>Nat.cast_one</a>, <a>Nat.zero_eq</a>]", [{"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "Int.ofNat_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [73, 17], "def_end_pos": [73, 27]}, {"full_name": "Int.ofNat_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [27, 9], "def_end_pos": [27, 19]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "MvPolynomial.C_1", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 12]}, {"full_name": "IsUnit.dvd", "def_path": "Mathlib/Algebra/Divisibility/Units.lean", "def_pos": [81, 9], "def_end_pos": [81, 12]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}]], "state_before": "case zero\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nIH : \u2200 m < 0, (map (Int.castRingHom \u211a)) (wittStructureInt p \u03a6 m) = wittStructureRat p ((map (Int.castRingHom \u211a)) \u03a6) m\n\u22a2 C \u2191(p ^ 0) \u2223\n    (bind\u2081 fun b => (rename fun i => (b, i)) (W_ \u2124 0)) \u03a6 -\n      \u2211 i \u2208 Finset.range 0, C (\u2191p ^ i) * wittStructureInt p \u03a6 i ^ p ^ (0 - i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 pow_add, \u2190 add_assoc]", "annotated_tactic": ["rw [\u2190 <a>pow_add</a>, \u2190 <a>add_assoc</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}]], "state_before": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 p ^ (n + 1) = p ^ k * p ^ (n - k + 1)", "state_after": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 p ^ (n + 1) = p ^ (k + (n - k) + 1)"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 p ^ (n + 1) = p ^ (k + (n - k) + 1)", "state_after": "case e_a.e_a\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 n = k + (n - k)"}, {"tactic": "rw [add_comm, \u2190 tsub_eq_iff_eq_add_of_le hk]", "annotated_tactic": ["rw [<a>add_comm</a>, \u2190 <a>tsub_eq_iff_eq_add_of_le</a> hk]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "tsub_eq_iff_eq_add_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [210, 9], "def_end_pos": [210, 33]}]], "state_before": "case e_a.e_a\np : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\n\u22a2 n = k + (n - k)", "state_after": "no goals"}, {"tactic": "rw [\u2190 pow_succ', \u2190 tsub_add_eq_add_tsub hk]", "annotated_tactic": ["rw [\u2190 <a>pow_succ'</a>, \u2190 <a>tsub_add_eq_add_tsub</a> hk]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "tsub_add_eq_add_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [219, 9], "def_end_pos": [219, 29]}]], "state_before": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u2124\nn k : \u2115\nhk : k \u2264 n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n\u22a2 p ^ (n + 1 - k) = p * p ^ (n - k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/Impartial.lean", "full_name": "SetTheory.PGame.Impartial.nonneg", "start": [130, 1], "end": [133, 27], "traced_tactics": [{"tactic": "have h' := neg_lt_neg_iff.2 h", "annotated_tactic": ["have h' := <a>neg_lt_neg_iff</a>.2 h", [{"full_name": "SetTheory.PGame.neg_lt_neg_iff", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 23]}]], "state_before": "G : PGame\ninst\u271d : G.Impartial\nh : G < 0\n\u22a2 False", "state_after": "G : PGame\ninst\u271d : G.Impartial\nh : G < 0\nh' : -0 < -G\n\u22a2 False"}, {"tactic": "rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h'", "annotated_tactic": ["rw [<a>neg_zero</a>, <a>lt_congr_right</a> (<a>Equiv.symm</a> (<a>neg_equiv_self</a> G))] at h'", [{"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}, {"full_name": "SetTheory.PGame.lt_congr_right", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [937, 9], "def_end_pos": [937, 23]}, {"full_name": "SetTheory.PGame.Equiv.symm", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [810, 19], "def_end_pos": [810, 29]}, {"full_name": "SetTheory.PGame.Impartial.neg_equiv_self", "def_path": "Mathlib/SetTheory/Game/Impartial.lean", "def_pos": [64, 9], "def_end_pos": [64, 23]}]], "state_before": "G : PGame\ninst\u271d : G.Impartial\nh : G < 0\nh' : -0 < -G\n\u22a2 False", "state_after": "G : PGame\ninst\u271d : G.Impartial\nh : G < 0\nh' : 0 < G\n\u22a2 False"}, {"tactic": "exact (h.trans h').false", "annotated_tactic": ["exact (h.trans h').<a>false</a>", [{"full_name": "LT.lt.false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [302, 19], "def_end_pos": [302, 24]}]], "state_before": "G : PGame\ninst\u271d : G.Impartial\nh : G < 0\nh' : 0 < G\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "isGreatest_union_iff", "start": [397, 1], "end": [400, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "full_name": "NNReal.ball_zero_eq_Ico'", "start": [123, 1], "end": [124, 67], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\u22650\n\u22a2 ball 0 \u2191c = Ico 0 c", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc x : \u211d\u22650\n\u22a2 x \u2208 ball 0 \u2191c \u2194 x \u2208 Ico 0 c"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc x : \u211d\u22650\n\u22a2 x \u2208 ball 0 \u2191c \u2194 x \u2208 Ico 0 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "ContDiffOn.comp", "start": [632, 1], "end": [660, 78], "traced_tactics": [{"tactic": "let Eu : Type max uE uF uG := ULift.{max uF uG} E", "annotated_tactic": ["let Eu : Type max uE uF uG := <a>ULift</a>.{max uF uG} E", [{"full_name": "ULift", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [809, 11], "def_end_pos": [809, 16]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "let Fu : Type max uE uF uG := ULift.{max uE uG} F", "annotated_tactic": ["let Fu : Type max uE uF uG := <a>ULift</a>.{max uE uG} F", [{"full_name": "ULift", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [809, 11], "def_end_pos": [809, 16]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "let Gu : Type max uE uF uG := ULift.{max uE uF} G", "annotated_tactic": ["let Gu : Type max uE uF uG := <a>ULift</a>.{max uE uF} G", [{"full_name": "ULift", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [809, 11], "def_end_pos": [809, 16]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have isoE : Eu \u2243L[\ud835\udd5c] E := ContinuousLinearEquiv.ulift", "annotated_tactic": ["have isoE : Eu \u2243L[\ud835\udd5c] E := <a>ContinuousLinearEquiv.ulift</a>", [{"full_name": "ContinuousLinearEquiv.ulift", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2324, 5], "def_end_pos": [2324, 10]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have isoF : Fu \u2243L[\ud835\udd5c] F := ContinuousLinearEquiv.ulift", "annotated_tactic": ["have isoF : Fu \u2243L[\ud835\udd5c] F := <a>ContinuousLinearEquiv.ulift</a>", [{"full_name": "ContinuousLinearEquiv.ulift", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2324, 5], "def_end_pos": [2324, 10]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have isoG : Gu \u2243L[\ud835\udd5c] G := ContinuousLinearEquiv.ulift", "annotated_tactic": ["have isoG : Gu \u2243L[\ud835\udd5c] G := <a>ContinuousLinearEquiv.ulift</a>", [{"full_name": "ContinuousLinearEquiv.ulift", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2324, 5], "def_end_pos": [2324, 10]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "let fu : Eu \u2192 Fu := (isoF.symm \u2218 f) \u2218 isoE", "annotated_tactic": ["let fu : Eu \u2192 Fu := (isoF.symm \u2218 f) \u2218 isoE", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have fu_diff : ContDiffOn \ud835\udd5c n fu (isoE \u207b\u00b9' s) := by\n  rwa [isoE.contDiffOn_comp_iff, isoF.symm.comp_contDiffOn_iff]", "annotated_tactic": ["have fu_diff : <a>ContDiffOn</a> \ud835\udd5c n fu (isoE \u207b\u00b9' s) := by\n    rwa [isoE.contDiffOn_comp_iff, isoF.symm.comp_contDiffOn_iff]", [{"full_name": "ContDiffOn", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [640, 5], "def_end_pos": [640, 15]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "let gu : Fu \u2192 Gu := (isoG.symm \u2218 g) \u2218 isoF", "annotated_tactic": ["let gu : Fu \u2192 Gu := (isoG.symm \u2218 g) \u2218 isoF", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have gu_diff : ContDiffOn \ud835\udd5c n gu (isoF \u207b\u00b9' t) := by\n  rwa [isoF.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff]", "annotated_tactic": ["have gu_diff : <a>ContDiffOn</a> \ud835\udd5c n gu (isoF \u207b\u00b9' t) := by\n    rwa [isoF.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff]", [{"full_name": "ContDiffOn", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [640, 5], "def_end_pos": [640, 15]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have main : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (isoE \u207b\u00b9' s) := by\n  apply ContDiffOn.comp_same_univ gu_diff fu_diff\n  intro y hy\n  simp only [fu, ContinuousLinearEquiv.coe_apply, Function.comp_apply, mem_preimage]\n  rw [isoF.apply_symm_apply (f (isoE y))]\n  exact st hy", "annotated_tactic": ["have main : <a>ContDiffOn</a> \ud835\udd5c n (gu \u2218 fu) (isoE \u207b\u00b9' s) := by\n    apply <a>ContDiffOn.comp_same_univ</a> gu_diff fu_diff\n    intro y hy\n    simp only [fu, <a>ContinuousLinearEquiv.coe_apply</a>, <a>Function.comp_apply</a>, <a>mem_preimage</a>]\n    rw [isoF.apply_symm_apply (f (isoE y))]\n    exact st hy", [{"full_name": "ContDiffOn", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [640, 5], "def_end_pos": [640, 15]}, {"full_name": "_private.Mathlib.Analysis.Calculus.ContDiff.Basic.0.ContDiffOn.comp_same_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [590, 17], "def_end_pos": [590, 42]}, {"full_name": "ContinuousLinearEquiv.coe_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "have : gu \u2218 fu = (isoG.symm \u2218 g \u2218 f) \u2218 isoE := by\n  ext y\n  simp only [fu, gu, Function.comp_apply]\n  rw [isoF.apply_symm_apply (f (isoE y))]", "annotated_tactic": ["have : gu \u2218 fu = (isoG.symm \u2218 g \u2218 f) \u2218 isoE := by\n    ext y\n    simp only [fu, gu, <a>Function.comp_apply</a>]\n    rw [isoF.apply_symm_apply (f (isoE y))]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\nthis : gu \u2218 fu = (\u21d1isoG.symm \u2218 g \u2218 f) \u2218 \u21d1isoE\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s"}, {"tactic": "rwa [this, isoE.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff] at main", "annotated_tactic": ["rwa [this, isoE.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff] at main", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\nthis : gu \u2218 fu = (\u21d1isoG.symm \u2218 g \u2218 f) \u2218 \u21d1isoE\n\u22a2 ContDiffOn \ud835\udd5c n (g \u2218 f) s", "state_after": "no goals"}, {"tactic": "rwa [isoE.contDiffOn_comp_iff, isoF.symm.comp_contDiffOn_iff]", "annotated_tactic": ["rwa [isoE.contDiffOn_comp_iff, isoF.symm.comp_contDiffOn_iff]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\n\u22a2 ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)", "state_after": "no goals"}, {"tactic": "rwa [isoF.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff]", "annotated_tactic": ["rwa [isoF.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\n\u22a2 ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)", "state_after": "no goals"}, {"tactic": "apply ContDiffOn.comp_same_univ gu_diff fu_diff", "annotated_tactic": ["apply <a>ContDiffOn.comp_same_univ</a> gu_diff fu_diff", [{"full_name": "_private.Mathlib.Analysis.Calculus.ContDiff.Basic.0.ContDiffOn.comp_same_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [590, 17], "def_end_pos": [590, 42]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\n\u22a2 ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\n\u22a2 \u21d1isoE \u207b\u00b9' s \u2286 fu \u207b\u00b9' (\u21d1isoF \u207b\u00b9' t)"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\n\u22a2 \u21d1isoE \u207b\u00b9' s \u2286 fu \u207b\u00b9' (\u21d1isoF \u207b\u00b9' t)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 y \u2208 fu \u207b\u00b9' (\u21d1isoF \u207b\u00b9' t)"}, {"tactic": "simp only [fu, ContinuousLinearEquiv.coe_apply, Function.comp_apply, mem_preimage]", "annotated_tactic": ["simp only [fu, <a>ContinuousLinearEquiv.coe_apply</a>, <a>Function.comp_apply</a>, <a>mem_preimage</a>]", [{"full_name": "ContinuousLinearEquiv.coe_apply", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 y \u2208 fu \u207b\u00b9' (\u21d1isoF \u207b\u00b9' t)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 isoF (isoF.symm (f (isoE y))) \u2208 t"}, {"tactic": "rw [isoF.apply_symm_apply (f (isoE y))]", "annotated_tactic": ["rw [isoF.apply_symm_apply (f (isoE y))]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 isoF (isoF.symm (f (isoE y))) \u2208 t", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 f (isoE y) \u2208 t"}, {"tactic": "exact st hy", "annotated_tactic": ["exact st hy", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\ny : Eu\nhy : y \u2208 \u21d1isoE \u207b\u00b9' s\n\u22a2 f (isoE y) \u2208 t", "state_after": "no goals"}, {"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\n\u22a2 gu \u2218 fu = (\u21d1isoG.symm \u2218 g \u2218 f) \u2218 \u21d1isoE", "state_after": "case h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\ny : Eu\n\u22a2 ((gu \u2218 fu) y).down = (((\u21d1isoG.symm \u2218 g \u2218 f) \u2218 \u21d1isoE) y).down"}, {"tactic": "simp only [fu, gu, Function.comp_apply]", "annotated_tactic": ["simp only [fu, gu, <a>Function.comp_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "case h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\ny : Eu\n\u22a2 ((gu \u2218 fu) y).down = (((\u21d1isoG.symm \u2218 g \u2218 f) \u2218 \u21d1isoE) y).down", "state_after": "case h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\ny : Eu\n\u22a2 (isoG.symm (g (isoF (isoF.symm (f (isoE y)))))).down = (isoG.symm (g (f (isoE y)))).down"}, {"tactic": "rw [isoF.apply_symm_apply (f (isoE y))]", "annotated_tactic": ["rw [isoF.apply_symm_apply (f (isoE y))]", []], "state_before": "case h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t\u271d u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nt : Set F\ng : F \u2192 G\nf : E \u2192 F\nhg : ContDiffOn \ud835\udd5c n g t\nhf : ContDiffOn \ud835\udd5c n f s\nst : s \u2286 f \u207b\u00b9' t\nEu : Type (max uE uF uG) := ULift.{max uF uG, uE} E\nFu : Type (max uE uF uG) := ULift.{max uE uG, uF} F\nGu : Type (max uE uF uG) := ULift.{max uE uF, uG} G\nisoE : Eu \u2243L[\ud835\udd5c] E\nisoF : Fu \u2243L[\ud835\udd5c] F\nisoG : Gu \u2243L[\ud835\udd5c] G\nfu : Eu \u2192 Fu := (\u21d1isoF.symm \u2218 f) \u2218 \u21d1isoE\nfu_diff : ContDiffOn \ud835\udd5c n fu (\u21d1isoE \u207b\u00b9' s)\ngu : Fu \u2192 Gu := (\u21d1isoG.symm \u2218 g) \u2218 \u21d1isoF\ngu_diff : ContDiffOn \ud835\udd5c n gu (\u21d1isoF \u207b\u00b9' t)\nmain : ContDiffOn \ud835\udd5c n (gu \u2218 fu) (\u21d1isoE \u207b\u00b9' s)\ny : Eu\n\u22a2 (isoG.symm (g (isoF (isoF.symm (f (isoE y)))))).down = (isoG.symm (g (f (isoE y)))).down", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.mgf_undef", "start": [174, 1], "end": [175, 37], "traced_tactics": [{"tactic": "simp only [mgf, integral_undef hX]", "annotated_tactic": ["simp only [<a>mgf</a>, <a>integral_undef</a> hX]", [{"full_name": "ProbabilityTheory.mgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [103, 5], "def_end_pos": [103, 8]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [833, 9], "def_end_pos": [833, 23]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\nt : \u211d\nhX : \u00acIntegrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 mgf X \u03bc t = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "Embedding.map_nhdsWithin_eq", "start": [1184, 1], "end": [1187, 60], "traced_tactics": [{"tactic": "rw [nhdsWithin, Filter.map_inf hf.inj, hf.map_nhds_eq, map_principal, \u2190 nhdsWithin_inter',\n  inter_eq_self_of_subset_right (image_subset_range _ _)]", "annotated_tactic": ["rw [<a>nhdsWithin</a>, <a>Filter.map_inf</a> hf.inj, hf.map_nhds_eq, <a>map_principal</a>, \u2190 <a>nhdsWithin_inter'</a>,\n    <a>inter_eq_self_of_subset_right</a> (<a>image_subset_range</a> _ _)]", [{"full_name": "nhdsWithin", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [123, 5], "def_end_pos": [123, 15]}, {"full_name": "Filter.map_inf", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2717, 9], "def_end_pos": [2717, 16]}, {"full_name": "Filter.map_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1913, 9], "def_end_pos": [1913, 22]}, {"full_name": "nhdsWithin_inter'", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [257, 9], "def_end_pos": [257, 26]}, {"full_name": "Set.inter_eq_self_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [960, 9], "def_end_pos": [960, 38]}, {"full_name": "Set.image_subset_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [710, 9], "def_end_pos": [710, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\nhf : _root_.Embedding f\ns : Set \u03b1\nx : \u03b1\n\u22a2 map f (\ud835\udcdd[s] x) = \ud835\udcdd[f '' s] f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "Polynomial.ker_mapRingHom", "start": [632, 1], "end": [638, 7], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 LinearMap.ker (mapRingHom f).toSemilinearMap = map C (RingHom.ker f)", "state_after": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 x\u271d \u2208 LinearMap.ker (mapRingHom f).toSemilinearMap \u2194 x\u271d \u2208 map C (RingHom.ker f)"}, {"tactic": "simp only [LinearMap.mem_ker, RingHom.toSemilinearMap_apply, coe_mapRingHom]", "annotated_tactic": ["simp only [<a>LinearMap.mem_ker</a>, <a>RingHom.toSemilinearMap_apply</a>, <a>coe_mapRingHom</a>]", [{"full_name": "LinearMap.mem_ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [65, 9], "def_end_pos": [65, 16]}, {"full_name": "RingHom.toSemilinearMap_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [522, 3], "def_end_pos": [522, 8]}, {"full_name": "Polynomial.coe_mapRingHom", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [775, 9], "def_end_pos": [775, 23]}]], "state_before": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 x\u271d \u2208 LinearMap.ker (mapRingHom f).toSemilinearMap \u2194 x\u271d \u2208 map C (RingHom.ker f)", "state_after": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2191\u2191(mapRingHom f)).toFun x\u271d = 0 \u2194 x\u271d \u2208 map C (RingHom.ker f)"}, {"tactic": "rw [mem_map_C_iff, Polynomial.ext_iff]", "annotated_tactic": ["rw [<a>mem_map_C_iff</a>, <a>Polynomial.ext_iff</a>]", [{"full_name": "Ideal.mem_map_C_iff", "def_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "def_pos": [607, 9], "def_end_pos": [607, 22]}, {"full_name": "Polynomial.ext_iff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [822, 9], "def_end_pos": [822, 16]}]], "state_before": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2191\u2191(mapRingHom f)).toFun x\u271d = 0 \u2194 x\u271d \u2208 map C (RingHom.ker f)", "state_after": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2200 (n : \u2115), ((\u2191\u2191(mapRingHom f)).toFun x\u271d).coeff n = coeff 0 n) \u2194 \u2200 (n : \u2115), x\u271d.coeff n \u2208 RingHom.ker f"}, {"tactic": "simp_rw [RingHom.mem_ker f]", "annotated_tactic": ["simp_rw [<a>RingHom.mem_ker</a> f]", [{"full_name": "RingHom.mem_ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [605, 9], "def_end_pos": [605, 16]}]], "state_before": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2200 (n : \u2115), ((\u2191\u2191(mapRingHom f)).toFun x\u271d).coeff n = coeff 0 n) \u2194 \u2200 (n : \u2115), x\u271d.coeff n \u2208 RingHom.ker f", "state_after": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2200 (n : \u2115), ((\u2191\u2191(mapRingHom f)).toFun x\u271d).coeff n = coeff 0 n) \u2194 \u2200 (n : \u2115), f (x\u271d.coeff n) = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nR : Type u\nS : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : R[X]\n\u22a2 (\u2200 (n : \u2115), ((\u2191\u2191(mapRingHom f)).toFun x\u271d).coeff n = coeff 0 n) \u2194 \u2200 (n : \u2115), f (x\u271d.coeff n) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.neg_ball", "start": [960, 1], "end": [962, 83], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr : \u211d\nx : E\n\u22a2 -p.ball x r = p.ball (-x) r", "state_after": "case h\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr : \u211d\nx x\u271d : E\n\u22a2 x\u271d \u2208 -p.ball x r \u2194 x\u271d \u2208 p.ball (-x) r"}, {"tactic": "rw [Set.mem_neg, mem_ball, mem_ball, \u2190 neg_add', sub_neg_eq_add, map_neg_eq_map]", "annotated_tactic": ["rw [<a>Set.mem_neg</a>, <a>mem_ball</a>, <a>mem_ball</a>, \u2190 <a>neg_add'</a>, <a>sub_neg_eq_add</a>, <a>map_neg_eq_map</a>]", [{"full_name": "Set.mem_neg", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 14]}, {"full_name": "Seminorm.mem_ball", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [674, 9], "def_end_pos": [674, 17]}, {"full_name": "Seminorm.mem_ball", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [674, 9], "def_end_pos": [674, 17]}, {"full_name": "neg_add'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [745, 3], "def_end_pos": [745, 14]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [697, 3], "def_end_pos": [697, 14]}, {"full_name": "AddGroupSeminormClass.map_neg_eq_map", "def_path": "Mathlib/Algebra/Order/Hom/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 17]}]], "state_before": "case h\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np\u271d p : Seminorm \ud835\udd5c E\nr : \u211d\nx x\u271d : E\n\u22a2 x\u271d \u2208 -p.ball x r \u2194 x\u271d \u2208 p.ball (-x) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Sigma.lean", "full_name": "Finset.sigmaLift_nonempty", "start": [198, 1], "end": [201, 32], "traced_tactics": [{"tactic": "simp_rw [nonempty_iff_ne_empty, sigmaLift]", "annotated_tactic": ["simp_rw [<a>nonempty_iff_ne_empty</a>, <a>sigmaLift</a>]", [{"full_name": "Finset.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [606, 9], "def_end_pos": [606, 30]}, {"full_name": "Finset.sigmaLift", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [151, 5], "def_end_pos": [151, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 (sigmaLift f a b).Nonempty \u2194 \u2203 (h : a.fst = b.fst), (f (h \u25b8 a.snd) b.snd).Nonempty", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) \u2260 \u2205 \u2194\n    \u2203 (h : a.fst = b.fst), f (\u22ef \u25b8 a.snd) b.snd \u2260 \u2205"}, {"tactic": "split_ifs with h <;> simp [h]", "annotated_tactic": ["split_ifs with h <;> simp [h]", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) \u2260 \u2205 \u2194\n    \u2203 (h : a.fst = b.fst), f (\u22ef \u25b8 a.snd) b.snd \u2260 \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "full_name": "Affine.Simplex.circumsphere_reindex", "start": [392, 1], "end": [396, 56], "traced_tactics": [{"tactic": "refine s.circumsphere_unique_dist_eq.2 _ \u27e8?_, ?_\u27e9 <;> rw [\u2190 s.reindex_range_points e]", "annotated_tactic": ["refine s.circumsphere_unique_dist_eq.2 _ \u27e8?_, ?_\u27e9 <;> rw [\u2190 s.reindex_range_points e]", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nm n : \u2115\ns : Simplex \u211d P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 (s.reindex e).circumsphere = s.circumsphere", "state_after": "case refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nm n : \u2115\ns : Simplex \u211d P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 (s.reindex e).circumsphere.center \u2208 affineSpan \u211d (Set.range (s.reindex e).points)\n\ncase refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nm n : \u2115\ns : Simplex \u211d P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 Set.range (s.reindex e).points \u2286 Metric.sphere (s.reindex e).circumsphere.center (s.reindex e).circumsphere.radius"}, {"tactic": "exact (s.reindex e).circumsphere_unique_dist_eq.1.1", "annotated_tactic": ["exact (s.reindex e).<a>circumsphere_unique_dist_eq</a>.1.1", [{"full_name": "Affine.Simplex.circumsphere_unique_dist_eq", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [253, 9], "def_end_pos": [253, 36]}]], "state_before": "case refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nm n : \u2115\ns : Simplex \u211d P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 (s.reindex e).circumsphere.center \u2208 affineSpan \u211d (Set.range (s.reindex e).points)", "state_after": "no goals"}, {"tactic": "exact (s.reindex e).circumsphere_unique_dist_eq.1.2", "annotated_tactic": ["exact (s.reindex e).<a>circumsphere_unique_dist_eq</a>.1.2", [{"full_name": "Affine.Simplex.circumsphere_unique_dist_eq", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [253, 9], "def_end_pos": [253, 36]}]], "state_before": "case refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nm n : \u2115\ns : Simplex \u211d P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 Set.range (s.reindex e).points \u2286 Metric.sphere (s.reindex e).circumsphere.center (s.reindex e).circumsphere.radius", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/StdBasis.lean", "full_name": "Pi.basisFun_repr", "start": [284, 1], "end": [284, 99], "traced_tactics": [{"tactic": "simp [basisFun]", "annotated_tactic": ["simp [<a>basisFun</a>]", [{"full_name": "Pi.basisFun", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [273, 19], "def_end_pos": [273, 27]}]], "state_before": "R : Type u_1\n\u03b7 : Type u_2\n\u03b9s : \u03b7 \u2192 Type u_3\nMs : \u03b7 \u2192 Type u_4\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : (i : \u03b7) \u2192 AddCommMonoid (Ms i)\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Module R (Ms i)\ninst\u271d : Finite \u03b7\nx : \u03b7 \u2192 R\ni : \u03b7\n\u22a2 ((basisFun R \u03b7).repr x) i = x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "full_name": "Set.projIic_eq_self", "start": [102, 1], "end": [102, 98], "traced_tactics": [{"tactic": "simp [projIic, Subtype.ext_iff]", "annotated_tactic": ["simp [<a>projIic</a>, <a>Subtype.ext_iff</a>]", [{"full_name": "Set.projIic", "def_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "def_pos": [44, 5], "def_end_pos": [44, 12]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\n\u22a2 projIic b x = \u27e8b, \u22ef\u27e9 \u2194 b \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/Basic.lean", "full_name": "SetTheory.PGame.mul_one_equiv", "start": [674, 1], "end": [675, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.nnnorm_approxOn_le", "start": [68, 1], "end": [74, 22], "traced_tactics": [{"tactic": "have := edist_approxOn_le hf h\u2080 x n", "annotated_tactic": ["have := <a>edist_approxOn_le</a> hf h\u2080 x n", [{"full_name": "MeasureTheory.SimpleFunc.edist_approxOn_le", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDense.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a"}, {"tactic": "rw [edist_comm y\u2080] at this", "annotated_tactic": ["rw [<a>edist_comm</a> y\u2080] at this", [{"full_name": "PseudoEMetricSpace.edist_comm", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [83, 3], "def_end_pos": [83, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist (f x) y\u2080\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a"}, {"tactic": "simp only [edist_nndist, nndist_eq_nnnorm] at this", "annotated_tactic": ["simp only [<a>edist_nndist</a>, <a>nndist_eq_nnnorm</a>] at this", [{"full_name": "edist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [306, 9], "def_end_pos": [306, 21]}, {"full_name": "nndist_eq_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [788, 7], "def_end_pos": [788, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist (f x) y\u2080\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2191\u2016f x - y\u2080\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a"}, {"tactic": "exact mod_cast this", "annotated_tactic": ["exact mod_cast this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2191\u2016f x - y\u2080\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Invertible.lean", "full_name": "invOf_le_one", "start": [39, 1], "end": [40, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.rel_refl_of_refl_on", "start": [2777, 1], "end": [2782, 87], "traced_tactics": [{"tactic": "refine m.induction_on ?_ ?_", "annotated_tactic": ["refine m.induction_on ?_ ?_", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 x \u2208 m, r x x) \u2192 Rel r m m", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 x \u2208 0, r x x) \u2192 Rel r 0 0\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1) (s : Multiset \u03b1), ((\u2200 x \u2208 s, r x x) \u2192 Rel r s s) \u2192 (\u2200 x \u2208 a ::\u2098 s, r x x) \u2192 Rel r (a ::\u2098 s) (a ::\u2098 s)"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 x \u2208 0, r x x) \u2192 Rel r 0 0", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u2200 x \u2208 0, r x x\n\u22a2 Rel r 0 0"}, {"tactic": "apply Rel.zero", "annotated_tactic": ["apply <a>Rel.zero</a>", [{"full_name": "Multiset.Rel.zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2763, 5], "def_end_pos": [2763, 9]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u2200 x \u2208 0, r x x\n\u22a2 Rel r 0 0", "state_after": "no goals"}, {"tactic": "intro a m ih h", "annotated_tactic": ["intro a m ih h", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1) (s : Multiset \u03b1), ((\u2200 x \u2208 s, r x x) \u2192 Rel r s s) \u2192 (\u2200 x \u2208 a ::\u2098 s, r x x) \u2192 Rel r (a ::\u2098 s) (a ::\u2098 s)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm\u271d : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nm : Multiset \u03b1\nih : (\u2200 x \u2208 m, r x x) \u2192 Rel r m m\nh : \u2200 x \u2208 a ::\u2098 m, r x x\n\u22a2 Rel r (a ::\u2098 m) (a ::\u2098 m)"}, {"tactic": "exact Rel.cons (h _ (mem_cons_self _ _)) (ih fun _ ha => h _ (mem_cons_of_mem ha))", "annotated_tactic": ["exact <a>Rel.cons</a> (h _ (<a>mem_cons_self</a> _ _)) (ih fun _ ha => h _ (<a>mem_cons_of_mem</a> ha))", [{"full_name": "Multiset.Rel.cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2764, 5], "def_end_pos": [2764, 9]}, {"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 22]}, {"full_name": "Multiset.mem_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm\u271d : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nm : Multiset \u03b1\nih : (\u2200 x \u2208 m, r x x) \u2192 Rel r m m\nh : \u2200 x \u2208 a ::\u2098 m, r x x\n\u22a2 Rel r (a ::\u2098 m) (a ::\u2098 m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.cos_pi_div_four", "start": [819, 1], "end": [823, 9], "traced_tactics": [{"tactic": "trans cos (\u03c0 / 2 ^ 2)", "annotated_tactic": ["trans <a>cos</a> (\u03c0 / 2 ^ 2)", [{"full_name": "Real.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [126, 12], "def_end_pos": [126, 15]}]], "state_before": "x : \u211d\n\u22a2 cos (\u03c0 / 4) = \u221a2 / 2", "state_after": "x : \u211d\n\u22a2 cos (\u03c0 / 4) = cos (\u03c0 / 2 ^ 2)\n\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 2) = \u221a2 / 2"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "x : \u211d\n\u22a2 cos (\u03c0 / 4) = cos (\u03c0 / 2 ^ 2)", "state_after": "case e_x.e_a\nx : \u211d\n\u22a2 4 = 2 ^ 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case e_x.e_a\nx : \u211d\n\u22a2 4 = 2 ^ 2", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 2) = \u221a2 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IsTensorProduct.lean", "full_name": "IsBaseChange.inductionOn", "start": [187, 8], "end": [189, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "full_name": "zero_le_two", "start": [32, 1], "end": [35, 43], "traced_tactics": [{"tactic": "rw [\u2190 one_add_one_eq_two]", "annotated_tactic": ["rw [\u2190 <a>one_add_one_eq_two</a>]", [{"full_name": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [231, 9], "def_end_pos": [231, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 1 + 1"}, {"tactic": "exact add_nonneg zero_le_one zero_le_one", "annotated_tactic": ["exact <a>add_nonneg</a> <a>zero_le_one</a> <a>zero_le_one</a>", [{"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 1 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "full_name": "UpperHalfPlane.im_inv_neg_coe_pos", "start": [189, 1], "end": [190, 48], "traced_tactics": [{"tactic": "simpa using div_pos z.property (normSq_pos z)", "annotated_tactic": ["simpa using <a>div_pos</a> z.property (<a>normSq_pos</a> z)", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [79, 7], "def_end_pos": [79, 14]}, {"full_name": "UpperHalfPlane.normSq_pos", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 19]}]], "state_before": "z : \u210d\n\u22a2 0 < (-\u2191z)\u207b\u00b9.im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Derivation/AdjointAction.lean", "full_name": "LieDerivation.coe_ad_apply_eq_ad_apply", "start": [53, 1], "end": [54, 96], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : L\n\u22a2 \u2191((ad R L) x) = (LieAlgebra.ad R L) x", "state_after": "case h\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx x\u271d : L\n\u22a2 \u2191((ad R L) x) x\u271d = ((LieAlgebra.ad R L) x) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx x\u271d : L\n\u22a2 \u2191((ad R L) x) x\u271d = ((LieAlgebra.ad R L) x) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Image.lean", "full_name": "StrictAntiOn.mapsTo_Ioc", "start": [200, 1], "end": [203, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/Hensel.lean", "full_name": "newton_seq_dist", "start": [322, 9], "end": [326, 39], "traced_tactics": [{"tactic": "have hex : \u2203 m, k = n + m := Nat.exists_eq_add_of_le hnk", "annotated_tactic": ["have hex : \u2203 m, k = n + m := <a>Nat.exists_eq_add_of_le</a> hnk", [{"full_name": "Nat.exists_eq_add_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [192, 19], "def_end_pos": [192, 38]}]], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n"}, {"tactic": "let \u27e8_, hex'\u27e9 := hex", "annotated_tactic": ["let \u27e8_, hex'\u27e9 := hex", []], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n"}, {"tactic": "rw [hex']", "annotated_tactic": ["rw [hex']", []], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq (n + w\u271d) - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n"}, {"tactic": "apply newton_seq_dist_aux", "annotated_tactic": ["apply <a>newton_seq_dist_aux</a>", [{"full_name": "_private.Mathlib.NumberTheory.Padics.Hensel.0.newton_seq_dist_aux", "def_path": "Mathlib/NumberTheory/Padics/Hensel.lean", "def_pos": [299, 17], "def_end_pos": [299, 36]}]], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq (n + w\u271d) - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean", "full_name": "CategoryTheory.EffectiveEpiFamily.transitive_of_finite", "start": [44, 1], "end": [68, 77], "traced_tactics": [{"tactic": "rw [\u2190 Sieve.effectiveEpimorphic_family]", "annotated_tactic": ["rw [\u2190 <a>Sieve.effectiveEpimorphic_family</a>]", [{"full_name": "CategoryTheory.Sieve.effectiveEpimorphic_family", "def_path": "Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean", "def_pos": [244, 9], "def_end_pos": [244, 41]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 EffectiveEpiFamily (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst).EffectiveEpimorphic"}, {"tactic": "suffices h\u2082 : (Sieve.generate (Presieve.ofArrows (fun (\u27e8a, b\u27e9 : \u03a3 _, \u03b2 _) => Y_n a b)\n      (fun \u27e8a,b\u27e9 => \u03c0_n a b \u226b \u03c0 a))) \u2208 GrothendieckTopology.sieves (coherentTopology C) X by\n  change Nonempty _\n  rw [\u2190 Sieve.forallYonedaIsSheaf_iff_colimit]\n  exact fun W => coherentTopology.isSheaf_yoneda_obj W _ h\u2082", "annotated_tactic": ["suffices h\u2082 : (<a>Sieve.generate</a> (<a>Presieve.ofArrows</a> (fun (\u27e8a, b\u27e9 : \u03a3 _, \u03b2 _) => Y_n a b)\n        (fun \u27e8a,b\u27e9 => \u03c0_n a b \u226b \u03c0 a))) \u2208 <a>GrothendieckTopology.sieves</a> (<a>coherentTopology</a> C) X by\n    change <a>Nonempty</a> _\n    rw [\u2190 <a>Sieve.forallYonedaIsSheaf_iff_colimit</a>]\n    exact fun W => <a>coherentTopology.isSheaf_yoneda_obj</a> W _ h\u2082", [{"full_name": "CategoryTheory.Sieve.generate", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [400, 5], "def_end_pos": [400, 13]}, {"full_name": "CategoryTheory.Presieve.ofArrows", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [137, 11], "def_end_pos": [137, 19]}, {"full_name": "CategoryTheory.GrothendieckTopology.sieves", "def_path": "Mathlib/CategoryTheory/Sites/Grothendieck.lean", "def_pos": [78, 3], "def_end_pos": [78, 9]}, {"full_name": "CategoryTheory.coherentTopology", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/Basic.lean", "def_pos": [81, 5], "def_end_pos": [81, 21]}, {"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}, {"full_name": "CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit", "def_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "def_pos": [181, 9], "def_end_pos": [181, 40]}, {"full_name": "CategoryTheory.coherentTopology.isSheaf_yoneda_obj", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean", "def_pos": [44, 9], "def_end_pos": [44, 27]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst).EffectiveEpimorphic", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X"}, {"tactic": "apply Coverage.saturate.transitive X (Sieve.generate (Presieve.ofArrows Y \u03c0))", "annotated_tactic": ["apply <a>Coverage.saturate.transitive</a> X (<a>Sieve.generate</a> (<a>Presieve.ofArrows</a> Y \u03c0))", [{"full_name": "CategoryTheory.Coverage.saturate.transitive", "def_path": "Mathlib/CategoryTheory/Sites/Coverage.lean", "def_pos": [184, 5], "def_end_pos": [184, 15]}, {"full_name": "CategoryTheory.Sieve.generate", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [400, 5], "def_end_pos": [400, 13]}, {"full_name": "CategoryTheory.Presieve.ofArrows", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [137, 11], "def_end_pos": [137, 19]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X", "state_after": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 (coherentCoverage C).saturate X (Sieve.generate (Presieve.ofArrows Y \u03c0))\n\ncase a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 \u2200 \u2983Y_1 : C\u2984 \u2983f : Y_1 \u27f6 X\u2984,\n    (Sieve.generate (Presieve.ofArrows Y \u03c0)).arrows f \u2192\n      (coherentCoverage C).saturate Y_1\n        (Sieve.pullback f\n          (Sieve.generate\n            (Presieve.ofArrows\n              (fun x =>\n                match x with\n                | \u27e8a, b\u27e9 => Y_n a b)\n              fun x =>\n              match x with\n              | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)))"}, {"tactic": "change Nonempty _", "annotated_tactic": ["change <a>Nonempty</a> _", [{"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nh\u2082 :\n  Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X\n\u22a2 (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst).EffectiveEpimorphic", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nh\u2082 :\n  Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X\n\u22a2 Nonempty\n    (Limits.IsColimit\n      (Sieve.generate (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst)).arrows.cocone)"}, {"tactic": "rw [\u2190 Sieve.forallYonedaIsSheaf_iff_colimit]", "annotated_tactic": ["rw [\u2190 <a>Sieve.forallYonedaIsSheaf_iff_colimit</a>]", [{"full_name": "CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit", "def_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "def_pos": [181, 9], "def_end_pos": [181, 40]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nh\u2082 :\n  Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X\n\u22a2 Nonempty\n    (Limits.IsColimit\n      (Sieve.generate (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst)).arrows.cocone)", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nh\u2082 :\n  Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X\n\u22a2 \u2200 (W : C),\n    Presieve.IsSheafFor (yoneda.obj W)\n      (Sieve.generate (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst)).arrows"}, {"tactic": "exact fun W => coherentTopology.isSheaf_yoneda_obj W _ h\u2082", "annotated_tactic": ["exact fun W => <a>coherentTopology.isSheaf_yoneda_obj</a> W _ h\u2082", [{"full_name": "CategoryTheory.coherentTopology.isSheaf_yoneda_obj", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean", "def_pos": [44, 9], "def_end_pos": [44, 27]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nh\u2082 :\n  Sieve.generate\n      (Presieve.ofArrows\n        (fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => Y_n a b)\n        fun x =>\n        match x with\n        | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a) \u2208\n    (coherentTopology C).sieves X\n\u22a2 \u2200 (W : C),\n    Presieve.IsSheafFor (yoneda.obj W)\n      (Sieve.generate (Presieve.ofArrows (fun c => Y_n c.fst c.snd) fun c => \u03c0_n c.fst c.snd \u226b \u03c0 c.fst)).arrows", "state_after": "no goals"}, {"tactic": "apply Coverage.saturate.of", "annotated_tactic": ["apply <a>Coverage.saturate.of</a>", [{"full_name": "CategoryTheory.Coverage.saturate.of", "def_path": "Mathlib/CategoryTheory/Sites/Coverage.lean", "def_pos": [182, 5], "def_end_pos": [182, 7]}]], "state_before": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 (coherentCoverage C).saturate X (Sieve.generate (Presieve.ofArrows Y \u03c0))", "state_after": "case a.hS\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 Presieve.ofArrows Y \u03c0 \u2208 (coherentCoverage C).covering X"}, {"tactic": "use \u03b1, inferInstance, Y, \u03c0", "annotated_tactic": ["use \u03b1, <a>inferInstance</a>, Y, \u03c0", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case a.hS\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 Presieve.ofArrows Y \u03c0 \u2208 (coherentCoverage C).covering X", "state_after": "no goals"}, {"tactic": "intro V f \u27e8Y\u2081, h, g, \u27e8hY, hf\u27e9\u27e9", "annotated_tactic": ["intro V f \u27e8Y\u2081, h, g, \u27e8hY, hf\u27e9\u27e9", []], "state_before": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\n\u22a2 \u2200 \u2983Y_1 : C\u2984 \u2983f : Y_1 \u27f6 X\u2984,\n    (Sieve.generate (Presieve.ofArrows Y \u03c0)).arrows f \u2192\n      (coherentCoverage C).saturate Y_1\n        (Sieve.pullback f\n          (Sieve.generate\n            (Presieve.ofArrows\n              (fun x =>\n                match x with\n                | \u27e8a, b\u27e9 => Y_n a b)\n              fun x =>\n              match x with\n              | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)))", "state_after": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 (coherentCoverage C).saturate V\n    (Sieve.pullback f\n      (Sieve.generate\n        (Presieve.ofArrows\n          (fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => Y_n a b)\n          fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)))"}, {"tactic": "rw [\u2190 hf, Sieve.pullback_comp]", "annotated_tactic": ["rw [\u2190 hf, <a>Sieve.pullback_comp</a>]", [{"full_name": "CategoryTheory.Sieve.pullback_comp", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [530, 9], "def_end_pos": [530, 22]}]], "state_before": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 (coherentCoverage C).saturate V\n    (Sieve.pullback f\n      (Sieve.generate\n        (Presieve.ofArrows\n          (fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => Y_n a b)\n          fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)))", "state_after": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 (coherentCoverage C).saturate V\n    (Sieve.pullback h\n      (Sieve.pullback g\n        (Sieve.generate\n          (Presieve.ofArrows\n            (fun x =>\n              match x with\n              | \u27e8a, b\u27e9 => Y_n a b)\n            fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))))"}, {"tactic": "apply (coherentTopology C).pullback_stable'", "annotated_tactic": ["apply (<a>coherentTopology</a> C).<a>pullback_stable'</a>", [{"full_name": "CategoryTheory.coherentTopology", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/Basic.lean", "def_pos": [81, 5], "def_end_pos": [81, 21]}, {"full_name": "CategoryTheory.GrothendieckTopology.pullback_stable'", "def_path": "Mathlib/CategoryTheory/Sites/Grothendieck.lean", "def_pos": [83, 3], "def_end_pos": [83, 19]}]], "state_before": "case a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 (coherentCoverage C).saturate V\n    (Sieve.pullback h\n      (Sieve.pullback g\n        (Sieve.generate\n          (Presieve.ofArrows\n            (fun x =>\n              match x with\n              | \u27e8a, b\u27e9 => Y_n a b)\n            fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))))", "state_after": "case a.a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 Sieve.pullback g\n      (Sieve.generate\n        (Presieve.ofArrows\n          (fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => Y_n a b)\n          fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)) \u2208\n    (coherentTopology C).sieves Y\u2081"}, {"tactic": "apply coherentTopology.mem_sieves_of_hasEffectiveEpiFamily", "annotated_tactic": ["apply <a>coherentTopology.mem_sieves_of_hasEffectiveEpiFamily</a>", [{"full_name": "CategoryTheory.coherentTopology.mem_sieves_of_hasEffectiveEpiFamily", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean", "def_pos": [29, 9], "def_end_pos": [29, 61]}]], "state_before": "case a.a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 Sieve.pullback g\n      (Sieve.generate\n        (Presieve.ofArrows\n          (fun x =>\n            match x with\n            | \u27e8a, b\u27e9 => Y_n a b)\n          fun x =>\n          match x with\n          | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a)) \u2208\n    (coherentTopology C).sieves Y\u2081", "state_after": "case a.a.a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 \u2203 \u03b1_1,\n    \u2203 (_ : Finite \u03b1_1),\n      \u2203 Y_1 \u03c0_1,\n        EffectiveEpiFamily Y_1 \u03c0_1 \u2227\n          \u2200 (a : \u03b1_1),\n            (Sieve.pullback g\n                  (Sieve.generate\n                    (Presieve.ofArrows\n                      (fun x =>\n                        match x with\n                        | \u27e8a, b\u27e9 => Y_n a b)\n                      fun x =>\n                      match x with\n                      | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))).arrows\n              (\u03c0_1 a)"}, {"tactic": "obtain \u27e8i\u27e9 := hY", "annotated_tactic": ["obtain \u27e8i\u27e9 := hY", []], "state_before": "case a.a.a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X\nh\u271d : EffectiveEpiFamily Y \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY\u2081 : C\nh : V \u27f6 Y\u2081\ng : Y\u2081 \u27f6 X\nhY : Presieve.ofArrows Y \u03c0 g\nhf : h \u226b g = f\n\u22a2 \u2203 \u03b1_1,\n    \u2203 (_ : Finite \u03b1_1),\n      \u2203 Y_1 \u03c0_1,\n        EffectiveEpiFamily Y_1 \u03c0_1 \u2227\n          \u2200 (a : \u03b1_1),\n            (Sieve.pullback g\n                  (Sieve.generate\n                    (Presieve.ofArrows\n                      (fun x =>\n                        match x with\n                        | \u27e8a, b\u27e9 => Y_n a b)\n                      fun x =>\n                      match x with\n                      | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))).arrows\n              (\u03c0_1 a)", "state_after": "case a.a.a.mk\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY\u271d : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y\u271d a \u27f6 X\nh\u271d : EffectiveEpiFamily Y\u271d \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y\u271d a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY : C\ni : \u03b1\nh : V \u27f6 Y\u271d i\nhf : h \u226b \u03c0 i = f\n\u22a2 \u2203 \u03b1_1,\n    \u2203 (_ : Finite \u03b1_1),\n      \u2203 Y \u03c0_1,\n        EffectiveEpiFamily Y \u03c0_1 \u2227\n          \u2200 (a : \u03b1_1),\n            (Sieve.pullback (\u03c0 i)\n                  (Sieve.generate\n                    (Presieve.ofArrows\n                      (fun x =>\n                        match x with\n                        | \u27e8a, b\u27e9 => Y_n a b)\n                      fun x =>\n                      match x with\n                      | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))).arrows\n              (\u03c0_1 a)"}, {"tactic": "exact \u27e8\u03b2 i, inferInstance, Y_n i, \u03c0_n i, H i, fun b \u21a6\n  \u27e8Y_n i b, (\ud835\udfd9 _), \u03c0_n i b \u226b \u03c0 i, \u27e8(\u27e8i, b\u27e9 : \u03a3 (i : \u03b1), \u03b2 i)\u27e9, by simp\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u03b2 i, <a>inferInstance</a>, Y_n i, \u03c0_n i, H i, fun b \u21a6\n      \u27e8Y_n i b, (\ud835\udfd9 _), \u03c0_n i b \u226b \u03c0 i, \u27e8(\u27e8i, b\u27e9 : \u03a3 (i : \u03b1), \u03b2 i)\u27e9, by simp\u27e9\u27e9", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case a.a.a.mk\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY\u271d : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y\u271d a \u27f6 X\nh\u271d : EffectiveEpiFamily Y\u271d \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y\u271d a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY : C\ni : \u03b1\nh : V \u27f6 Y\u271d i\nhf : h \u226b \u03c0 i = f\n\u22a2 \u2203 \u03b1_1,\n    \u2203 (_ : Finite \u03b1_1),\n      \u2203 Y \u03c0_1,\n        EffectiveEpiFamily Y \u03c0_1 \u2227\n          \u2200 (a : \u03b1_1),\n            (Sieve.pullback (\u03c0 i)\n                  (Sieve.generate\n                    (Presieve.ofArrows\n                      (fun x =>\n                        match x with\n                        | \u27e8a, b\u27e9 => Y_n a b)\n                      fun x =>\n                      match x with\n                      | \u27e8a, b\u27e9 => \u03c0_n a b \u226b \u03c0 a))).arrows\n              (\u03c0_1 a)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Precoherent C\nX : C\n\u03b1 : Type\ninst\u271d\u00b9 : Finite \u03b1\nY\u271d : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 Y\u271d a \u27f6 X\nh\u271d : EffectiveEpiFamily Y\u271d \u03c0\n\u03b2 : \u03b1 \u2192 Type\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nY_n : (a : \u03b1) \u2192 \u03b2 a \u2192 C\n\u03c0_n : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 Y_n a b \u27f6 Y\u271d a\nH : \u2200 (a : \u03b1), EffectiveEpiFamily (Y_n a) (\u03c0_n a)\nV : C\nf : V \u27f6 X\nY : C\ni : \u03b1\nh : V \u27f6 Y\u271d i\nhf : h \u226b \u03c0 i = f\nb : \u03b2 i\n\u22a2 \ud835\udfd9 (Y_n i b) \u226b \u03c0_n i b \u226b \u03c0 i = \u03c0_n i b \u226b \u03c0 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "full_name": "MvPowerSeries.constantCoeff_zero", "start": [484, 1], "end": [485, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_bit1", "start": [260, 1], "end": [263, 31], "traced_tactics": [{"tactic": "rw [bit1, zpow_add_one_of_ne_neg_one, zpow_bit0]", "annotated_tactic": ["rw [<a>bit1</a>, <a>zpow_add_one_of_ne_neg_one</a>, <a>zpow_bit0</a>]", [{"full_name": "bit1", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [41, 5], "def_end_pos": [41, 9]}, {"full_name": "Matrix.zpow_add_one_of_ne_neg_one", "def_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "def_pos": [249, 9], "def_end_pos": [249, 35]}, {"full_name": "Matrix.zpow_bit0", "def_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "def_pos": [243, 9], "def_end_pos": [243, 18]}]], "state_before": "n' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\n\u22a2 A ^ bit1 n = A ^ n * A ^ n * A", "state_after": "case a\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\n\u22a2 bit0 n \u2260 -1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case a\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\n\u22a2 bit0 n \u2260 -1", "state_after": "case a\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\nh : bit0 n = -1\n\u22a2 False"}, {"tactic": "simpa using congr_arg bodd h", "annotated_tactic": ["simpa using <a>congr_arg</a> <a>bodd</a> h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Int.bodd", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [32, 5], "def_end_pos": [32, 9]}]], "state_before": "case a\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\nh : bit0 n = -1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.kroneckerMap_assoc\u2081", "start": [186, 1], "end": [192, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.Measure.exists_measure_inter_spanningSets_pos", "start": [736, 1], "end": [740, 52], "traced_tactics": [{"tactic": "rw [\u2190 not_iff_not]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [447, 9], "def_end_pos": [447, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2203 n, 0 < \u03bc (s \u2229 spanningSets \u03bc n)) \u2194 0 < \u03bc s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u00ac\u2203 n, 0 < \u03bc (s \u2229 spanningSets \u03bc n)) \u2194 \u00ac0 < \u03bc s"}, {"tactic": "simp only [not_exists, not_lt, nonpos_iff_eq_zero]", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>not_lt</a>, <a>nonpos_iff_eq_zero</a>]", [{"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u00ac\u2203 n, 0 < \u03bc (s \u2229 spanningSets \u03bc n)) \u2194 \u00ac0 < \u03bc s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u2115), \u03bc (s \u2229 spanningSets \u03bc x) = 0) \u2194 \u03bc s = 0"}, {"tactic": "exact forall_measure_inter_spanningSets_eq_zero s", "annotated_tactic": ["exact <a>forall_measure_inter_spanningSets_eq_zero</a> s", [{"full_name": "MeasureTheory.Measure.forall_measure_inter_spanningSets_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [727, 9], "def_end_pos": [727, 50]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u2115), \u03bc (s \u2229 spanningSets \u03bc x) = 0) \u2194 \u03bc s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.image_sInter_subset", "start": [1484, 1], "end": [1486, 29], "traced_tactics": [{"tactic": "rw [sInter_eq_biInter]", "annotated_tactic": ["rw [<a>sInter_eq_biInter</a>]", [{"full_name": "Set.sInter_eq_biInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 f '' \u22c2\u2080 S \u2286 \u22c2 s \u2208 S, f '' s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 f '' \u22c2 i \u2208 S, i \u2286 \u22c2 s \u2208 S, f '' s"}, {"tactic": "apply image_iInter\u2082_subset", "annotated_tactic": ["apply <a>image_iInter\u2082_subset</a>", [{"full_name": "Set.image_iInter\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1479, 9], "def_end_pos": [1479, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 f '' \u22c2 i \u2208 S, i \u2286 \u22c2 s \u2208 S, f '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.setIntegral_mono_ae_restrict", "start": [811, 1], "end": [813, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/ModuleCat/Biproducts.lean", "full_name": "ModuleCat.binaryProductLimitCone_cone_\u03c0_app_left", "start": [58, 1], "end": [60, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "full_name": "UniformOnFun.gen_eq_preimage_restrict", "start": [601, 11], "end": [606, 59], "traced_tactics": [{"tactic": "ext uv", "annotated_tactic": ["ext uv", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\n\ud835\udd16 : Set (Set \u03b1)\nS : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\n\u22a2 UniformOnFun.gen \ud835\udd16 S V =\n    Prod.map (S.restrict \u2218 \u21d1UniformFun.toFun) (S.restrict \u2218 \u21d1UniformFun.toFun) \u207b\u00b9' UniformFun.gen (\u2191S) \u03b2 V", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\n\ud835\udd16 : Set (Set \u03b1)\nS : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nuv : (\u03b1 \u2192\u1d64[\ud835\udd16] \u03b2) \u00d7 (\u03b1 \u2192\u1d64[\ud835\udd16] \u03b2)\n\u22a2 uv \u2208 UniformOnFun.gen \ud835\udd16 S V \u2194\n    uv \u2208 Prod.map (S.restrict \u2218 \u21d1UniformFun.toFun) (S.restrict \u2218 \u21d1UniformFun.toFun) \u207b\u00b9' UniformFun.gen (\u2191S) \u03b2 V"}, {"tactic": "exact \u27e8fun h \u27e8x, hx\u27e9 => h x hx, fun h x hx => h \u27e8x, hx\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun h \u27e8x, hx\u27e9 => h x hx, fun h x hx => h \u27e8x, hx\u27e9\u27e9", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\n\ud835\udd16 : Set (Set \u03b1)\nS : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nuv : (\u03b1 \u2192\u1d64[\ud835\udd16] \u03b2) \u00d7 (\u03b1 \u2192\u1d64[\ud835\udd16] \u03b2)\n\u22a2 uv \u2208 UniformOnFun.gen \ud835\udd16 S V \u2194\n    uv \u2208 Prod.map (S.restrict \u2218 \u21d1UniformFun.toFun) (S.restrict \u2218 \u21d1UniformFun.toFun) \u207b\u00b9' UniformFun.gen (\u2191S) \u03b2 V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "full_name": "MvPolynomial.psum_def", "start": [310, 1], "end": [310, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Monotone.lean", "full_name": "MonotoneOn.Ioc", "start": 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TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nhf : OpenEmbedding f\nl : Filter Z\nx : X\n\u22a2 Tendsto (g \u2218 f) (\ud835\udcdd x) l \u2194 Tendsto g (\ud835\udcdd (f x)) l", "state_after": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nhf : OpenEmbedding f\nl : Filter Z\nx : X\n\u22a2 map g (\ud835\udcdd (f x)) \u2264 l \u2194 Tendsto g (\ud835\udcdd (f x)) l"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nhf : OpenEmbedding f\nl : Filter Z\nx : X\n\u22a2 map g (\ud835\udcdd (f x)) \u2264 l \u2194 Tendsto g (\ud835\udcdd (f x)) l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Inv.lean", "full_name": "ENNReal.inv_le_inv'", "start": [250, 21], "end": [251, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.Embedding.comp_inj", "start": [743, 1], "end": [744, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Div.lean", "full_name": "Polynomial.pow_mul_divByMonic_rootMultiplicity_eq", "start": [578, 1], "end": [584, 7], "traced_tactics": [{"tactic": "have : Monic ((X - C a) ^ rootMultiplicity a p) := (monic_X_sub_C _).pow _", "annotated_tactic": ["have : <a>Monic</a> ((<a>X</a> - <a>C</a> a) ^ <a>rootMultiplicity</a> a p) := (<a>monic_X_sub_C</a> _).<a>pow</a> _", [{"full_name": "Polynomial.Monic", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [77, 5], "def_end_pos": [77, 10]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 6]}, {"full_name": "Polynomial.C", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [501, 5], "def_end_pos": [501, 6]}, {"full_name": "Polynomial.rootMultiplicity", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [524, 5], "def_end_pos": [524, 21]}, {"full_name": "Polynomial.monic_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [388, 9], "def_end_pos": [388, 22]}, {"full_name": "Polynomial.Monic.pow", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [128, 9], "def_end_pos": [128, 18]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Ring R\np\u271d q p : R[X]\na : R\n\u22a2 (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) = p", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Ring R\np\u271d q p : R[X]\na : R\nthis : ((X - C a) ^ rootMultiplicity a p).Monic\n\u22a2 (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) = p"}, {"tactic": "conv_rhs =>\n    rw [\u2190 modByMonic_add_div p this,\n      (modByMonic_eq_zero_iff_dvd this).2 (pow_rootMultiplicity_dvd _ _)]", "annotated_tactic": ["conv_rhs =>\n      rw [\u2190 <a>modByMonic_add_div</a> p this,\n        (<a>modByMonic_eq_zero_iff_dvd</a> this).2 (<a>pow_rootMultiplicity_dvd</a> _ _)]", [{"full_name": "Polynomial.modByMonic_add_div", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [262, 9], "def_end_pos": [262, 27]}, {"full_name": "Polynomial.modByMonic_eq_zero_iff_dvd", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [398, 9], "def_end_pos": [398, 35]}, {"full_name": "Polynomial.pow_rootMultiplicity_dvd", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [570, 9], "def_end_pos": [570, 33]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Ring R\np\u271d q p : R[X]\na : R\nthis : ((X - C a) ^ rootMultiplicity a p).Monic\n\u22a2 (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) = p", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Ring R\np\u271d q p : R[X]\na : R\nthis : ((X - C a) ^ rootMultiplicity a p).Monic\n\u22a2 (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) =\n    0 + (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Ring R\np\u271d q p : R[X]\na : R\nthis : ((X - C a) ^ rootMultiplicity a p).Monic\n\u22a2 (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) =\n    0 + (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Homotopy/Basic.lean", "full_name": "ContinuousMap.Homotopic.piMap", "start": [394, 11], "end": [397, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "full_name": "WeierstrassCurve.Affine.evalEval_polynomial_zero", "start": [183, 1], "end": [185, 96], "traced_tactics": [{"tactic": "simp only [evalEval_polynomial, zero_add, zero_sub, mul_zero, zero_pow <| Nat.succ_ne_zero _]", "annotated_tactic": ["simp only [<a>evalEval_polynomial</a>, <a>zero_add</a>, <a>zero_sub</a>, <a>mul_zero</a>, <a>zero_pow</a> <| <a>Nat.succ_ne_zero</a> _]", [{"full_name": "WeierstrassCurve.Affine.evalEval_polynomial", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [176, 7], "def_end_pos": [176, 26]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [160, 15], "def_end_pos": [160, 23]}, {"full_name": "Nat.succ_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [708, 17], "def_end_pos": [708, 29]}]], "state_before": "R : Type u\ninst\u271d : CommRing R\nW : Affine R\n\u22a2 evalEval 0 0 W.polynomial = -W.a\u2086", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.coe_map", "start": [229, 1], "end": [230, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Pointwise.lean", "full_name": "Subgroup.Normal.conjAct", "start": [428, 1], "end": [432, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "full_name": "UpperHalfPlane.im_smul_eq_div_normSq", "start": [396, 1], "end": [398, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.min_val", "start": [258, 1], "end": [258, 59], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n m : \u2115\na : Fin n\n\u22a2 min (\u2191a) n = \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SumOverResidueClass.lean", "full_name": "not_summable_indicator_mod_of_antitone_of_neg", "start": [60, 1], "end": [68, 94], "traced_tactics": [{"tactic": "rw [\u2190 ZMod.natCast_zmod_val k, summable_indicator_mod_iff_summable]", "annotated_tactic": ["rw [\u2190 <a>ZMod.natCast_zmod_val</a> k, <a>summable_indicator_mod_iff_summable</a>]", [{"full_name": "ZMod.natCast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 25]}, {"full_name": "summable_indicator_mod_iff_summable", "def_path": "Mathlib/Analysis/SumOverResidueClass.lean", "def_pos": [31, 7], "def_end_pos": [31, 42]}]], "state_before": "m : \u2115\nhm : NeZero m\nf : \u2115 \u2192 \u211d\nhf : Antitone f\nn : \u2115\nhn : f n < 0\nk : ZMod m\n\u22a2 \u00acSummable ({n | \u2191n = k}.indicator f)", "state_after": "m : \u2115\nhm : NeZero m\nf : \u2115 \u2192 \u211d\nhf : Antitone f\nn : \u2115\nhn : f n < 0\nk : ZMod m\n\u22a2 \u00acSummable fun n => f (m * n + k.val)"}, {"tactic": "exact not_summable_of_antitone_of_neg\n  (hf.comp_monotone <| (Covariant.monotone_of_const m).add_const k.val) <|\n  (hf <| (Nat.le_mul_of_pos_left n Fin.size_pos').trans <| Nat.le_add_right ..).trans_lt hn", "annotated_tactic": ["exact <a>not_summable_of_antitone_of_neg</a>\n    (hf.comp_monotone <| (<a>Covariant.monotone_of_const</a> m).<a>add_const</a> k.val) <|\n    (hf <| (<a>Nat.le_mul_of_pos_left</a> n <a>Fin.size_pos'</a>).<a>trans</a> <| <a>Nat.le_add_right</a> ..).<a>trans_lt</a> hn", [{"full_name": "not_summable_of_antitone_of_neg", "def_path": "Mathlib/Analysis/SumOverResidueClass.lean", "def_pos": [49, 7], "def_end_pos": [49, 38]}, {"full_name": "Covariant.monotone_of_const", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean", "def_pos": [249, 9], "def_end_pos": [249, 36]}, {"full_name": "Monotone.add_const", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1327, 15], "def_end_pos": [1327, 24]}, {"full_name": "Nat.le_mul_of_pos_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [465, 19], "def_end_pos": [465, 37]}, {"full_name": "Fin.size_pos'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 18]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "Nat.le_add_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [395, 9], "def_end_pos": [395, 21]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "m : \u2115\nhm : NeZero m\nf : \u2115 \u2192 \u211d\nhf : Antitone f\nn : \u2115\nhn : f n < 0\nk : ZMod m\n\u22a2 \u00acSummable fun n => f (m * n + k.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "lcm_dvd_iff", "start": [744, 1], "end": [753, 42], "traced_tactics": [{"tactic": "by_cases h : a = 0 \u2228 b = 0", "annotated_tactic": ["by_cases h : a = 0 \u2228 b = 0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : a = 0 \u2228 b = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c"}, {"tactic": "rcases h with (rfl | rfl) <;>\n  simp (config := { contextual := true }) only [iff_def, lcm_zero_left, lcm_zero_right,\n    zero_dvd_iff, dvd_zero, eq_self_iff_true, and_true_iff, imp_true_iff]", "annotated_tactic": ["rcases h with (rfl | rfl) <;>\n      simp (config := { contextual := <a>true</a> }) only [<a>iff_def</a>, <a>lcm_zero_left</a>, <a>lcm_zero_right</a>,\n        <a>zero_dvd_iff</a>, <a>dvd_zero</a>, <a>eq_self_iff_true</a>, <a>and_true_iff</a>, <a>imp_true_iff</a>]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "iff_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1388, 9], "def_end_pos": [1388, 16]}, {"full_name": "GCDMonoid.lcm_zero_left", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [288, 3], "def_end_pos": [288, 16]}, {"full_name": "GCDMonoid.lcm_zero_right", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [290, 3], "def_end_pos": [290, 17]}, {"full_name": "zero_dvd_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}, {"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "imp_true_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1412, 9], "def_end_pos": [1412, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : a = 0 \u2228 b = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c", "state_after": "no goals"}, {"tactic": "obtain \u27e8h1, h2\u27e9 := not_or.1 h", "annotated_tactic": ["obtain \u27e8h1, h2\u27e9 := <a>not_or</a>.1 h", [{"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c", "state_after": "case neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c"}, {"tactic": "have h : gcd a b \u2260 0 := fun H => h1 ((gcd_eq_zero_iff _ _).1 H).1", "annotated_tactic": ["have h : <a>gcd</a> a b \u2260 0 := fun H => h1 ((<a>gcd_eq_zero_iff</a> _ _).1 H).1", [{"full_name": "GCDMonoid.gcd", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [276, 3], "def_end_pos": [276, 6]}, {"full_name": "gcd_eq_zero_iff", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 24]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c", "state_after": "case neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh\u271d : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\nh : gcd a b \u2260 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c"}, {"tactic": "rw [\u2190 mul_dvd_mul_iff_left h, (gcd_mul_lcm a b).dvd_iff_dvd_left, \u2190\n  (gcd_mul_right' c a b).dvd_iff_dvd_right, dvd_gcd_iff, mul_comm b c, mul_dvd_mul_iff_left h1,\n  mul_dvd_mul_iff_right h2, and_comm]", "annotated_tactic": ["rw [\u2190 <a>mul_dvd_mul_iff_left</a> h, (<a>gcd_mul_lcm</a> a b).<a>dvd_iff_dvd_left</a>, \u2190\n      (<a>gcd_mul_right'</a> c a b).<a>dvd_iff_dvd_right</a>, <a>dvd_gcd_iff</a>, <a>mul_comm</a> b c, <a>mul_dvd_mul_iff_left</a> h1,\n      <a>mul_dvd_mul_iff_right</a> h2, <a>and_comm</a>]", [{"full_name": "mul_dvd_mul_iff_left", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [49, 9], "def_end_pos": [49, 29]}, {"full_name": "gcd_mul_lcm", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [333, 9], "def_end_pos": [333, 20]}, {"full_name": "Associated.dvd_iff_dvd_left", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [604, 9], "def_end_pos": [604, 36]}, {"full_name": "gcd_mul_right'", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 23]}, {"full_name": "Associated.dvd_iff_dvd_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}, {"full_name": "dvd_gcd_iff", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "mul_dvd_mul_iff_left", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [49, 9], "def_end_pos": [49, 29]}, {"full_name": "mul_dvd_mul_iff_right", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [56, 9], "def_end_pos": [56, 30]}, {"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh\u271d : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\nh : gcd a b \u2260 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean", "full_name": "ProbabilityTheory.IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction", "start": [390, 1], "end": [392, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isLittleO_bot", "start": [654, 1], "end": [655, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Inverse.lean", "full_name": "FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos", "start": [501, 1], "end": [571, 52], "traced_tactics": [{"tactic": "obtain \u27e8C, r, Cpos, rpos, ple\u27e9 :\n  \u2203 (C r : _) (_ : 0 < C) (_ : 0 < r), \u2200 n : \u2115, \u2016p n\u2016 \u2264 C * r ^ n :=\n  le_mul_pow_of_radius_pos p hp", "annotated_tactic": ["obtain \u27e8C, r, Cpos, rpos, ple\u27e9 :\n    \u2203 (C r : _) (_ : 0 < C) (_ : 0 < r), \u2200 n : \u2115, \u2016p n\u2016 \u2264 C * r ^ n :=\n    <a>le_mul_pow_of_radius_pos</a> p hp", [{"full_name": "FormalMultilinearSeries.le_mul_pow_of_radius_pos", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [329, 9], "def_end_pos": [329, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\n\u22a2 0 < (p.rightInv i).radius"}, {"tactic": "let I := \u2016(i.symm : F \u2192L[\ud835\udd5c] E)\u2016", "annotated_tactic": ["let I := \u2016(i.symm : F \u2192L[\ud835\udd5c] E)\u2016", []], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\n\u22a2 0 < (p.rightInv i).radius"}, {"tactic": "obtain \u27e8a, apos, ha1, ha2\u27e9 :\n  \u2203 (a : _) (apos : 0 < a),\n    2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2 := by\n  have :\n    Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0)\n      (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n    tendsto_const_nhds.mul tendsto_id\n  have A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by\n    apply (tendsto_order.1 this).2; simp [zero_lt_one]\n  have : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) :=\n    tendsto_const_nhds.mul tendsto_id\n  have B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by\n    apply (tendsto_order.1 this).2; simp [zero_lt_one]\n  have C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by\n    filter_upwards [self_mem_nhdsWithin] with _ ha using ha\n  rcases (C.and ((A.and B).filter_mono inf_le_left)).exists with \u27e8a, ha\u27e9\n  exact \u27e8a, ha.1, ha.2.1.le, ha.2.2.le\u27e9", "annotated_tactic": ["obtain \u27e8a, apos, ha1, ha2\u27e9 :\n    \u2203 (a : _) (apos : 0 < a),\n      2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2 := by\n    have :\n      <a>Tendsto</a> (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0)\n        (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n      tendsto_const_nhds.mul <a>tendsto_id</a>\n    have A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by\n      apply (<a>tendsto_order</a>.1 this).2; simp [<a>zero_lt_one</a>]\n    have : <a>Tendsto</a> (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) :=\n      tendsto_const_nhds.mul <a>tendsto_id</a>\n    have B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by\n      apply (<a>tendsto_order</a>.1 this).2; simp [<a>zero_lt_one</a>]\n    have C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with _ ha using ha\n    rcases (C.and ((A.and B).<a>filter_mono</a> <a>inf_le_left</a>)).<a>exists</a> with \u27e8a, ha\u27e9\n    exact \u27e8a, ha.1, ha.2.1.<a>le</a>, ha.2.2.<a>le</a>\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "Filter.Eventually.filter_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 31]}, {"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [358, 9], "def_end_pos": [358, 20]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1328, 9], "def_end_pos": [1328, 26]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\n\u22a2 0 < (p.rightInv i).radius"}, {"tactic": "let S n := \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016", "annotated_tactic": ["let S n := \u2211 k \u2208 <a>Ico</a> 1 n, a ^ k * \u2016p.rightInv i k\u2016", [{"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 0 < (p.rightInv i).radius"}, {"tactic": "let a' : NNReal := \u27e8a, apos.le\u27e9", "annotated_tactic": ["let a' : <a>NNReal</a> := \u27e8a, apos.le\u27e9", [{"full_name": "NNReal", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 11]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\n\u22a2 0 < (p.rightInv i).radius"}, {"tactic": "suffices H : (a' : ENNReal) \u2264 (p.rightInv i).radius by\n  apply lt_of_lt_of_le _ H\n  simpa only [ENNReal.coe_pos]", "annotated_tactic": ["suffices H : (a' : <a>ENNReal</a>) \u2264 (p.rightInv i).<a>radius</a> by\n    apply <a>lt_of_lt_of_le</a> _ H\n    -- Prior to leanprover/lean4#2734, this was `exact_mod_cast apos`.\n    simpa only [<a>ENNReal.coe_pos</a>]", [{"full_name": "ENNReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [98, 5], "def_end_pos": [98, 12]}, {"full_name": "FormalMultilinearSeries.radius", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [123, 5], "def_end_pos": [123, 11]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [402, 28], "def_end_pos": [402, 35]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\n\u22a2 0 < (p.rightInv i).radius", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\n\u22a2 \u2191a' \u2264 (p.rightInv i).radius"}, {"tactic": "apply le_radius_of_bound _ ((I + 1) * a) fun n => ?_", "annotated_tactic": ["apply <a>le_radius_of_bound</a> _ ((I + 1) * a) fun n => ?_", [{"full_name": "FormalMultilinearSeries.le_radius_of_bound", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\n\u22a2 \u2191a' \u2264 (p.rightInv i).radius", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a"}, {"tactic": "by_cases hn : n = 0", "annotated_tactic": ["by_cases hn : n = 0", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a", "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a"}, {"tactic": "have :\n  Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0)\n    (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n  tendsto_const_nhds.mul tendsto_id", "annotated_tactic": ["have :\n      <a>Tendsto</a> (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0)\n        (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n      tendsto_const_nhds.mul <a>tendsto_id</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "have A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by\n  apply (tendsto_order.1 this).2; simp [zero_lt_one]", "annotated_tactic": ["have A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by\n      apply (<a>tendsto_order</a>.1 this).2; simp [<a>zero_lt_one</a>]", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "have : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) :=\n  tendsto_const_nhds.mul tendsto_id", "annotated_tactic": ["have : <a>Tendsto</a> (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) :=\n      tendsto_const_nhds.mul <a>tendsto_id</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "have B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by\n  apply (tendsto_order.1 this).2; simp [zero_lt_one]", "annotated_tactic": ["have B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by\n      apply (<a>tendsto_order</a>.1 this).2; simp [<a>zero_lt_one</a>]", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "have C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by\n  filter_upwards [self_mem_nhdsWithin] with _ ha using ha", "annotated_tactic": ["have C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with _ ha using ha", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[>] 0, 0 < a\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "rcases (C.and ((A.and B).filter_mono inf_le_left)).exists with \u27e8a, ha\u27e9", "annotated_tactic": ["rcases (C.and ((A.and B).<a>filter_mono</a> <a>inf_le_left</a>)).<a>exists</a> with \u27e8a, ha\u27e9", [{"full_name": "Filter.Eventually.filter_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 31]}, {"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [358, 9], "def_end_pos": [358, 20]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1328, 9], "def_end_pos": [1328, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[>] 0, 0 < a\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "case intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[>] 0, 0 < a\na : \u211d\nha : 0 < a \u2227 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1 \u2227 r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2"}, {"tactic": "exact \u27e8a, ha.1, ha.2.1.le, ha.2.2.le\u27e9", "annotated_tactic": ["exact \u27e8a, ha.1, ha.2.1.<a>le</a>, ha.2.2.<a>le</a>\u27e9", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[>] 0, 0 < a\na : \u211d\nha : 0 < a \u2227 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1 \u2227 r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a, \u2203 (_ : 0 < a), 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2", "state_after": "no goals"}, {"tactic": "apply (tendsto_order.1 this).2", "annotated_tactic": ["apply (<a>tendsto_order</a>.1 this).2", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 1 > 2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0"}, {"tactic": "simp [zero_lt_one]", "annotated_tactic": ["simp [<a>zero_lt_one</a>]", [{"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 1 > 2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0", "state_after": "no goals"}, {"tactic": "apply (tendsto_order.1 this).2", "annotated_tactic": ["apply (<a>tendsto_order</a>.1 this).2", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2", "state_after": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 1 / 2 > r * (I + 1) * 0"}, {"tactic": "simp [zero_lt_one]", "annotated_tactic": ["simp [<a>zero_lt_one</a>]", [{"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 1 / 2 > r * (I + 1) * 0", "state_after": "no goals"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with _ ha using ha", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with _ ha using ha", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd[>] 0, 0 < a", "state_after": "no goals"}, {"tactic": "apply Nat.le_induction", "annotated_tactic": ["apply <a>Nat.le_induction</a>", [{"full_name": "Nat.le_induction", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [953, 7], "def_end_pos": [953, 19]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a", "state_after": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 S 1 \u2264 (I + 1) * a\n\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a \u2192 S (n + 1) \u2264 (I + 1) * a"}, {"tactic": "simp only [S]", "annotated_tactic": ["simp only [S]", []], "state_before": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 S 1 \u2264 (I + 1) * a", "state_after": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 \u2211 k \u2208 Ico 1 1, a ^ k * \u2016p.rightInv i k\u2016 \u2264 (I + 1) * a"}, {"tactic": "rw [Ico_eq_empty_of_le (le_refl 1), sum_empty]", "annotated_tactic": ["rw [<a>Ico_eq_empty_of_le</a> (<a>le_refl</a> 1), <a>sum_empty</a>]", [{"full_name": "Finset.Ico_eq_empty_of_le", "def_path": "Mathlib/Order/Interval/Finset/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 27]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}]], "state_before": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 \u2211 k \u2208 Ico 1 1, a ^ k * \u2016p.rightInv i k\u2016 \u2264 (I + 1) * a", "state_after": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 0 \u2264 (I + 1) * a"}, {"tactic": "exact mul_nonneg (add_nonneg (norm_nonneg _) zero_le_one) apos.le", "annotated_tactic": ["exact <a>mul_nonneg</a> (<a>add_nonneg</a> (<a>norm_nonneg</a> _) <a>zero_le_one</a>) apos.le", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "case base\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 0 \u2264 (I + 1) * a", "state_after": "no goals"}, {"tactic": "intro n one_le_n hn", "annotated_tactic": ["intro n one_le_n hn", []], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a \u2192 S (n + 1) \u2264 (I + 1) * a", "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\n\u22a2 S (n + 1) \u2264 (I + 1) * a"}, {"tactic": "have In : 2 \u2264 n + 1 := by linarith only [one_le_n]", "annotated_tactic": ["have In : 2 \u2264 n + 1 := by linarith only [one_le_n]", []], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\n\u22a2 S (n + 1) \u2264 (I + 1) * a", "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\n\u22a2 S (n + 1) \u2264 (I + 1) * a"}, {"tactic": "have rSn : r * S n \u2264 1 / 2 :=\n  calc\n    r * S n \u2264 r * ((I + 1) * a) := by gcongr\n    _ \u2264 1 / 2 := by rwa [\u2190 mul_assoc]", "annotated_tactic": ["have rSn : r * S n \u2264 1 / 2 :=\n        calc\n          r * S n \u2264 r * ((I + 1) * a) := by gcongr\n          _ \u2264 1 / 2 := by rwa [\u2190 <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\n\u22a2 S (n + 1) \u2264 (I + 1) * a", "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 S (n + 1) \u2264 (I + 1) * a"}, {"tactic": "linarith only [one_le_n]", "annotated_tactic": ["linarith only [one_le_n]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\n\u22a2 2 \u2264 n + 1", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\n\u22a2 r * S n \u2264 r * ((I + 1) * a)", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mul_assoc]", "annotated_tactic": ["rwa [\u2190 <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\n\u22a2 r * ((I + 1) * a) \u2264 1 / 2", "state_after": "no goals"}, {"tactic": "rw [geom_sum_Ico' _ In]", "annotated_tactic": ["rw [<a>geom_sum_Ico'</a> _ In]", [{"full_name": "geom_sum_Ico'", "def_path": "Mathlib/Algebra/GeomSum.lean", "def_pos": [370, 9], "def_end_pos": [370, 22]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * \u2211 k \u2208 Ico 2 (n + 1), (r * S n) ^ k =\n    I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 r * S n \u2260 1"}, {"tactic": "exact ne_of_lt (rSn.trans_lt (by norm_num))", "annotated_tactic": ["exact <a>ne_of_lt</a> (rSn.trans_lt (by norm_num))", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 r * S n \u2260 1", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 1 / 2 < 1", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) \u2264 I * a + I * C * ((r * S n) ^ 2 / (1 / 2))", "state_after": "case bc.h.hac\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 (r * S n) ^ 2 - (r * S n) ^ (n + 1) \u2264 (r * S n) ^ 2\n\ncase bc.h.hbd\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 1 / 2 \u2264 1 - r * S n"}, {"tactic": "simp only [sub_le_self_iff]", "annotated_tactic": ["simp only [<a>sub_le_self_iff</a>]", [{"full_name": "sub_le_self_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [359, 3], "def_end_pos": [359, 14]}]], "state_before": "case bc.h.hac\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 (r * S n) ^ 2 - (r * S n) ^ (n + 1) \u2264 (r * S n) ^ 2", "state_after": "case bc.h.hac\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 0 \u2264 (r * S n) ^ (n + 1)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case bc.h.hac\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 0 \u2264 (r * S n) ^ (n + 1)", "state_after": "no goals"}, {"tactic": "linarith only [rSn]", "annotated_tactic": ["linarith only [rSn]", []], "state_before": "case bc.h.hbd\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 1 / 2 \u2264 1 - r * S n", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) = I * a + 2 * I * C * (r * S n) ^ 2", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + 2 * I * C * (r * S n) ^ 2 \u2264 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nrSn : r * S n \u2264 1 / 2\n\u22a2 (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a \u2264 (I + 1) * a", "state_after": "no goals"}, {"tactic": "apply lt_of_lt_of_le _ H", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> _ H", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nH : \u2191a' \u2264 (p.rightInv i).radius\n\u22a2 0 < (p.rightInv i).radius", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nH : \u2191a' \u2264 (p.rightInv i).radius\n\u22a2 0 < \u2191a'"}, {"tactic": "simpa only [ENNReal.coe_pos]", "annotated_tactic": ["simpa only [<a>ENNReal.coe_pos</a>]", [{"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [402, 28], "def_end_pos": [402, 35]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nH : \u2191a' \u2264 (p.rightInv i).radius\n\u22a2 0 < \u2191a'", "state_after": "no goals"}, {"tactic": "have : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016 := by congr <;> try rw [hn]", "annotated_tactic": ["have : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016 := by congr <;> try rw [hn]", []], "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a", "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\nthis : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a"}, {"tactic": "simp only [this, norm_zero, zero_mul, rightInv_coeff_zero]", "annotated_tactic": ["simp only [this, <a>norm_zero</a>, <a>zero_mul</a>, <a>rightInv_coeff_zero</a>]", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [507, 30], "def_end_pos": [507, 39]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "FormalMultilinearSeries.rightInv_coeff_zero", "def_path": "Mathlib/Analysis/Analytic/Inverse.lean", "def_pos": [177, 9], "def_end_pos": [177, 28]}]], "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\nthis : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a", "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\nthis : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016\n\u22a2 0 \u2264 (I + 1) * a"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\nthis : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016\n\u22a2 0 \u2264 (I + 1) * a", "state_after": "no goals"}, {"tactic": "congr <;> try rw [hn]", "annotated_tactic": ["congr <;> try rw [hn]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\n\u22a2 \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016", "state_after": "no goals"}, {"tactic": "rw [hn]", "annotated_tactic": ["rw [hn]", []], "state_before": "case h.e_1.h.e_10\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => UniformSpace.toTopologicalSpace) fun i => UniformSpace.toTopologicalSpace", "state_after": "no goals"}, {"tactic": "have one_le_n : 1 \u2264 n := bot_lt_iff_ne_bot.2 hn", "annotated_tactic": ["have one_le_n : 1 \u2264 n := <a>bot_lt_iff_ne_bot</a>.2 hn", [{"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [339, 9], "def_end_pos": [339, 26]}]], "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a", "state_after": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a"}, {"tactic": "calc\n  \u2016p.rightInv i n\u2016 * (a' : \u211d) ^ n = a ^ n * \u2016p.rightInv i n\u2016 := mul_comm _ _\n  _ \u2264 \u2211 k \u2208 Ico 1 (n + 1), a ^ k * \u2016p.rightInv i k\u2016 :=\n    (haveI : \u2200 k \u2208 Ico 1 (n + 1), 0 \u2264 a ^ k * \u2016p.rightInv i k\u2016 := fun k _ => by positivity\n    single_le_sum this (by simp [one_le_n]))\n  _ \u2264 (I + 1) * a := IRec (n + 1) (by norm_num)", "annotated_tactic": ["calc\n      \u2016p.rightInv i n\u2016 * (a' : \u211d) ^ n = a ^ n * \u2016p.rightInv i n\u2016 := <a>mul_comm</a> _ _\n      _ \u2264 \u2211 k \u2208 <a>Ico</a> 1 (n + 1), a ^ k * \u2016p.rightInv i k\u2016 :=\n        (haveI : \u2200 k \u2208 <a>Ico</a> 1 (n + 1), 0 \u2264 a ^ k * \u2016p.rightInv i k\u2016 := fun k _ => by positivity\n        <a>single_le_sum</a> this (by simp [one_le_n]))\n      _ \u2264 (I + 1) * a := IRec (n + 1) (by norm_num)", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}, {"full_name": "Finset.single_le_sum", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [198, 15], "def_end_pos": [198, 28]}]], "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\n\u22a2 \u2016p.rightInv i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\nk : \u2115\nx\u271d : k \u2208 Ico 1 (n + 1)\n\u22a2 0 \u2264 a ^ k * \u2016p.rightInv i k\u2016", "state_after": "no goals"}, {"tactic": "simp [one_le_n]", "annotated_tactic": ["simp [one_le_n]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\nthis : \u2200 k \u2208 Ico 1 (n + 1), 0 \u2264 a ^ k * \u2016p.rightInv i k\u2016\n\u22a2 n \u2208 Ico 1 (n + 1)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < p.radius\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191i.symm\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k \u2208 Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := \u27e8a, \u22ef\u27e9\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\n\u22a2 1 \u2264 n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "tprod_eq_mul_tprod_ite'", "start": [637, 1], "end": [649, 82], "traced_tactics": [{"tactic": "split_ifs with h <;> simp [update_apply, h]", "annotated_tactic": ["split_ifs with h <;> simp [<a>update_apply</a>, h]", [{"full_name": "Function.update_apply", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [568, 9], "def_end_pos": [568, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : ContinuousMul \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Multipliable (update f b 1)\nn : \u03b2\n\u22a2 f n = (if n = b then f n else 1) * update f b 1 n", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : ContinuousMul \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Multipliable (update f b 1)\n\u22a2 (\u220f' (x : \u03b2), if x = b then f x else 1) * \u220f' (x : \u03b2), update f b 1 x =\n    (if b = b then f b else 1) * \u220f' (x : \u03b2), update f b 1 x", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : ContinuousMul \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Multipliable (update f b 1)\n\u22a2 (\u220f' (x : \u03b2), if x = b then f x else 1) = if b = b then f b else 1"}, {"tactic": "exact tprod_eq_mulSingle b fun b' hb' \u21a6 if_neg hb'", "annotated_tactic": ["exact <a>tprod_eq_mulSingle</a> b fun b' hb' \u21a6 <a>if_neg</a> hb'", [{"full_name": "tprod_eq_mulSingle", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 27]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}]], "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : ContinuousMul \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Multipliable (update f b 1)\n\u22a2 (\u220f' (x : \u03b2), if x = b then f x else 1) = if b = b then f b else 1", "state_after": "no goals"}, {"tactic": "simp only [update, eq_self_iff_true, if_true, eq_rec_constant, dite_eq_ite]", "annotated_tactic": ["simp only [<a>update</a>, <a>eq_self_iff_true</a>, <a>if_true</a>, <a>eq_rec_constant</a>, <a>dite_eq_ite</a>]", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [553, 5], "def_end_pos": [553, 11]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "if_true", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [24, 17], "def_end_pos": [24, 24]}, {"full_name": "eq_rec_constant", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [39, 17], "def_end_pos": [39, 32]}, {"full_name": "dite_eq_ite", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [55, 17], "def_end_pos": [55, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : ContinuousMul \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : Multipliable (update f b 1)\n\u22a2 (if b = b then f b else 1) * \u220f' (x : \u03b2), update f b 1 x = f b * \u220f' (x : \u03b2), if x = b then 1 else f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_congr", "start": [162, 1], "end": [163, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "full_name": "Submodule.coe_smul_of_tower", "start": [309, 1], "end": [311, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.mono_of_isZero_kernel'", "start": [244, 1], "end": [247, 48], "traced_tactics": [{"tactic": "obtain \u27e8a, ha\u27e9 := KernelFork.IsLimit.lift' hc _ hg", "annotated_tactic": ["obtain \u27e8a, ha\u27e9 := <a>KernelFork.IsLimit.lift'</a> hc _ hg", [{"full_name": "CategoryTheory.Limits.KernelFork.IsLimit.lift'", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [136, 5], "def_end_pos": [136, 29]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng\u271d g' : Q \u27f6 R\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nh : IsZero c.pt\nP\u271d : C\ng : P\u271d \u27f6 X\nhg : g \u226b f = 0\n\u22a2 g = 0", "state_after": "case mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng\u271d g' : Q \u27f6 R\nX Y : C\nf : X \u27f6 Y\nc : KernelFork f\nhc : IsLimit c\nh : IsZero c.pt\nP\u271d : C\ng : P\u271d \u27f6 X\nhg : g \u226b f = 0\na : P\u271d \u27f6 c.pt\nha : a \u226b Fork.\u03b9 c = g\n\u22a2 g = 0"}, {"tactic": "rw [\u2190 ha, h.eq_of_tgt a 0, Limits.zero_comp]", "annotated_tactic": ["rw [\u2190 ha, h.eq_of_tgt a 0, <a>Limits.zero_comp</a>]", [{"full_name": "CategoryTheory.Limits.zero_comp", "def_path": 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"start": [168, 1], "end": [169, 73], "traced_tactics": [{"tactic": "simpa only [one_smul] using mem_balancedCoreAux_iff.1 hx 1 norm_one.ge", "annotated_tactic": ["simpa only [<a>one_smul</a>] using <a>mem_balancedCoreAux_iff</a>.1 hx 1 norm_one.ge", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}, {"full_name": "mem_balancedCoreAux_iff", "def_path": "Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean", "def_pos": [104, 9], "def_end_pos": [104, 32]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t s : Set E\nx : E\nhx : x \u2208 balancedCoreAux \ud835\udd5c s\n\u22a2 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/DropRight.lean", "full_name": "List.rtakeWhile_idempotent", "start": [244, 1], "end": [246, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.swap_smul_self_smul", "start": [546, 1], "end": [547, 55], "traced_tactics": [{"tactic": "simp [smul_smul]", "annotated_tactic": ["simp [<a>smul_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [446, 7], "def_end_pos": [446, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : MulAction (Perm \u03b1) \u03b2\ni j : \u03b1\nx : \u03b2\n\u22a2 swap i j \u2022 swap i j \u2022 x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "BddBelow.isBoundedUnder", "start": [129, 1], "end": [132, 74], "traced_tactics": [{"tactic": "simpa [mem_lowerBounds] using hb", "annotated_tactic": ["simpa [<a>mem_lowerBounds</a>] using hb", [{"full_name": "mem_lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : Filter \u03b1\ninst\u271d : Preorder \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\nb : \u03b1\nhb : b \u2208 lowerBounds (range u)\n\u22a2 \u2200 (x : \u03b2), u x \u2265 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "PadicSeq.stationaryPoint_spec", "start": [109, 1], "end": [112, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed_encode", "start": [882, 1], "end": [884, 47], "traced_tactics": [{"tactic": "rw [ennrealRatEmbed, Encodable.encodek]", "annotated_tactic": ["rw [<a>ennrealRatEmbed</a>, <a>Encodable.encodek</a>]", [{"full_name": "MeasureTheory.SimpleFunc.ennrealRatEmbed", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [878, 5], "def_end_pos": [878, 20]}, {"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [52, 3], "def_end_pos": [52, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nq : \u211a\n\u22a2 ennrealRatEmbed (Encodable.encode q) = \u2191(\u2191q).toNNReal", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nq : \u211a\n\u22a2 ENNReal.ofReal \u2191((some q).getD 0) = \u2191(\u2191q).toNNReal"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\nK : Type u_5\nq : \u211a\n\u22a2 ENNReal.ofReal \u2191((some q).getD 0) = \u2191(\u2191q).toNNReal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "full_name": "CauSeq.coe_mul", "start": [303, 1], "end": [304, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Seq.lean", "full_name": "Stream'.Seq.not_terminates_iff", "start": [129, 1], "end": [130, 101], "traced_tactics": [{"tactic": "simp only [Terminates, TerminatedAt, \u2190 Ne.eq_def, Option.ne_none_iff_isSome, not_exists, iff_self]", "annotated_tactic": ["simp only [<a>Terminates</a>, <a>TerminatedAt</a>, \u2190 Ne.eq_def, <a>Option.ne_none_iff_isSome</a>, <a>not_exists</a>, <a>iff_self</a>]", [{"full_name": "Stream'.Seq.Terminates", "def_path": "Mathlib/Data/Seq/Seq.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Stream'.Seq.TerminatedAt", "def_path": "Mathlib/Data/Seq/Seq.lean", "def_pos": [115, 5], "def_end_pos": [115, 17]}, {"full_name": "Option.ne_none_iff_isSome", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "iff_self", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [126, 17], "def_end_pos": [126, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u00acs.Terminates \u2194 \u2200 (n : \u2115), (s.get? n).isSome = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformContinuous_snd", "start": [1622, 1], "end": [1624, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Constructions.lean", "full_name": "Units.continuous_val", "start": [167, 1], "end": [168, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "full_name": "CategoryTheory.Sieve.functorPullback_monotone", "start": [730, 1], "end": [732, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "full_name": "MulAction.card_orbit_mul_card_stabilizer_eq_card_group", "start": [210, 1], "end": [213, 83], "traced_tactics": [{"tactic": "rw [\u2190 Fintype.card_prod, Fintype.card_congr (orbitProdStabilizerEquivGroup \u03b1 b)]", "annotated_tactic": ["rw [\u2190 <a>Fintype.card_prod</a>, <a>Fintype.card_congr</a> (<a>orbitProdStabilizerEquivGroup</a> \u03b1 b)]", [{"full_name": "Fintype.card_prod", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [59, 9], "def_end_pos": [59, 26]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [151, 9], "def_end_pos": [151, 19]}, {"full_name": "MulAction.orbitProdStabilizerEquivGroup", "def_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "def_pos": [202, 19], "def_end_pos": [202, 48]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2075 inst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : MulAction \u03b1 \u03b2\nx b : \u03b2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : Fintype \u2191(orbit \u03b1 b)\ninst\u271d : Fintype \u21a5(stabilizer \u03b1 b)\n\u22a2 Fintype.card \u2191(orbit \u03b1 b) * Fintype.card \u21a5(stabilizer \u03b1 b) = Fintype.card \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_indexOf", "start": [1144, 1], "end": [1145, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_parallelepiped", "start": [581, 1], "end": [590, 58], "traced_tactics": [{"tactic": "have : FiniteDimensional \u211d G := FiniteDimensional.of_fintype_basis b", "annotated_tactic": ["have : <a>FiniteDimensional</a> \u211d G := <a>FiniteDimensional.of_fintype_basis</a> b", [{"full_name": "FiniteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [79, 8], "def_end_pos": [79, 25]}, {"full_name": "FiniteDimensional.of_fintype_basis", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\n\u22a2 b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|", "state_after": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|"}, {"tactic": "have A : parallelepiped v = b.constr \u2115 v '' parallelepiped b := by\n  rw [image_parallelepiped]\n  refine congr_arg _ <| funext fun i \u21a6 ?_\n  exact (b.constr_basis \u2115 v i).symm", "annotated_tactic": ["have A : <a>parallelepiped</a> v = b.constr \u2115 v '' <a>parallelepiped</a> b := by\n    rw [<a>image_parallelepiped</a>]\n    -- Porting note: was `congr 1 with i` but Lean 4 `congr` applies `ext` first\n    refine <a>congr_arg</a> _ <| <a>funext</a> fun i \u21a6 ?_\n    exact (b.constr_basis \u2115 v i).<a>symm</a>", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [48, 5], "def_end_pos": [48, 19]}, {"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [48, 5], "def_end_pos": [48, 19]}, {"full_name": "image_parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [67, 9], "def_end_pos": [67, 29]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|", "state_after": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\nA : parallelepiped v = \u21d1((b.constr \u2115) v) '' parallelepiped \u21d1b\n\u22a2 b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|"}, {"tactic": "rw [A, addHaar_image_linearMap, b.addHaar_self, mul_one, \u2190 LinearMap.det_toMatrix b,\n  \u2190 Basis.toMatrix_eq_toMatrix_constr, Basis.det_apply]", "annotated_tactic": ["rw [A, <a>addHaar_image_linearMap</a>, b.addHaar_self, <a>mul_one</a>, \u2190 <a>LinearMap.det_toMatrix</a> b,\n    \u2190 <a>Basis.toMatrix_eq_toMatrix_constr</a>, <a>Basis.det_apply</a>]", [{"full_name": "MeasureTheory.Measure.addHaar_image_linearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [313, 9], "def_end_pos": [313, 32]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "LinearMap.det_toMatrix", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "Basis.toMatrix_eq_toMatrix_constr", "def_path": "Mathlib/LinearAlgebra/Matrix/Basis.lean", "def_pos": [66, 9], "def_end_pos": [66, 36]}, {"full_name": "Basis.det_apply", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [526, 9], "def_end_pos": [526, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\nA : parallelepiped v = \u21d1((b.constr \u2115) v) '' parallelepiped \u21d1b\n\u22a2 b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|", "state_after": "no goals"}, {"tactic": "rw [image_parallelepiped]", "annotated_tactic": ["rw [<a>image_parallelepiped</a>]", [{"full_name": "image_parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [67, 9], "def_end_pos": [67, 29]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 parallelepiped v = \u21d1((b.constr \u2115) v) '' parallelepiped \u21d1b", "state_after": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 parallelepiped v = parallelepiped (\u21d1((b.constr \u2115) v) \u2218 \u21d1b)"}, {"tactic": "refine congr_arg _ <| funext fun i \u21a6 ?_", "annotated_tactic": ["refine <a>congr_arg</a> _ <| <a>funext</a> fun i \u21a6 ?_", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 parallelepiped v = parallelepiped (\u21d1((b.constr \u2115) v) \u2218 \u21d1b)", "state_after": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\ni : \u03b9\n\u22a2 v i = (\u21d1((b.constr \u2115) v) \u2218 \u21d1b) i"}, {"tactic": "exact (b.constr_basis \u2115 v i).symm", "annotated_tactic": ["exact (b.constr_basis \u2115 v i).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\ni : \u03b9\n\u22a2 v i = (\u21d1((b.constr \u2115) v) \u2218 \u21d1b) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.toRingEquiv_symm", "start": [390, 1], "end": [391, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/IsometricSMul.lean", "full_name": "edist_mul_right", "start": [114, 1], "end": [116, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Product.lean", "full_name": "ContMDiffWithinAt.prod_mk", "start": [59, 1], "end": [63, 41], "traced_tactics": [{"tactic": "rw [contMDiffWithinAt_iff] at *", "annotated_tactic": ["rw [<a>contMDiffWithinAt_iff</a>] at *", [{"full_name": "contMDiffWithinAt_iff", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [319, 9], "def_end_pos": [319, 30]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf\u271d f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx : M\nm n : \u2115\u221e\nf : M \u2192 M'\ng : M \u2192 N'\nhf : ContMDiffWithinAt I I' n f s x\nhg : ContMDiffWithinAt I J' n g s x\n\u22a2 ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf\u271d f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx : M\nm n : \u2115\u221e\nf : M \u2192 M'\ng : M \u2192 N'\nhf :\n  ContinuousWithinAt f s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt I' (f x)) \u2218 f \u2218 \u2191(extChartAt I x).symm) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I)\n      (\u2191(extChartAt I x) x)\nhg :\n  ContinuousWithinAt g s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt J' (g x)) \u2218 g \u2218 \u2191(extChartAt I x).symm) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I)\n      (\u2191(extChartAt I x) x)\n\u22a2 ContinuousWithinAt (fun x => (f x, g x)) s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt (I'.prod J') (f x, g x)) \u2218 (fun x => (f x, g x)) \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)"}, {"tactic": "exact \u27e8hf.1.prod hg.1, hf.2.prod hg.2\u27e9", "annotated_tactic": ["exact \u27e8hf.1.<a>prod</a> hg.1, hf.2.<a>prod</a> hg.2\u27e9", [{"full_name": "ContinuousWithinAt.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1144, 9], "def_end_pos": [1144, 32]}, {"full_name": "ContDiffWithinAt.prod", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [530, 9], "def_end_pos": [530, 30]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf\u271d f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx : M\nm n : \u2115\u221e\nf : M \u2192 M'\ng : M \u2192 N'\nhf :\n  ContinuousWithinAt f s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt I' (f x)) \u2218 f \u2218 \u2191(extChartAt I x).symm) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I)\n      (\u2191(extChartAt I x) x)\nhg :\n  ContinuousWithinAt g s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt J' (g x)) \u2218 g \u2218 \u2191(extChartAt I x).symm) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I)\n      (\u2191(extChartAt I x) x)\n\u22a2 ContinuousWithinAt (fun x => (f x, g x)) s x \u2227\n    ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt (I'.prod J') (f x, g x)) \u2218 (fun x => (f x, g x)) \u2218 \u2191(extChartAt I x).symm)\n      (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.coordChange_coordChange", "start": [718, 1], "end": [722, 46], "traced_tactics": [{"tactic": "rw [coordChange, e\u2081.mk_coordChange _ h\u2081 h\u2082, \u2190 e\u2082.coe_coe, e\u2082.left_inv, coordChange]", "annotated_tactic": ["rw [<a>coordChange</a>, e\u2081.mk_coordChange _ h\u2081 h\u2082, \u2190 e\u2082.coe_coe, e\u2082.left_inv, <a>coordChange</a>]", [{"full_name": "Trivialization.coordChange", "def_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "def_pos": [691, 5], "def_end_pos": [691, 16]}, {"full_name": "Trivialization.coordChange", "def_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "def_pos": [691, 5], "def_end_pos": [691, 16]}]], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 e\u2083 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 e\u2082.coordChange e\u2083 b (e\u2081.coordChange e\u2082 b x) = e\u2081.coordChange e\u2083 b x", "state_after": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 e\u2083 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 \u2191e\u2081.symm (b, x) \u2208 e\u2082.source"}, {"tactic": "rwa [e\u2082.mem_source, e\u2081.proj_symm_apply' h\u2081]", "annotated_tactic": ["rwa [e\u2082.mem_source, e\u2081.proj_symm_apply' h\u2081]", []], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 e\u2083 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 \u2191e\u2081.symm (b, x) \u2208 e\u2082.source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.RightInverse.leftInverse_of_surjective", "start": [382, 1], "end": [384, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/TensorProduct.lean", "full_name": "TensorProduct.LieModule.lie_tmul_right", "start": [80, 1], "end": [83, 51], "traced_tactics": [{"tactic": "simp only [hasBracketAux, LinearMap.rTensor_tmul, toEnd_apply_apply,\n  LinearMap.add_apply, LinearMap.lTensor_tmul]", "annotated_tactic": ["simp only [<a>hasBracketAux</a>, <a>LinearMap.rTensor_tmul</a>, <a>toEnd_apply_apply</a>,\n      <a>LinearMap.add_apply</a>, <a>LinearMap.lTensor_tmul</a>]", [{"full_name": "TensorProduct.LieModule.hasBracketAux", "def_path": "Mathlib/Algebra/Lie/TensorProduct.lean", "def_pos": [46, 5], "def_end_pos": [46, 18]}, {"full_name": "LinearMap.rTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1181, 9], "def_end_pos": [1181, 21]}, {"full_name": "LieModule.toEnd_apply_apply", "def_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "def_pos": [195, 3], "def_end_pos": [195, 8]}, {"full_name": "LinearMap.add_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [910, 9], "def_end_pos": [910, 18]}, {"full_name": "LinearMap.lTensor_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 21]}]], "state_before": "R : Type u\ninst\u271d\u00b9\u2078 : CommRing R\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\nQ : Type w\u2083\ninst\u271d\u00b9\u2077 : LieRing L\ninst\u271d\u00b9\u2076 : LieAlgebra R L\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : LieRingModule L N\ninst\u271d\u2078 : LieModule R L N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L P\ninst\u271d\u2074 : LieModule R L P\ninst\u271d\u00b3 : AddCommGroup Q\ninst\u271d\u00b2 : Module R Q\ninst\u271d\u00b9 : LieRingModule L Q\ninst\u271d : LieModule R L Q\nx : L\nm : M\nn : N\n\u22a2 (hasBracketAux x) (m \u2297\u209c[R] n) = \u2045x, m\u2046 \u2297\u209c[R] n + m \u2297\u209c[R] \u2045x, n\u2046", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Abelian.lean", "full_name": "LieModule.coe_linearMap_maxTrivLinearMapEquivLieModuleHom", "start": [239, 1], "end": [240, 87], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : \u21a5(maxTrivSubmodule R L (M \u2192\u2097[R] N))\n\u22a2 \u2191(maxTrivLinearMapEquivLieModuleHom f) = \u2191f", "state_after": "case h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : \u21a5(maxTrivSubmodule R L (M \u2192\u2097[R] N))\nx\u271d : M\n\u22a2 \u2191(maxTrivLinearMapEquivLieModuleHom f) x\u271d = \u2191f x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : \u21a5(maxTrivSubmodule R L (M \u2192\u2097[R] N))\nx\u271d : M\n\u22a2 \u2191(maxTrivLinearMapEquivLieModuleHom f) x\u271d = \u2191f x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.IsNormal.bsup_eq", "start": [1958, 1], "end": [1960, 73], "traced_tactics": [{"tactic": "rw [\u2190 IsNormal.bsup.{u, u, v} H (fun x _ => x) h.1, bsup_id_limit h.2]", "annotated_tactic": ["rw [\u2190 <a>IsNormal.bsup</a>.{u, u, v} H (fun x _ => x) h.1, <a>bsup_id_limit</a> h.2]", [{"full_name": "Ordinal.IsNormal.bsup", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1505, 9], "def_end_pos": [1505, 22]}, {"full_name": "Ordinal.bsup_id_limit", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1916, 9], "def_end_pos": [1916, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nf : Ordinal.{u} \u2192 Ordinal.{max u v}\nH : IsNormal f\no : Ordinal.{u}\nh : o.IsLimit\n\u22a2 (o.bsup fun x x_1 => f x) = f o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.sign_mul", "start": [1397, 1], "end": [1405, 63], "traced_tactics": [{"tactic": "induction x, y using induction\u2082_symm_neg with\n| top_zero => simp only [zero_mul, mul_zero, sign_zero]\n| top_top => rfl\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => simp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n| top_pos _ h =>\n  rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]\n| neg_left h => rw [neg_mul, sign_neg, sign_neg, h, neg_mul]", "annotated_tactic": ["induction x, y using <a>induction\u2082_symm_neg</a> with\n  | top_zero => simp only [<a>zero_mul</a>, <a>mul_zero</a>, <a>sign_zero</a>]\n  | top_top => rfl\n  | symm h => rwa [<a>mul_comm</a>, <a>EReal.mul_comm</a>]\n  | coe_coe => simp only [\u2190 <a>coe_mul</a>, <a>sign_coe</a>, <a>_root_.sign_mul</a>, <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _)]\n  | top_pos _ h =>\n    rw [<a>top_mul_coe_of_pos</a> h, <a>sign_top</a>, <a>one_mul</a>, <a>sign_pos</a> (<a>EReal.coe_pos</a>.2 h)]\n  | neg_left h => rw [<a>neg_mul</a>, <a>sign_neg</a>, <a>sign_neg</a>, h, <a>neg_mul</a>]", [{"full_name": "EReal.induction\u2082_symm_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1216, 7], "def_end_pos": [1216, 26]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "sign_zero", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [346, 9], "def_end_pos": [346, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "EReal.mul_comm", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [219, 19], "def_end_pos": [219, 27]}, {"full_name": "EReal.coe_mul", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [168, 9], "def_end_pos": [168, 16]}, {"full_name": "EReal.sign_coe", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1384, 9], "def_end_pos": [1384, 17]}, {"full_name": "sign_mul", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [439, 9], "def_end_pos": [439, 17]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [353, 9], "def_end_pos": [353, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [122, 30], "def_end_pos": [122, 40]}, {"full_name": "EReal.top_mul_coe_of_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1106, 7], "def_end_pos": [1106, 25]}, {"full_name": "EReal.sign_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1377, 9], "def_end_pos": [1377, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "sign_pos", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [350, 9], "def_end_pos": [350, 17]}, {"full_name": "EReal.coe_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [395, 19], "def_end_pos": [395, 26]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "EReal.sign_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1391, 17], "def_end_pos": [1391, 25]}, {"full_name": "EReal.sign_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1391, 17], "def_end_pos": [1391, 25]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "x y : EReal\n\u22a2 sign (x * y) = sign x * sign y", "state_after": "no goals"}, {"tactic": "simp only [zero_mul, mul_zero, sign_zero]", "annotated_tactic": ["simp only [<a>zero_mul</a>, <a>mul_zero</a>, <a>sign_zero</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "sign_zero", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [346, 9], "def_end_pos": [346, 18]}]], "state_before": "case top_zero\n\n\u22a2 sign (\u22a4 * 0) = sign \u22a4 * sign 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case top_top\n\n\u22a2 sign (\u22a4 * \u22a4) = sign \u22a4 * sign \u22a4", "state_after": "no goals"}, {"tactic": "rwa [mul_comm, EReal.mul_comm]", "annotated_tactic": ["rwa [<a>mul_comm</a>, <a>EReal.mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "EReal.mul_comm", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [219, 19], "def_end_pos": [219, 27]}]], "state_before": "case symm\nx\u271d y\u271d : EReal\nh : sign (x\u271d * y\u271d) = sign x\u271d * sign y\u271d\n\u22a2 sign (y\u271d * x\u271d) = sign y\u271d * sign x\u271d", "state_after": "no goals"}, {"tactic": "simp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]", "annotated_tactic": ["simp only [\u2190 <a>coe_mul</a>, <a>sign_coe</a>, <a>_root_.sign_mul</a>, <a>ENNReal.ofReal_mul</a> (<a>abs_nonneg</a> _)]", [{"full_name": "EReal.coe_mul", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [168, 9], "def_end_pos": [168, 16]}, {"full_name": "EReal.sign_coe", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1384, 9], "def_end_pos": [1384, 17]}, {"full_name": "sign_mul", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [439, 9], "def_end_pos": [439, 17]}, {"full_name": "ENNReal.ofReal_mul", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [353, 9], "def_end_pos": [353, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [122, 30], "def_end_pos": [122, 40]}]], "state_before": "case coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 sign (\u2191x\u271d * \u2191y\u271d) = sign \u2191x\u271d * sign \u2191y\u271d", "state_after": "no goals"}, {"tactic": "rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]", "annotated_tactic": ["rw [<a>top_mul_coe_of_pos</a> h, <a>sign_top</a>, <a>one_mul</a>, <a>sign_pos</a> (<a>EReal.coe_pos</a>.2 h)]", [{"full_name": "EReal.top_mul_coe_of_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1106, 7], "def_end_pos": [1106, 25]}, {"full_name": "EReal.sign_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1377, 9], "def_end_pos": [1377, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "sign_pos", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [350, 9], "def_end_pos": [350, 17]}, {"full_name": "EReal.coe_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [395, 19], "def_end_pos": [395, 26]}]], "state_before": "case top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 sign (\u22a4 * \u2191x\u271d) = sign \u22a4 * sign \u2191x\u271d", "state_after": "no goals"}, {"tactic": "rw [neg_mul, sign_neg, sign_neg, h, neg_mul]", "annotated_tactic": ["rw [<a>neg_mul</a>, <a>sign_neg</a>, <a>sign_neg</a>, h, <a>neg_mul</a>]", [{"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "EReal.sign_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1391, 17], "def_end_pos": [1391, 25]}, {"full_name": "EReal.sign_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1391, 17], "def_end_pos": [1391, 25]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}]], "state_before": "case neg_left\nx\u271d y\u271d : EReal\nh : sign (x\u271d * y\u271d) = sign x\u271d * sign y\u271d\n\u22a2 sign (-x\u271d * y\u271d) = sign (-x\u271d) * sign y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sheaves/Presheaf.lean", "full_name": "TopCat.Presheaf.ext", "start": [59, 1], "end": [66, 10], "traced_tactics": [{"tactic": "apply NatTrans.ext", "annotated_tactic": ["apply <a>NatTrans.ext</a>", [{"full_name": "CategoryTheory.NatTrans.ext", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [47, 3], "def_end_pos": [47, 6]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\n\u22a2 f = g", "state_after": "case app\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\n\u22a2 f.app = g.app"}, {"tactic": "ext U", "annotated_tactic": ["ext U", []], "state_before": "case app\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\n\u22a2 f.app = g.app", "state_after": "case app.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 f.app U = g.app U"}, {"tactic": "induction U with | _ U => ?_", "annotated_tactic": ["induction U with | _ U => ?_", []], "state_before": "case app.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 f.app U = g.app U", "state_after": "case app.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\nU : Opens \u2191X\n\u22a2 f.app { unop := U } = g.app { unop := U }"}, {"tactic": "apply w", "annotated_tactic": ["apply w", []], "state_before": "case app.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), f.app { unop := U } = g.app { unop := U }\nU : Opens \u2191X\n\u22a2 f.app { unop := U } = g.app { unop := U }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.isGLB_mem", "start": [1964, 1], "end": [1967, 46], "traced_tactics": [{"tactic": "rw [\u2190 mem_coe]", "annotated_tactic": ["rw [\u2190 <a>mem_coe</a>]", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ni : \u03b1\ns : Finset \u03b1\nhis : IsGLB (\u2191s) i\nhs : s.Nonempty\n\u22a2 i \u2208 s", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ni : \u03b1\ns : Finset \u03b1\nhis : IsGLB (\u2191s) i\nhs : s.Nonempty\n\u22a2 i \u2208 \u2191s"}, {"tactic": "exact ((isGLB_iff_isLeast i s hs).mp his).1", "annotated_tactic": ["exact ((<a>isGLB_iff_isLeast</a> i s hs).<a>mp</a> his).1", [{"full_name": "Finset.isGLB_iff_isLeast", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1949, 9], "def_end_pos": [1949, 26]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ni : \u03b1\ns : Finset \u03b1\nhis : IsGLB (\u2191s) i\nhs : s.Nonempty\n\u22a2 i \u2208 \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "full_name": "fourierCoeffOn_of_hasDerivAt", "start": [553, 1], "end": [560, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "full_name": "pow_eq_one_iff_cases", "start": [232, 1], "end": [233, 32], "traced_tactics": [{"tactic": "simp [\u2190 pow_eq_pow_iff_cases]", "annotated_tactic": ["simp [\u2190 <a>pow_eq_pow_iff_cases</a>]", [{"full_name": "pow_eq_pow_iff_cases", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [226, 7], "def_end_pos": [226, 27]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d : LinearOrderedRing R\na b : R\nn : \u2115\n\u22a2 a ^ n = 1 \u2194 n = 0 \u2228 a = 1 \u2228 a = -1 \u2227 Even n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Coxeter/Basic.lean", "full_name": "CoxeterSystem.simple_mul_simple_cancel_left", "start": [208, 1], "end": [209, 21], "traced_tactics": [{"tactic": "simp [\u2190 mul_assoc]", "annotated_tactic": ["simp [\u2190 <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "B : Type u_1\nB' : Type u_2\ne : B \u2243 B'\nW : Type u_3\nH : Type u_4\ninst\u271d\u00b9 : Group W\ninst\u271d : Group H\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n\u22a2 cs.simple i * (cs.simple i * w) = w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "full_name": "lt_tsub_of_add_lt_left", "start": [375, 1], "end": [376, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Alternating/Basic.lean", "full_name": "ContinuousAlternatingMap.smul_apply", "start": [173, 1], "end": [174, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.div_union", "start": [751, 1], "end": [752, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Yoneda.lean", "full_name": "CategoryTheory.yonedaEvaluation_map_down", "start": [359, 1], "end": [362, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sheaves/Stalks.lean", "full_name": "TopCat.Presheaf.germ_res", "start": [104, 1], "end": [107, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.ofReal_def", "start": [105, 1], "end": [106, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "Associates.factors_subsingleton", "start": [1491, 1], "end": [1493, 23], "traced_tactics": [{"tactic": "have : Subsingleton (Associates \u03b1) := inferInstance", "annotated_tactic": ["have : <a>Subsingleton</a> (<a>Associates</a> \u03b1) := <a>inferInstance</a>", [{"full_name": "Subsingleton", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1009, 7], "def_end_pos": [1009, 19]}, {"full_name": "Associates", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [825, 8], "def_end_pos": [825, 18]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\na : Associates \u03b1\n\u22a2 a.factors = \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\na : Associates \u03b1\nthis : Subsingleton (Associates \u03b1)\n\u22a2 a.factors = \u22a4"}, {"tactic": "convert factors_zero", "annotated_tactic": ["convert <a>factors_zero</a>", [{"full_name": "Associates.factors_zero", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\na : Associates \u03b1\nthis : Subsingleton (Associates \u03b1)\n\u22a2 a.factors = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.I_ne_zero", "start": [306, 1], "end": [306, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "full_name": "YoungDiagram.mem_sup", "start": [128, 1], "end": [129, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fintype/BigOperators.lean", "full_name": "Fintype.card_piFinset", "start": [130, 1], "end": [131, 72], "traced_tactics": [{"tactic": "simp [piFinset, card_map]", "annotated_tactic": ["simp [<a>piFinset</a>, <a>card_map</a>]", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.card_map", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : DecidableEq \u03ba\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ns : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 (piFinset s).card = \u220f i : \u03b9, (s i).card", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.BalancedSz.symm", "start": [192, 1], "end": [193, 47], "traced_tactics": [{"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\u03b1 : Type u_1\nl r : \u2115\n\u22a2 l + r \u2264 1 \u2192 r + l \u2264 1", "state_after": "\u03b1 : Type u_1\nl r : \u2115\n\u22a2 r + l \u2264 1 \u2192 r + l \u2264 1"}, {"tactic": "exact id", "annotated_tactic": ["exact <a>id</a>", [{"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\nl r : \u2115\n\u22a2 r + l \u2264 1 \u2192 r + l \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.map_inf", "start": [2717, 1], "end": [2722, 45], "traced_tactics": [{"tactic": "refine map_inf_le.antisymm ?_", "annotated_tactic": ["refine map_inf_le.antisymm ?_", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\n\u22a2 map m (f \u2293 g) = map m f \u2293 map m g", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\n\u22a2 map m f \u2293 map m g \u2264 map m (f \u2293 g)"}, {"tactic": "rintro t \u27e8s\u2081, hs\u2081, s\u2082, hs\u2082, ht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\u27e9", "annotated_tactic": ["rintro t \u27e8s\u2081, hs\u2081, s\u2082, hs\u2082, ht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\n\u22a2 map m f \u2293 map m g \u2264 map m (f \u2293 g)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\nt : Set \u03b2\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2208 f\ns\u2082 : Set \u03b1\nhs\u2082 : s\u2082 \u2208 g\nht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\n\u22a2 t \u2208 map m f \u2293 map m g"}, {"tactic": "refine mem_inf_of_inter (image_mem_map hs\u2081) (image_mem_map hs\u2082) ?_", "annotated_tactic": ["refine <a>mem_inf_of_inter</a> (<a>image_mem_map</a> hs\u2081) (<a>image_mem_map</a> hs\u2082) ?_", [{"full_name": "Filter.mem_inf_of_inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 25]}, {"full_name": "Filter.image_mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1938, 9], "def_end_pos": [1938, 22]}, {"full_name": "Filter.image_mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1938, 9], "def_end_pos": [1938, 22]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\nt : Set \u03b2\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2208 f\ns\u2082 : Set \u03b1\nhs\u2082 : s\u2082 \u2208 g\nht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\n\u22a2 t \u2208 map m f \u2293 map m g", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\nt : Set \u03b2\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2208 f\ns\u2082 : Set \u03b1\nhs\u2082 : s\u2082 \u2208 g\nht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\n\u22a2 m '' s\u2081 \u2229 m '' s\u2082 \u2286 t"}, {"tactic": "rw [\u2190 image_inter h, image_subset_iff, ht]", "annotated_tactic": ["rw [\u2190 <a>image_inter</a> h, <a>image_subset_iff</a>, ht]", [{"full_name": "Set.image_inter", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [326, 9], "def_end_pos": [326, 20]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [481, 9], "def_end_pos": [481, 25]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl f g : Filter \u03b1\nm : \u03b1 \u2192 \u03b2\nh : Injective m\nt : Set \u03b2\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2208 f\ns\u2082 : Set \u03b1\nhs\u2082 : s\u2082 \u2208 g\nht : m \u207b\u00b9' t = s\u2081 \u2229 s\u2082\n\u22a2 m '' s\u2081 \u2229 m '' s\u2082 \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.isUnit_of_left_inverse", "start": [168, 1], "end": [169, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/WSeq.lean", "full_name": "Stream'.WSeq.Equiv.trans", "start": [614, 1], "end": [615, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Num/Bitwise.lean", "full_name": "PosNum.shiftl_succ_eq_bit0_shiftl", "start": [123, 1], "end": [125, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.image2_iUnion_left", "start": [1867, 1], "end": [1869, 60], "traced_tactics": [{"tactic": "simp only [\u2190 image_prod, iUnion_prod_const, image_iUnion]", "annotated_tactic": ["simp only [\u2190 <a>image_prod</a>, <a>iUnion_prod_const</a>, <a>image_iUnion</a>]", [{"full_name": "Set.image_prod", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [79, 7], "def_end_pos": [79, 17]}, {"full_name": "Set.iUnion_prod_const", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1780, 9], "def_end_pos": [1780, 26]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f (\u22c3 i, s i) t = \u22c3 i, image2 f (s i) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_erase", "start": [2400, 1], "end": [2402, 16], "traced_tactics": [{"tactic": "rw [\u2190 sdiff_singleton_eq_erase, sdiff_sdiff_eq_sdiff_union (singleton_subset_iff.2 h), insert_eq,\n  union_comm]", "annotated_tactic": ["rw [\u2190 <a>sdiff_singleton_eq_erase</a>, <a>sdiff_sdiff_eq_sdiff_union</a> (<a>singleton_subset_iff</a>.2 h), <a>insert_eq</a>,\n    <a>union_comm</a>]", [{"full_name": "Finset.sdiff_singleton_eq_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2292, 9], "def_end_pos": [2292, 33]}, {"full_name": "Finset.sdiff_sdiff_eq_sdiff_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2386, 9], "def_end_pos": [2386, 35]}, {"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 29]}, {"full_name": "Finset.insert_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1517, 9], "def_end_pos": [1517, 18]}, {"full_name": "Finset.union_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1459, 9], "def_end_pos": [1459, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nh : a \u2208 s\n\u22a2 s \\ t.erase a = insert a (s \\ t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.valMinAbs_eq_zero", "start": [1228, 1], "end": [1232, 35], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "n : \u2115\nx : ZMod n\n\u22a2 x.valMinAbs = 0 \u2194 x = 0", "state_after": "case zero\nx : ZMod 0\n\u22a2 x.valMinAbs = 0 \u2194 x = 0\n\ncase succ\nn : \u2115\nx : ZMod (n + 1)\n\u22a2 x.valMinAbs = 0 \u2194 x = 0"}, {"tactic": "rw [\u2190 valMinAbs_zero n.succ]", "annotated_tactic": ["rw [\u2190 <a>valMinAbs_zero</a> n.succ]", [{"full_name": "ZMod.valMinAbs_zero", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1222, 9], "def_end_pos": [1222, 23]}]], "state_before": "case succ\nn : \u2115\nx : ZMod (n + 1)\n\u22a2 x.valMinAbs = 0 \u2194 x = 0", "state_after": "case succ\nn : \u2115\nx : ZMod (n + 1)\n\u22a2 x.valMinAbs = valMinAbs 0 \u2194 x = 0"}, {"tactic": "apply injective_valMinAbs.eq_iff", "annotated_tactic": ["apply injective_valMinAbs.eq_iff", []], "state_before": "case succ\nn : \u2115\nx : ZMod (n + 1)\n\u22a2 x.valMinAbs = valMinAbs 0 \u2194 x = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nx : ZMod 0\n\u22a2 x.valMinAbs = 0 \u2194 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.le", "start": [174, 11], "end": [175, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalHom.lean", "full_name": "NonUnitalAlgHom.restrictScalars_injective", "start": [557, 1], "end": [559, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fintype/BigOperators.lean", "full_name": "Fintype.prod_bool", "start": [37, 1], "end": [37, 89], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u220f b : Bool, f b = f true * f false", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/NormNum/Basic.lean", "full_name": "Mathlib.Meta.NormNum.isRat_div", "start": [435, 1], "end": [437, 55], "traced_tactics": [{"tactic": "simpa [div_eq_mul_inv] using h", "annotated_tactic": ["simpa [<a>div_eq_mul_inv</a>] using h", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DivisionRing \u03b1\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1\nx\u271d\u00b9 : \u2124\nx\u271d : \u2115\nh : IsRat (x\u271d\u00b3 * x\u271d\u00b2\u207b\u00b9) x\u271d\u00b9 x\u271d\n\u22a2 IsRat (x\u271d\u00b3 / x\u271d\u00b2) x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_subtype", "start": [1140, 1], "end": [1142, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_iff", "start": [843, 1], "end": [848, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "full_name": "Finset.weightedVSub_const_smul", "start": [353, 1], "end": [355, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "full_name": "ContinuousMultilinearMap.opNorm_le_bound", "start": [415, 1], "end": [416, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Subfield.lean", "full_name": "Subfield.not_mem_of_not_mem_closure", "start": [676, 1], "end": [677, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.Tendsto.fst", "start": [149, 1], "end": [151, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "ENNReal.rpow_lt_top_iff_of_pos", "start": [516, 1], "end": [517, 63], "traced_tactics": [{"tactic": "simp only [lt_top_iff_ne_top, Ne, rpow_eq_top_iff_of_pos hy]", "annotated_tactic": ["simp only [<a>lt_top_iff_ne_top</a>, <a>Ne</a>, <a>rpow_eq_top_iff_of_pos</a> hy]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [141, 9], "def_end_pos": [141, 26]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "ENNReal.rpow_eq_top_iff_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [512, 9], "def_end_pos": [512, 31]}]], "state_before": "x : \u211d\u22650\u221e\ny : \u211d\nhy : 0 < y\n\u22a2 x ^ y < \u22a4 \u2194 x < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean", "full_name": "AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos", "start": [153, 1], "end": [233, 43], "traced_tactics": [{"tactic": "obtain \u27e8c\u2081, _, c\u2082, _, h\u27e9 := hf (1/2) (by norm_num)", "annotated_tactic": ["obtain \u27e8c\u2081, _, c\u2082, _, h\u27e9 := hf (1/2) (by norm_num)", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case intro.intro.intro.intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0"}, {"tactic": "match lt_trichotomy c\u2081 c\u2082 with\n| .inl hlt => left\n  filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n  have h' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x := by\n    rw [Set.mem_Icc]\n    exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9\n  have hu := hx (3/4 * x) h'\n  have hu := Set.nonempty_of_mem hu\n  rw [Set.nonempty_Icc] at hu\n  have hu' : 0 \u2264 (c\u2082 - c\u2081) * f x := by linarith\n  exact nonneg_of_mul_nonneg_right hu' (by linarith)\n| .inr (.inr hgt) => right\n  filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n  have h' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x := by\n    rw [Set.mem_Icc]\n    exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9\n  have hu := hx (3/4 * x) h'\n  have hu := Set.nonempty_of_mem hu\n  rw [Set.nonempty_Icc] at hu\n  have hu' : (c\u2081 - c\u2082) * f x \u2264 0 := by linarith\n  exact nonpos_of_mul_nonpos_right hu' (by linarith)\n| .inr (.inl heq) => have hmain : \u2203 c, \u2200\u1da0 x in atTop, f x = c := by\n    simp only [heq, Set.Icc_self, Set.mem_singleton_iff, one_mul] at h\n    rw [eventually_atTop] at h\n    obtain \u27e8n\u2080, hn\u2080\u27e9 := h\n    refine \u27e8f (max n\u2080 2), ?_\u27e9\n    rw [eventually_atTop]\n    refine \u27e8max n\u2080 2, ?_\u27e9\n    refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step\n    case base =>\n      intro x \u27e8hxlb, hxub\u27e9\n      have h\u2081 := calc n\u2080 \u2264 1 * max n\u2080 2 := by simp\n                      _ \u2264 2 * max n\u2080 2 := by gcongr; norm_num\n      have h\u2082 := hn\u2080 (2 * max n\u2080 2) h\u2081 (max n\u2080 2) \u27e8by simp [-max_le_iff, hxlb], by linarith\u27e9\n      rw [h\u2082]\n      exact hn\u2080 (2 * max n\u2080 2) h\u2081 x \u27e8by simp [-max_le_iff, hxlb], le_of_lt hxub\u27e9\n    case step =>\n      intro n hn hyp_ind z hz\n      have z_nonneg : 0 \u2264 z := by\n        calc (0:\u211d) \u2264 (2:\u211d)^n * max n\u2080 2 := by\n                      exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)\n                _ \u2264 z := by exact_mod_cast hz.1\n      have le_2n : max n\u2080 2 \u2264 (2:\u211d)^n * max n\u2080 2 := by\n        nth_rewrite 1 [\u2190 one_mul (max n\u2080 2)]\n        gcongr\n        exact one_le_pow_of_one_le (by norm_num : (1:\u211d) \u2264 2) _\n      have n\u2080_le_z : n\u2080 \u2264 z := by\n        calc n\u2080 \u2264 max n\u2080 2 := by simp\n              _ \u2264 (2:\u211d)^n * max n\u2080 2 := le_2n\n              _ \u2264 _ := by exact_mod_cast hz.1\n      have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, le_rfl\u27e9\n      have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8le_rfl, by linarith\u27e9\n      have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n      have half_z_to_base : f (1/2 * z) = f (max n\u2080 2) := by\n        refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n        case lb =>\n          calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n                      _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n                      _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1\n        case ub =>\n          have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n            rw [one_div, pow_add, pow_one]\n            ring\n          rw [h\u2081, mul_assoc]\n          gcongr\n          exact_mod_cast hz.2\n      rw [\u2190 z_to_half_z, half_z_to_base]\n  obtain \u27e8c, hc\u27e9 := hmain\n  cases le_or_lt 0 c with\n  | inl hpos =>\n    exact Or.inl <| by filter_upwards [hc] with _ hc; simpa only [hc]\n  | inr hneg =>\n    right\n    filter_upwards [hc] with x hc\n    exact le_of_lt <| by simpa only [hc]", "annotated_tactic": ["match <a>lt_trichotomy</a> c\u2081 c\u2082 with\n  | .inl hlt => -- c\u2081 < c\u2082\n    left\n    filter_upwards [h, <a>eventually_ge_atTop</a> 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x \u2208 <a>Set.Icc</a> (1 / 2 * x) x := by\n      rw [<a>Set.mem_Icc</a>]\n      exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9\n    have hu := hx (3/4 * x) h'\n    have hu := <a>Set.nonempty_of_mem</a> hu\n    rw [<a>Set.nonempty_Icc</a>] at hu\n    have hu' : 0 \u2264 (c\u2082 - c\u2081) * f x := by linarith\n    exact <a>nonneg_of_mul_nonneg_right</a> hu' (by linarith)\n  | .inr (.inr hgt) => -- c\u2082 < c\u2081\n    right\n    filter_upwards [h, <a>eventually_ge_atTop</a> 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x \u2208 <a>Set.Icc</a> (1 / 2 * x) x := by\n      rw [<a>Set.mem_Icc</a>]\n      exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9\n    have hu := hx (3/4 * x) h'\n    have hu := <a>Set.nonempty_of_mem</a> hu\n    rw [<a>Set.nonempty_Icc</a>] at hu\n    have hu' : (c\u2081 - c\u2082) * f x \u2264 0 := by linarith\n    exact <a>nonpos_of_mul_nonpos_right</a> hu' (by linarith)\n  | .inr (.inl heq) => -- c\u2081 = c\u2082\n    have hmain : \u2203 c, \u2200\u1da0 x in <a>atTop</a>, f x = c := by\n      simp only [heq, <a>Set.Icc_self</a>, <a>Set.mem_singleton_iff</a>, <a>one_mul</a>] at h\n      rw [<a>eventually_atTop</a>] at h\n      obtain \u27e8n\u2080, hn\u2080\u27e9 := h\n      refine \u27e8f (<a>max</a> n\u2080 2), ?_\u27e9\n      rw [<a>eventually_atTop</a>]\n      refine \u27e8<a>max</a> n\u2080 2, ?_\u27e9\n      refine <a>Real.induction_Ico_mul</a> _ 2 (by norm_num) (by positivity) ?base ?step\n      case base =>\n        intro x \u27e8hxlb, hxub\u27e9\n        have h\u2081 := calc n\u2080 \u2264 1 * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 2 * <a>max</a> n\u2080 2 := by gcongr; norm_num\n        have h\u2082 := hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 (<a>max</a> n\u2080 2) \u27e8by simp [-<a>max_le_iff</a>, hxlb], by linarith\u27e9\n        rw [h\u2082]\n        exact hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 x \u27e8by simp [-<a>max_le_iff</a>, hxlb], <a>le_of_lt</a> hxub\u27e9\n      case step =>\n        intro n hn hyp_ind z hz\n        have z_nonneg : 0 \u2264 z := by\n          calc (0:\u211d) \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n                        exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)\n                  _ \u2264 z := by exact_mod_cast hz.1\n        have le_2n : <a>max</a> n\u2080 2 \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n          nth_rewrite 1 [\u2190 <a>one_mul</a> (<a>max</a> n\u2080 2)]\n          gcongr\n          exact <a>one_le_pow_of_one_le</a> (by norm_num : (1:\u211d) \u2264 2) _\n        have n\u2080_le_z : n\u2080 \u2264 z := by\n          calc n\u2080 \u2264 <a>max</a> n\u2080 2 := by simp\n                _ \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := le_2n\n                _ \u2264 _ := by exact_mod_cast hz.1\n        have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, <a>le_rfl</a>\u27e9\n        have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8<a>le_rfl</a>, by linarith\u27e9\n        have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n        have half_z_to_base : f (1/2 * z) = f (<a>max</a> n\u2080 2) := by\n          refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n          case lb =>\n            calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1\n          case ub =>\n            have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring\n            rw [h\u2081, <a>mul_assoc</a>]\n            gcongr\n            exact_mod_cast hz.2\n        rw [\u2190 z_to_half_z, half_z_to_base]\n    obtain \u27e8c, hc\u27e9 := hmain\n    cases <a>le_or_lt</a> 0 c with\n    | <a>inl</a> hpos =>\n      exact <a>Or.inl</a> <| by filter_upwards [hc] with _ hc; simpa only [hc]\n    | <a>inr</a> hneg =>\n      right\n      filter_upwards [hc] with x hc\n      exact <a>le_of_lt</a> <| by simpa only [hc]", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}, {"full_name": "Set.nonempty_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 21]}, {"full_name": "nonneg_of_mul_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [868, 9], "def_end_pos": [868, 35]}, {"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}, {"full_name": "Set.nonempty_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 21]}, {"full_name": "nonpos_of_mul_nonpos_right", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [884, 9], "def_end_pos": [884, 35]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.Icc_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [773, 9], "def_end_pos": [773, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Real.induction_Ico_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [464, 7], "def_end_pos": [464, 29]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case intro.intro.intro.intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\n\u22a2 1 / 2 \u2208 Set.Ioo 0 1", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x"}, {"tactic": "filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg", "annotated_tactic": ["filter_upwards [h, <a>eventually_ge_atTop</a> 0] with x hx hx_nonneg", [{"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 0 \u2264 f x"}, {"tactic": "have h' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x := by\n  rw [Set.mem_Icc]\n  exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", "annotated_tactic": ["have h' : 3 / 4 * x \u2208 <a>Set.Icc</a> (1 / 2 * x) x := by\n      rw [<a>Set.mem_Icc</a>]\n      exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\n\u22a2 0 \u2264 f x"}, {"tactic": "have hu := hx (3/4 * x) h'", "annotated_tactic": ["have hu := hx (3/4 * x) h'", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\n\u22a2 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 0 \u2264 f x"}, {"tactic": "have hu := Set.nonempty_of_mem hu", "annotated_tactic": ["have hu := <a>Set.nonempty_of_mem</a> hu", [{"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : (Set.Icc (c\u2081 * f x) (c\u2082 * f x)).Nonempty\n\u22a2 0 \u2264 f x"}, {"tactic": "rw [Set.nonempty_Icc] at hu", "annotated_tactic": ["rw [<a>Set.nonempty_Icc</a>] at hu", [{"full_name": "Set.nonempty_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 21]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : (Set.Icc (c\u2081 * f x) (c\u2082 * f x)).Nonempty\n\u22a2 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 0 \u2264 f x"}, {"tactic": "have hu' : 0 \u2264 (c\u2082 - c\u2081) * f x := by linarith", "annotated_tactic": ["have hu' : 0 \u2264 (c\u2082 - c\u2081) * f x := by linarith", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : 0 \u2264 (c\u2082 - c\u2081) * f x\n\u22a2 0 \u2264 f x"}, {"tactic": "exact nonneg_of_mul_nonneg_right hu' (by linarith)", "annotated_tactic": ["exact <a>nonneg_of_mul_nonneg_right</a> hu' (by linarith)", [{"full_name": "nonneg_of_mul_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [868, 9], "def_end_pos": [868, 35]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : 0 \u2264 (c\u2082 - c\u2081) * f x\n\u22a2 0 \u2264 f x", "state_after": "no goals"}, {"tactic": "rw [Set.mem_Icc]", "annotated_tactic": ["rw [<a>Set.mem_Icc</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x \u2227 3 / 4 * x \u2264 x"}, {"tactic": "exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", "annotated_tactic": ["exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x \u2227 3 / 4 * x \u2264 x", "state_after": "no goals"}, {"tactic": "gcongr ?_ * x", "annotated_tactic": ["gcongr ?_ * x", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 \u2264 3 / 4"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 \u2264 3 / 4", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 3 / 4 * x \u2264 x", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 0 \u2264 (c\u2082 - c\u2081) * f x", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhlt : c\u2081 < c\u2082\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : 0 \u2264 (c\u2082 - c\u2081) * f x\n\u22a2 0 < c\u2082 - c\u2081", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0"}, {"tactic": "filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg", "annotated_tactic": ["filter_upwards [h, <a>eventually_ge_atTop</a> 0] with x hx hx_nonneg", [{"full_name": "Filter.eventually_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 f x \u2264 0"}, {"tactic": "have h' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x := by\n  rw [Set.mem_Icc]\n  exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", "annotated_tactic": ["have h' : 3 / 4 * x \u2208 <a>Set.Icc</a> (1 / 2 * x) x := by\n      rw [<a>Set.mem_Icc</a>]\n      exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\n\u22a2 f x \u2264 0"}, {"tactic": "have hu := hx (3/4 * x) h'", "annotated_tactic": ["have hu := hx (3/4 * x) h'", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\n\u22a2 f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 f x \u2264 0"}, {"tactic": "have hu := Set.nonempty_of_mem hu", "annotated_tactic": ["have hu := <a>Set.nonempty_of_mem</a> hu", [{"full_name": "Set.nonempty_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\n\u22a2 f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : (Set.Icc (c\u2081 * f x) (c\u2082 * f x)).Nonempty\n\u22a2 f x \u2264 0"}, {"tactic": "rw [Set.nonempty_Icc] at hu", "annotated_tactic": ["rw [<a>Set.nonempty_Icc</a>] at hu", [{"full_name": "Set.nonempty_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 21]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : (Set.Icc (c\u2081 * f x) (c\u2082 * f x)).Nonempty\n\u22a2 f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 f x \u2264 0"}, {"tactic": "have hu' : (c\u2081 - c\u2082) * f x \u2264 0 := by linarith", "annotated_tactic": ["have hu' : (c\u2081 - c\u2082) * f x \u2264 0 := by linarith", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : (c\u2081 - c\u2082) * f x \u2264 0\n\u22a2 f x \u2264 0"}, {"tactic": "exact nonpos_of_mul_nonpos_right hu' (by linarith)", "annotated_tactic": ["exact <a>nonpos_of_mul_nonpos_right</a> hu' (by linarith)", [{"full_name": "nonpos_of_mul_nonpos_right", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [884, 9], "def_end_pos": [884, 35]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : (c\u2081 - c\u2082) * f x \u2264 0\n\u22a2 f x \u2264 0", "state_after": "no goals"}, {"tactic": "rw [Set.mem_Icc]", "annotated_tactic": ["rw [<a>Set.mem_Icc</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x \u2227 3 / 4 * x \u2264 x"}, {"tactic": "exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", "annotated_tactic": ["exact \u27e8by gcongr ?_ * x; norm_num, by linarith\u27e9", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x \u2227 3 / 4 * x \u2264 x", "state_after": "no goals"}, {"tactic": "gcongr ?_ * x", "annotated_tactic": ["gcongr ?_ * x", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 * x \u2264 3 / 4 * x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 \u2264 3 / 4"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 1 / 2 \u2264 3 / 4", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\n\u22a2 3 / 4 * x \u2264 x", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\n\u22a2 (c\u2081 - c\u2082) * f x \u2264 0", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhgt : c\u2082 < c\u2081\nx : \u211d\nhx : \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhx_nonneg : 0 \u2264 x\nh' : 3 / 4 * x \u2208 Set.Icc (1 / 2 * x) x\nhu\u271d : f (3 / 4 * x) \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nhu : c\u2081 * f x \u2264 c\u2082 * f x\nhu' : (c\u2081 - c\u2082) * f x \u2264 0\n\u22a2 0 < c\u2081 - c\u2082", "state_after": "no goals"}, {"tactic": "have hmain : \u2203 c, \u2200\u1da0 x in atTop, f x = c := by\n  simp only [heq, Set.Icc_self, Set.mem_singleton_iff, one_mul] at h\n  rw [eventually_atTop] at h\n  obtain \u27e8n\u2080, hn\u2080\u27e9 := h\n  refine \u27e8f (max n\u2080 2), ?_\u27e9\n  rw [eventually_atTop]\n  refine \u27e8max n\u2080 2, ?_\u27e9\n  refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step\n  case base =>\n    intro x \u27e8hxlb, hxub\u27e9\n    have h\u2081 := calc n\u2080 \u2264 1 * max n\u2080 2 := by simp\n                    _ \u2264 2 * max n\u2080 2 := by gcongr; norm_num\n    have h\u2082 := hn\u2080 (2 * max n\u2080 2) h\u2081 (max n\u2080 2) \u27e8by simp [-max_le_iff, hxlb], by linarith\u27e9\n    rw [h\u2082]\n    exact hn\u2080 (2 * max n\u2080 2) h\u2081 x \u27e8by simp [-max_le_iff, hxlb], le_of_lt hxub\u27e9\n  case step =>\n    intro n hn hyp_ind z hz\n    have z_nonneg : 0 \u2264 z := by\n      calc (0:\u211d) \u2264 (2:\u211d)^n * max n\u2080 2 := by\n                    exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)\n              _ \u2264 z := by exact_mod_cast hz.1\n    have le_2n : max n\u2080 2 \u2264 (2:\u211d)^n * max n\u2080 2 := by\n      nth_rewrite 1 [\u2190 one_mul (max n\u2080 2)]\n      gcongr\n      exact one_le_pow_of_one_le (by norm_num : (1:\u211d) \u2264 2) _\n    have n\u2080_le_z : n\u2080 \u2264 z := by\n      calc n\u2080 \u2264 max n\u2080 2 := by simp\n            _ \u2264 (2:\u211d)^n * max n\u2080 2 := le_2n\n            _ \u2264 _ := by exact_mod_cast hz.1\n    have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, le_rfl\u27e9\n    have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8le_rfl, by linarith\u27e9\n    have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n    have half_z_to_base : f (1/2 * z) = f (max n\u2080 2) := by\n      refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n      case lb =>\n        calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n                    _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n                    _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1\n      case ub =>\n        have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n          rw [one_div, pow_add, pow_one]\n          ring\n        rw [h\u2081, mul_assoc]\n        gcongr\n        exact_mod_cast hz.2\n    rw [\u2190 z_to_half_z, half_z_to_base]", "annotated_tactic": ["have hmain : \u2203 c, \u2200\u1da0 x in <a>atTop</a>, f x = c := by\n      simp only [heq, <a>Set.Icc_self</a>, <a>Set.mem_singleton_iff</a>, <a>one_mul</a>] at h\n      rw [<a>eventually_atTop</a>] at h\n      obtain \u27e8n\u2080, hn\u2080\u27e9 := h\n      refine \u27e8f (<a>max</a> n\u2080 2), ?_\u27e9\n      rw [<a>eventually_atTop</a>]\n      refine \u27e8<a>max</a> n\u2080 2, ?_\u27e9\n      refine <a>Real.induction_Ico_mul</a> _ 2 (by norm_num) (by positivity) ?base ?step\n      case base =>\n        intro x \u27e8hxlb, hxub\u27e9\n        have h\u2081 := calc n\u2080 \u2264 1 * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 2 * <a>max</a> n\u2080 2 := by gcongr; norm_num\n        have h\u2082 := hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 (<a>max</a> n\u2080 2) \u27e8by simp [-<a>max_le_iff</a>, hxlb], by linarith\u27e9\n        rw [h\u2082]\n        exact hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 x \u27e8by simp [-<a>max_le_iff</a>, hxlb], <a>le_of_lt</a> hxub\u27e9\n      case step =>\n        intro n hn hyp_ind z hz\n        have z_nonneg : 0 \u2264 z := by\n          calc (0:\u211d) \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n                        exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)\n                  _ \u2264 z := by exact_mod_cast hz.1\n        have le_2n : <a>max</a> n\u2080 2 \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n          nth_rewrite 1 [\u2190 <a>one_mul</a> (<a>max</a> n\u2080 2)]\n          gcongr\n          exact <a>one_le_pow_of_one_le</a> (by norm_num : (1:\u211d) \u2264 2) _\n        have n\u2080_le_z : n\u2080 \u2264 z := by\n          calc n\u2080 \u2264 <a>max</a> n\u2080 2 := by simp\n                _ \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := le_2n\n                _ \u2264 _ := by exact_mod_cast hz.1\n        have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, <a>le_rfl</a>\u27e9\n        have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8<a>le_rfl</a>, by linarith\u27e9\n        have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n        have half_z_to_base : f (1/2 * z) = f (<a>max</a> n\u2080 2) := by\n          refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n          case lb =>\n            calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1\n          case ub =>\n            have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring\n            rw [h\u2081, <a>mul_assoc</a>]\n            gcongr\n            exact_mod_cast hz.2\n        rw [\u2190 z_to_half_z, half_z_to_base]", [{"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.Icc_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [773, 9], "def_end_pos": [773, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Real.induction_Ico_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [464, 7], "def_end_pos": [464, 29]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nhmain : \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0"}, {"tactic": "obtain \u27e8c, hc\u27e9 := hmain", "annotated_tactic": ["obtain \u27e8c, hc\u27e9 := hmain", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nhmain : \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0"}, {"tactic": "cases le_or_lt 0 c with\n| inl hpos =>\n  exact Or.inl <| by filter_upwards [hc] with _ hc; simpa only [hc]\n| inr hneg =>\n  right\n  filter_upwards [hc] with x hc\n  exact le_of_lt <| by simpa only [hc]", "annotated_tactic": ["cases <a>le_or_lt</a> 0 c with\n    | <a>inl</a> hpos =>\n      exact <a>Or.inl</a> <| by filter_upwards [hc] with _ hc; simpa only [hc]\n    | <a>inr</a> hneg =>\n      right\n      filter_upwards [hc] with x hc\n      exact <a>le_of_lt</a> <| by simpa only [hc]", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "no goals"}, {"tactic": "simp only [heq, Set.Icc_self, Set.mem_singleton_iff, one_mul] at h", "annotated_tactic": ["simp only [heq, <a>Set.Icc_self</a>, <a>Set.mem_singleton_iff</a>, <a>one_mul</a>] at h", [{"full_name": "Set.Icc_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [773, 9], "def_end_pos": [773, 17]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u = c\u2082 * f x\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c"}, {"tactic": "rw [eventually_atTop] at h", "annotated_tactic": ["rw [<a>eventually_atTop</a>] at h", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u = c\u2082 * f x\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nh : \u2203 a, \u2200 b \u2265 a, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c"}, {"tactic": "obtain \u27e8n\u2080, hn\u2080\u27e9 := h", "annotated_tactic": ["obtain \u27e8n\u2080, hn\u2080\u27e9 := h", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nh : \u2203 a, \u2200 b \u2265 a, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c", "state_after": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c"}, {"tactic": "refine \u27e8f (max n\u2080 2), ?_\u27e9", "annotated_tactic": ["refine \u27e8f (<a>max</a> n\u2080 2), ?_\u27e9", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, f x = c", "state_after": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x = f (max n\u2080 2)"}, {"tactic": "rw [eventually_atTop]", "annotated_tactic": ["rw [<a>eventually_atTop</a>]", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x = f (max n\u2080 2)", "state_after": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 a, \u2200 b \u2265 a, f b = f (max n\u2080 2)"}, {"tactic": "refine \u27e8max n\u2080 2, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>max</a> n\u2080 2, ?_\u27e9", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2203 a, \u2200 b \u2265 a, f b = f (max n\u2080 2)", "state_after": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 b \u2265 max n\u2080 2, f b = f (max n\u2080 2)"}, {"tactic": "refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step", "annotated_tactic": ["refine <a>Real.induction_Ico_mul</a> _ 2 (by norm_num) (by positivity) ?base ?step", [{"full_name": "Real.induction_Ico_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [464, 7], "def_end_pos": [464, 29]}]], "state_before": "case intro\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 b \u2265 max n\u2080 2, f b = f (max n\u2080 2)", "state_after": "case base\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 x \u2208 Set.Ico (max n\u2080 2) (2 * max n\u2080 2), f x = f (max n\u2080 2)\n\ncase step\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 n \u2265 1,\n    (\u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)) \u2192\n      \u2200 z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2), f z = f (max n\u2080 2)"}, {"tactic": "case base =>\n  intro x \u27e8hxlb, hxub\u27e9\n  have h\u2081 := calc n\u2080 \u2264 1 * max n\u2080 2 := by simp\n                  _ \u2264 2 * max n\u2080 2 := by gcongr; norm_num\n  have h\u2082 := hn\u2080 (2 * max n\u2080 2) h\u2081 (max n\u2080 2) \u27e8by simp [-max_le_iff, hxlb], by linarith\u27e9\n  rw [h\u2082]\n  exact hn\u2080 (2 * max n\u2080 2) h\u2081 x \u27e8by simp [-max_le_iff, hxlb], le_of_lt hxub\u27e9", "annotated_tactic": ["case base =>\n        intro x \u27e8hxlb, hxub\u27e9\n        have h\u2081 := calc n\u2080 \u2264 1 * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 2 * <a>max</a> n\u2080 2 := by gcongr; norm_num\n        have h\u2082 := hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 (<a>max</a> n\u2080 2) \u27e8by simp [-<a>max_le_iff</a>, hxlb], by linarith\u27e9\n        rw [h\u2082]\n        exact hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 x \u27e8by simp [-<a>max_le_iff</a>, hxlb], <a>le_of_lt</a> hxub\u27e9", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case base\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 x \u2208 Set.Ico (max n\u2080 2) (2 * max n\u2080 2), f x = f (max n\u2080 2)\n\ncase step\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 n \u2265 1,\n    (\u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)) \u2192\n      \u2200 z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2), f z = f (max n\u2080 2)", "state_after": "case step\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 n \u2265 1,\n    (\u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)) \u2192\n      \u2200 z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2), f z = f (max n\u2080 2)"}, {"tactic": "case step =>\n  intro n hn hyp_ind z hz\n  have z_nonneg : 0 \u2264 z := by\n    calc (0:\u211d) \u2264 (2:\u211d)^n * max n\u2080 2 := by\n                  exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)\n            _ \u2264 z := by exact_mod_cast hz.1\n  have le_2n : max n\u2080 2 \u2264 (2:\u211d)^n * max n\u2080 2 := by\n    nth_rewrite 1 [\u2190 one_mul (max n\u2080 2)]\n    gcongr\n    exact one_le_pow_of_one_le (by norm_num : (1:\u211d) \u2264 2) _\n  have n\u2080_le_z : n\u2080 \u2264 z := by\n    calc n\u2080 \u2264 max n\u2080 2 := by simp\n          _ \u2264 (2:\u211d)^n * max n\u2080 2 := le_2n\n          _ \u2264 _ := by exact_mod_cast hz.1\n  have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, le_rfl\u27e9\n  have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8le_rfl, by linarith\u27e9\n  have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n  have half_z_to_base : f (1/2 * z) = f (max n\u2080 2) := by\n    refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n    case lb =>\n      calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n                  _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n                  _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1\n    case ub =>\n      have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n        rw [one_div, pow_add, pow_one]\n        ring\n      rw [h\u2081, mul_assoc]\n      gcongr\n      exact_mod_cast hz.2\n  rw [\u2190 z_to_half_z, half_z_to_base]", "annotated_tactic": ["case step =>\n        intro n hn hyp_ind z hz\n        have z_nonneg : 0 \u2264 z := by\n          calc (0:\u211d) \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n                        exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)\n                  _ \u2264 z := by exact_mod_cast hz.1\n        have le_2n : <a>max</a> n\u2080 2 \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n          nth_rewrite 1 [\u2190 <a>one_mul</a> (<a>max</a> n\u2080 2)]\n          gcongr\n          exact <a>one_le_pow_of_one_le</a> (by norm_num : (1:\u211d) \u2264 2) _\n        have n\u2080_le_z : n\u2080 \u2264 z := by\n          calc n\u2080 \u2264 <a>max</a> n\u2080 2 := by simp\n                _ \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := le_2n\n                _ \u2264 _ := by exact_mod_cast hz.1\n        have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, <a>le_rfl</a>\u27e9\n        have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8<a>le_rfl</a>, by linarith\u27e9\n        have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'\n        have half_z_to_base : f (1/2 * z) = f (<a>max</a> n\u2080 2) := by\n          refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n          case lb =>\n            calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1\n          case ub =>\n            have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring\n            rw [h\u2081, <a>mul_assoc</a>]\n            gcongr\n            exact_mod_cast hz.2\n        rw [\u2190 z_to_half_z, half_z_to_base]", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case step\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 n \u2265 1,\n    (\u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)) \u2192\n      \u2200 z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2), f z = f (max n\u2080 2)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 1 < 2", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 0 < max n\u2080 2", "state_after": "no goals"}, {"tactic": "intro x \u27e8hxlb, hxub\u27e9", "annotated_tactic": ["intro x \u27e8hxlb, hxub\u27e9", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 x \u2208 Set.Ico (max n\u2080 2) (2 * max n\u2080 2), f x = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 f x = f (max n\u2080 2)"}, {"tactic": "have h\u2081 := calc n\u2080 \u2264 1 * max n\u2080 2 := by simp\n                _ \u2264 2 * max n\u2080 2 := by gcongr; norm_num", "annotated_tactic": ["have h\u2081 := calc n\u2080 \u2264 1 * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 2 * <a>max</a> n\u2080 2 := by gcongr; norm_num", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 f x = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\n\u22a2 f x = f (max n\u2080 2)"}, {"tactic": "have h\u2082 := hn\u2080 (2 * max n\u2080 2) h\u2081 (max n\u2080 2) \u27e8by simp [-max_le_iff, hxlb], by linarith\u27e9", "annotated_tactic": ["have h\u2082 := hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 (<a>max</a> n\u2080 2) \u27e8by simp [-<a>max_le_iff</a>, hxlb], by linarith\u27e9", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\n\u22a2 f x = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\nh\u2082 : f (max n\u2080 2) = c\u2082 * f (2 * max n\u2080 2)\n\u22a2 f x = f (max n\u2080 2)"}, {"tactic": "rw [h\u2082]", "annotated_tactic": ["rw [h\u2082]", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\nh\u2082 : f (max n\u2080 2) = c\u2082 * f (2 * max n\u2080 2)\n\u22a2 f x = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\nh\u2082 : f (max n\u2080 2) = c\u2082 * f (2 * max n\u2080 2)\n\u22a2 f x = c\u2082 * f (2 * max n\u2080 2)"}, {"tactic": "exact hn\u2080 (2 * max n\u2080 2) h\u2081 x \u27e8by simp [-max_le_iff, hxlb], le_of_lt hxub\u27e9", "annotated_tactic": ["exact hn\u2080 (2 * <a>max</a> n\u2080 2) h\u2081 x \u27e8by simp [-<a>max_le_iff</a>, hxlb], <a>le_of_lt</a> hxub\u27e9", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\nh\u2082 : f (max n\u2080 2) = c\u2082 * f (2 * max n\u2080 2)\n\u22a2 f x = c\u2082 * f (2 * max n\u2080 2)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 n\u2080 \u2264 1 * max n\u2080 2", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 1 * max n\u2080 2 \u2264 2 * max n\u2080 2", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 1 \u2264 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\n\u22a2 1 \u2264 2", "state_after": "no goals"}, {"tactic": "simp [-max_le_iff, hxlb]", "annotated_tactic": ["simp [-<a>max_le_iff</a>, hxlb]", [{"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\n\u22a2 1 / 2 * (2 * max n\u2080 2) \u2264 max n\u2080 2", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\n\u22a2 max n\u2080 2 \u2264 2 * max n\u2080 2", "state_after": "no goals"}, {"tactic": "simp [-max_le_iff, hxlb]", "annotated_tactic": ["simp [-<a>max_le_iff</a>, hxlb]", [{"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nx : \u211d\nhxlb : max n\u2080 2 \u2264 x\nhxub : x < 2 * max n\u2080 2\nh\u2081 : n\u2080 \u2264 2 * max n\u2080 2\nh\u2082 : f (max n\u2080 2) = c\u2082 * f (2 * max n\u2080 2)\n\u22a2 1 / 2 * (2 * max n\u2080 2) \u2264 x", "state_after": "no goals"}, {"tactic": "intro n hn hyp_ind z hz", "annotated_tactic": ["intro n hn hyp_ind z hz", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\n\u22a2 \u2200 n \u2265 1,\n    (\u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)) \u2192\n      \u2200 z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2), f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have z_nonneg : 0 \u2264 z := by\n  calc (0:\u211d) \u2264 (2:\u211d)^n * max n\u2080 2 := by\n                exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)\n          _ \u2264 z := by exact_mod_cast hz.1", "annotated_tactic": ["have z_nonneg : 0 \u2264 z := by\n          calc (0:\u211d) \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n                        exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)\n                  _ \u2264 z := by exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have le_2n : max n\u2080 2 \u2264 (2:\u211d)^n * max n\u2080 2 := by\n  nth_rewrite 1 [\u2190 one_mul (max n\u2080 2)]\n  gcongr\n  exact one_le_pow_of_one_le (by norm_num : (1:\u211d) \u2264 2) _", "annotated_tactic": ["have le_2n : <a>max</a> n\u2080 2 \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n          nth_rewrite 1 [\u2190 <a>one_mul</a> (<a>max</a> n\u2080 2)]\n          gcongr\n          exact <a>one_le_pow_of_one_le</a> (by norm_num : (1:\u211d) \u2264 2) _", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have n\u2080_le_z : n\u2080 \u2264 z := by\n  calc n\u2080 \u2264 max n\u2080 2 := by simp\n        _ \u2264 (2:\u211d)^n * max n\u2080 2 := le_2n\n        _ \u2264 _ := by exact_mod_cast hz.1", "annotated_tactic": ["have n\u2080_le_z : n\u2080 \u2264 z := by\n          calc n\u2080 \u2264 <a>max</a> n\u2080 2 := by simp\n                _ \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := le_2n\n                _ \u2264 _ := by exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, le_rfl\u27e9", "annotated_tactic": ["have fz_eq_c\u2082fz : f z = c\u2082 * f z := hn\u2080 z n\u2080_le_z z \u27e8by linarith, <a>le_rfl</a>\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8le_rfl, by linarith\u27e9", "annotated_tactic": ["have z_to_half_z' : f (1/2 * z) = c\u2082 * f z := hn\u2080 z n\u2080_le_z (1/2 * z) \u27e8<a>le_rfl</a>, by linarith\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'", "annotated_tactic": ["have z_to_half_z : f (1/2 * z) = f z := by rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "have half_z_to_base : f (1/2 * z) = f (max n\u2080 2) := by\n  refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n  case lb =>\n    calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n                _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n                _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1\n  case ub =>\n    have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n      rw [one_div, pow_add, pow_one]\n      ring\n    rw [h\u2081, mul_assoc]\n    gcongr\n    exact_mod_cast hz.2", "annotated_tactic": ["have half_z_to_base : f (1/2 * z) = f (<a>max</a> n\u2080 2) := by\n          refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9\n          case lb =>\n            calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1\n          case ub =>\n            have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring\n            rw [h\u2081, <a>mul_assoc</a>]\n            gcongr\n            exact_mod_cast hz.2", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 f z = f (max n\u2080 2)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nhalf_z_to_base : f (1 / 2 * z) = f (max n\u2080 2)\n\u22a2 f z = f (max n\u2080 2)"}, {"tactic": "rw [\u2190 z_to_half_z, half_z_to_base]", "annotated_tactic": ["rw [\u2190 z_to_half_z, half_z_to_base]", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nhalf_z_to_base : f (1 / 2 * z) = f (max n\u2080 2)\n\u22a2 f z = f (max n\u2080 2)", "state_after": "no goals"}, {"tactic": "calc (0:\u211d) \u2264 (2:\u211d)^n * max n\u2080 2 := by\n              exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)\n        _ \u2264 z := by exact_mod_cast hz.1", "annotated_tactic": ["calc (0:\u211d) \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := by\n                        exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)\n                  _ \u2264 z := by exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 0 \u2264 z", "state_after": "no goals"}, {"tactic": "exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)", "annotated_tactic": ["exact <a>mul_nonneg</a> (<a>pow_nonneg</a> (by norm_num) _) (by norm_num)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [241, 9], "def_end_pos": [241, 19]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 0 \u2264 2 ^ n * max n\u2080 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 0 \u2264 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 0 \u2264 max n\u2080 2", "state_after": "no goals"}, {"tactic": "exact_mod_cast hz.1", "annotated_tactic": ["exact_mod_cast hz.1", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\n\u22a2 2 ^ n * max n\u2080 2 \u2264 z", "state_after": "no goals"}, {"tactic": "nth_rewrite 1 [\u2190 one_mul (max n\u2080 2)]", "annotated_tactic": ["nth_rewrite 1 [\u2190 <a>one_mul</a> (<a>max</a> n\u2080 2)]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 max n\u2080 2 \u2264 2 ^ n * max n\u2080 2", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 1 * max n\u2080 2 \u2264 2 ^ n * max n\u2080 2"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 1 * max n\u2080 2 \u2264 2 ^ n * max n\u2080 2", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 1 \u2264 2 ^ n"}, {"tactic": "exact one_le_pow_of_one_le (by norm_num : (1:\u211d) \u2264 2) _", "annotated_tactic": ["exact <a>one_le_pow_of_one_le</a> (by norm_num : (1:\u211d) \u2264 2) _", [{"full_name": "one_le_pow_of_one_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 1 \u2264 2 ^ n", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\n\u22a2 1 \u2264 2", "state_after": "no goals"}, {"tactic": "calc n\u2080 \u2264 max n\u2080 2 := by simp\n      _ \u2264 (2:\u211d)^n * max n\u2080 2 := le_2n\n      _ \u2264 _ := by exact_mod_cast hz.1", "annotated_tactic": ["calc n\u2080 \u2264 <a>max</a> n\u2080 2 := by simp\n                _ \u2264 (2:\u211d)^n * <a>max</a> n\u2080 2 := le_2n\n                _ \u2264 _ := by exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\n\u22a2 n\u2080 \u2264 z", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\n\u22a2 n\u2080 \u2264 max n\u2080 2", "state_after": "no goals"}, {"tactic": "exact_mod_cast hz.1", "annotated_tactic": ["exact_mod_cast hz.1", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\n\u22a2 2 ^ n * max n\u2080 2 \u2264 z", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\n\u22a2 1 / 2 * z \u2264 z", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\n\u22a2 1 / 2 * z \u2264 z", "state_after": "no goals"}, {"tactic": "rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'", "annotated_tactic": ["rwa [\u2190 fz_eq_c\u2082fz] at z_to_half_z'", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\n\u22a2 f (1 / 2 * z) = f z", "state_after": "no goals"}, {"tactic": "refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9", "annotated_tactic": ["refine hyp_ind (1/2 * z) \u27e8?lb, ?ub\u27e9", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 f (1 / 2 * z) = f (max n\u2080 2)", "state_after": "case lb\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 max n\u2080 2 \u2264 1 / 2 * z\n\ncase ub\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2"}, {"tactic": "case lb =>\n  calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n              _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n              _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1", "annotated_tactic": ["case lb =>\n            calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case lb\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 max n\u2080 2 \u2264 1 / 2 * z\n\ncase ub\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2", "state_after": "case ub\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2"}, {"tactic": "case ub =>\n  have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n    rw [one_div, pow_add, pow_one]\n    ring\n  rw [h\u2081, mul_assoc]\n  gcongr\n  exact_mod_cast hz.2", "annotated_tactic": ["case ub =>\n            have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring\n            rw [h\u2081, <a>mul_assoc</a>]\n            gcongr\n            exact_mod_cast hz.2", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case ub\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2", "state_after": "no goals"}, {"tactic": "calc max n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * max n\u2080 2 := by simp\n            _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * max n\u2080 2 := by gcongr; norm_num\n            _ \u2264 _ := by rw [mul_assoc]; gcongr; exact_mod_cast hz.1", "annotated_tactic": ["calc <a>max</a> n\u2080 2 \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ 1  * <a>max</a> n\u2080 2 := by simp\n                        _ \u2264 ((1:\u211d)/(2:\u211d)) * (2:\u211d) ^ n * <a>max</a> n\u2080 2 := by gcongr; norm_num\n                        _ \u2264 _ := by rw [<a>mul_assoc</a>]; gcongr; exact_mod_cast hz.1", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 max n\u2080 2 \u2264 1 / 2 * z", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 max n\u2080 2 \u2264 1 / 2 * 2 ^ 1 * max n\u2080 2", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * 2 ^ 1 * max n\u2080 2 \u2264 1 / 2 * 2 ^ n * max n\u2080 2", "state_after": "case h.h.ha\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 \u2264 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h.h.ha\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 \u2264 2", "state_after": "no goals"}, {"tactic": "rw [mul_assoc]", "annotated_tactic": ["rw [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * 2 ^ n * max n\u2080 2 \u2264 1 / 2 * z", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * (2 ^ n * max n\u2080 2) \u2264 1 / 2 * z"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * (2 ^ n * max n\u2080 2) \u2264 1 / 2 * z", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 2 ^ n * max n\u2080 2 \u2264 z"}, {"tactic": "exact_mod_cast hz.1", "annotated_tactic": ["exact_mod_cast hz.1", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 2 ^ n * max n\u2080 2 \u2264 z", "state_after": "no goals"}, {"tactic": "have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n  rw [one_div, pow_add, pow_one]\n  ring", "annotated_tactic": ["have h\u2081 : (2:\u211d)^n = ((1:\u211d)/(2:\u211d)) * (2:\u211d)^(n+1) := by\n              rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]\n              ring", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2"}, {"tactic": "rw [h\u2081, mul_assoc]", "annotated_tactic": ["rw [h\u2081, <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 1 / 2 * z < 2 ^ n * max n\u2080 2", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 1 / 2 * z < 1 / 2 * (2 ^ (n + 1) * max n\u2080 2)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 1 / 2 * z < 1 / 2 * (2 ^ (n + 1) * max n\u2080 2)", "state_after": "case bc\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 z < 2 ^ (n + 1) * max n\u2080 2"}, {"tactic": "exact_mod_cast hz.2", "annotated_tactic": ["exact_mod_cast hz.2", []], "state_before": "case bc\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\nh\u2081 : 2 ^ n = 1 / 2 * 2 ^ (n + 1)\n\u22a2 z < 2 ^ (n + 1) * max n\u2080 2", "state_after": "no goals"}, {"tactic": "rw [one_div, pow_add, pow_one]", "annotated_tactic": ["rw [<a>one_div</a>, <a>pow_add</a>, <a>pow_one</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 2 ^ n = 1 / 2 * 2 ^ (n + 1)", "state_after": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 2 ^ n = 2\u207b\u00b9 * (2 ^ n * 2)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nheq : c\u2081 = c\u2082\nn\u2080 : \u211d\nhn\u2080 : \u2200 b \u2265 n\u2080, \u2200 u \u2208 Set.Icc (1 / 2 * b) b, f u = c\u2082 * f b\nn : \u2115\nhn : n \u2265 1\nhyp_ind : \u2200 z \u2208 Set.Ico (max n\u2080 2) (2 ^ n * max n\u2080 2), f z = f (max n\u2080 2)\nz : \u211d\nhz : z \u2208 Set.Ico (2 ^ n * max n\u2080 2) (2 ^ (n + 1) * max n\u2080 2)\nz_nonneg : 0 \u2264 z\nle_2n : max n\u2080 2 \u2264 2 ^ n * max n\u2080 2\nn\u2080_le_z : n\u2080 \u2264 z\nfz_eq_c\u2082fz : f z = c\u2082 * f z\nz_to_half_z' : f (1 / 2 * z) = c\u2082 * f z\nz_to_half_z : f (1 / 2 * z) = f z\n\u22a2 2 ^ n = 2\u207b\u00b9 * (2 ^ n * 2)", "state_after": "no goals"}, {"tactic": "exact Or.inl <| by filter_upwards [hc] with _ hc; simpa only [hc]", "annotated_tactic": ["exact <a>Or.inl</a> <| by filter_upwards [hc] with _ hc; simpa only [hc]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case intro.inl\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhpos : 0 \u2264 c\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "no goals"}, {"tactic": "filter_upwards [hc] with _ hc", "annotated_tactic": ["filter_upwards [hc] with _ hc", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhpos : 0 \u2264 c\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc\u271d : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhpos : 0 \u2264 c\na\u271d : \u211d\nhc : f a\u271d = c\n\u22a2 0 \u2264 f a\u271d"}, {"tactic": "simpa only [hc]", "annotated_tactic": ["simpa only [hc]", []], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc\u271d : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhpos : 0 \u2264 c\na\u271d : \u211d\nhc : f a\u271d = c\n\u22a2 0 \u2264 f a\u271d", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case intro.inr\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop, 0 \u2264 f x) \u2228 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case intro.inr.h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0"}, {"tactic": "filter_upwards [hc] with x hc", "annotated_tactic": ["filter_upwards [hc] with x hc", []], "state_before": "case intro.inr.h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, f x \u2264 0", "state_after": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc\u271d : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\nx : \u211d\nhc : f x = c\n\u22a2 f x \u2264 0"}, {"tactic": "exact le_of_lt <| by simpa only [hc]", "annotated_tactic": ["exact <a>le_of_lt</a> <| by simpa only [hc]", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\nf : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc\u271d : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\nx : \u211d\nhc : f x = c\n\u22a2 f x \u2264 0", "state_after": "no goals"}, {"tactic": "simpa only [hc]", "annotated_tactic": ["simpa only [hc]", []], "state_before": "f : \u211d \u2192 \u211d\nhf : GrowsPolynomially f\nc\u2081 : \u211d\nleft\u271d\u00b9 : c\u2081 > 0\nc\u2082 : \u211d\nleft\u271d : c\u2082 > 0\nh : \u2200\u1da0 (x : \u211d) in atTop, \u2200 u \u2208 Set.Icc (1 / 2 * x) x, f u \u2208 Set.Icc (c\u2081 * f x) (c\u2082 * f x)\nheq : c\u2081 = c\u2082\nc : \u211d\nhc\u271d : \u2200\u1da0 (x : \u211d) in atTop, f x = c\nhneg : c < 0\nx : \u211d\nhc : f x = c\n\u22a2 f x < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean", "full_name": "IsBoundedBilinearMap.fderivWithin", "start": [106, 11], "end": [109, 19], "traced_tactics": [{"tactic": "rw [DifferentiableAt.fderivWithin (h.differentiableAt p) hxs]", "annotated_tactic": ["rw [<a>DifferentiableAt.fderivWithin</a> (h.differentiableAt p) hxs]", [{"full_name": "DifferentiableAt.fderivWithin", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [656, 19], "def_end_pos": [656, 48]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nb : E \u00d7 F \u2192 G\nu : Set (E \u00d7 F)\nh : IsBoundedBilinearMap \ud835\udd5c b\np : E \u00d7 F\nhxs : UniqueDiffWithinAt \ud835\udd5c u p\n\u22a2 fderivWithin \ud835\udd5c b u p = h.deriv p", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nb : E \u00d7 F \u2192 G\nu : Set (E \u00d7 F)\nh : IsBoundedBilinearMap \ud835\udd5c b\np : E \u00d7 F\nhxs : UniqueDiffWithinAt \ud835\udd5c u p\n\u22a2 fderiv \ud835\udd5c b p = h.deriv p"}, {"tactic": "exact h.fderiv p", "annotated_tactic": ["exact h.fderiv p", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nb : E \u00d7 F \u2192 G\nu : Set (E \u00d7 F)\nh : IsBoundedBilinearMap \ud835\udd5c b\np : E \u00d7 F\nhxs : UniqueDiffWithinAt \ud835\udd5c u p\n\u22a2 fderiv \ud835\udd5c b p = h.deriv p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.toReal_nonneg_iff_sign_nonneg", "start": [911, 1], "end": [919, 62], "traced_tactics": [{"tactic": "rcases lt_trichotomy \u03b8.toReal 0 with (h | h | h)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> \u03b8.toReal 0 with (h | h | h)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}]], "state_before": "\u03b8 : Angle\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign", "state_after": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign\n\ncase inr.inl\n\u03b8 : Angle\nh : \u03b8.toReal = 0\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign\n\ncase inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign"}, {"tactic": "refine \u27e8fun hn => False.elim (h.not_le hn), fun hn => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun hn => <a>False.elim</a> (h.not_le hn), fun hn => ?_\u27e9", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign", "state_after": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\nhn : 0 \u2264 \u03b8.sign\n\u22a2 0 \u2264 \u03b8.toReal"}, {"tactic": "rw [toReal_neg_iff_sign_neg.1 h] at hn", "annotated_tactic": ["rw [<a>toReal_neg_iff_sign_neg</a>.1 h] at hn", [{"full_name": "Real.Angle.toReal_neg_iff_sign_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [901, 9], "def_end_pos": [901, 32]}]], "state_before": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\nhn : 0 \u2264 \u03b8.sign\n\u22a2 0 \u2264 \u03b8.toReal", "state_after": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\nhn : 0 \u2264 -1\n\u22a2 0 \u2264 \u03b8.toReal"}, {"tactic": "exact False.elim (hn.not_lt (by decide))", "annotated_tactic": ["exact <a>False.elim</a> (hn.not_lt (by decide))", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case inl\n\u03b8 : Angle\nh : \u03b8.toReal < 0\nhn : 0 \u2264 -1\n\u22a2 0 \u2264 \u03b8.toReal", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u03b8 : Angle\nh : \u03b8.toReal < 0\nhn : 0 \u2264 -1\n\u22a2 -1 < 0", "state_after": "no goals"}, {"tactic": "simp [h, sign, \u2190 sin_toReal]", "annotated_tactic": ["simp [h, <a>sign</a>, \u2190 <a>sin_toReal</a>]", [{"full_name": "Real.Angle.sign", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [852, 5], "def_end_pos": [852, 9]}, {"full_name": "Real.Angle.sin_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [727, 9], "def_end_pos": [727, 19]}]], "state_before": "case inr.inl\n\u03b8 : Angle\nh : \u03b8.toReal = 0\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign", "state_after": "no goals"}, {"tactic": "refine \u27e8fun _ => ?_, fun _ => h.le\u27e9", "annotated_tactic": ["refine \u27e8fun _ => ?_, fun _ => h.le\u27e9", []], "state_before": "case inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\n\u22a2 0 \u2264 \u03b8.toReal \u2194 0 \u2264 \u03b8.sign", "state_after": "case inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\nx\u271d : 0 \u2264 \u03b8.toReal\n\u22a2 0 \u2264 \u03b8.sign"}, {"tactic": "rw [sign, \u2190 sin_toReal, sign_nonneg_iff]", "annotated_tactic": ["rw [<a>sign</a>, \u2190 <a>sin_toReal</a>, <a>sign_nonneg_iff</a>]", [{"full_name": "Real.Angle.sign", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [852, 5], "def_end_pos": [852, 9]}, {"full_name": "Real.Angle.sin_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [727, 9], "def_end_pos": [727, 19]}, {"full_name": "sign_nonneg_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [397, 9], "def_end_pos": [397, 24]}]], "state_before": "case inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\nx\u271d : 0 \u2264 \u03b8.toReal\n\u22a2 0 \u2264 \u03b8.sign", "state_after": "case inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\nx\u271d : 0 \u2264 \u03b8.toReal\n\u22a2 0 \u2264 Real.sin \u03b8.toReal"}, {"tactic": "exact sin_nonneg_of_nonneg_of_le_pi h.le (toReal_le_pi \u03b8)", "annotated_tactic": ["exact <a>sin_nonneg_of_nonneg_of_le_pi</a> h.le (<a>toReal_le_pi</a> \u03b8)", [{"full_name": "Real.sin_nonneg_of_nonneg_of_le_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 38]}, {"full_name": "Real.Angle.toReal_le_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [564, 9], "def_end_pos": [564, 21]}]], "state_before": "case inr.inr\n\u03b8 : Angle\nh : 0 < \u03b8.toReal\nx\u271d : 0 \u2264 \u03b8.toReal\n\u22a2 0 \u2264 Real.sin \u03b8.toReal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectSum/Basic.lean", "full_name": "DirectSum.IsInternal.addSubmonoid_iSup_eq_top", "start": [395, 1], "end": [398, 40], "traced_tactics": [{"tactic": "rw [AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom, AddMonoidHom.mrange_top_iff_surjective]", "annotated_tactic": ["rw [<a>AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom</a>, <a>AddMonoidHom.mrange_top_iff_surjective</a>]", [{"full_name": "AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 62]}, {"full_name": "AddMonoidHom.mrange_top_iff_surjective", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [994, 3], "def_end_pos": [994, 14]}]], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nA : \u03b9 \u2192 AddSubmonoid M\nh : IsInternal A\n\u22a2 iSup A = \u22a4", "state_after": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nA : \u03b9 \u2192 AddSubmonoid M\nh : IsInternal A\n\u22a2 Surjective \u21d1(DFinsupp.sumAddHom fun i => (A i).subtype)"}, {"tactic": "exact Function.Bijective.surjective h", "annotated_tactic": ["exact <a>Function.Bijective.surjective</a> h", [{"full_name": "Function.Bijective.surjective", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [82, 19], "def_end_pos": [82, 39]}]], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nA : \u03b9 \u2192 AddSubmonoid M\nh : IsInternal A\n\u22a2 Surjective \u21d1(DFinsupp.sumAddHom fun i => (A i).subtype)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Blocks.lean", "full_name": "MulAction.IsBlock.def_mem", "start": [158, 1], "end": [160, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/ENNReal.lean", "full_name": "ENNReal.log_pow", "start": [184, 1], "end": [195, 8], "traced_tactics": [{"tactic": "cases' Nat.eq_zero_or_pos n with n_zero n_pos", "annotated_tactic": ["cases' <a>Nat.eq_zero_or_pos</a> n with n_zero n_pos", [{"full_name": "Nat.eq_zero_or_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 23]}]], "state_before": "x : \u211d\u22650\u221e\nn : \u2115\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inl\nx : \u211d\u22650\u221e\nn : \u2115\nn_zero : n = 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log\n\ncase inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log"}, {"tactic": "rcases ENNReal.trichotomy x with (rfl | rfl | x_real)", "annotated_tactic": ["rcases <a>ENNReal.trichotomy</a> x with (rfl | rfl | x_real)", [{"full_name": "ENNReal.trichotomy", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [467, 19], "def_end_pos": [467, 29]}]], "state_before": "case inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 (0 ^ n).log = \u2191\u2191n * log 0\n\ncase inr.inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 (\u22a4 ^ n).log = \u2191\u2191n * \u22a4.log\n\ncase inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x.toReal\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log"}, {"tactic": "simp [n_zero, pow_zero x]", "annotated_tactic": ["simp [n_zero, <a>pow_zero</a> x]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}]], "state_before": "case inl\nx : \u211d\u22650\u221e\nn : \u2115\nn_zero : n = 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "no goals"}, {"tactic": "rw [zero_pow (Ne.symm (ne_of_lt n_pos)), log_zero, EReal.mul_bot_of_pos]", "annotated_tactic": ["rw [<a>zero_pow</a> (<a>Ne.symm</a> (<a>ne_of_lt</a> n_pos)), <a>log_zero</a>, <a>EReal.mul_bot_of_pos</a>]", [{"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [160, 15], "def_end_pos": [160, 23]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "ENNReal.log_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/ENNReal.lean", "def_pos": [53, 9], "def_end_pos": [53, 17]}, {"full_name": "EReal.mul_bot_of_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1162, 7], "def_end_pos": [1162, 21]}]], "state_before": "case inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 (0 ^ n).log = \u2191\u2191n * log 0", "state_after": "case inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 0 < \u2191\u2191n"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 0 < \u2191\u2191n", "state_after": "no goals"}, {"tactic": "rw [ENNReal.top_pow n_pos, log_top, EReal.mul_top_of_pos]", "annotated_tactic": ["rw [<a>ENNReal.top_pow</a> n_pos, <a>log_top</a>, <a>EReal.mul_top_of_pos</a>]", [{"full_name": "ENNReal.top_pow", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}, {"full_name": "ENNReal.log_top", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/ENNReal.lean", "def_pos": [59, 9], "def_end_pos": [59, 16]}, {"full_name": "EReal.mul_top_of_pos", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [1114, 7], "def_end_pos": [1114, 21]}]], "state_before": "case inr.inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 (\u22a4 ^ n).log = \u2191\u2191n * \u22a4.log", "state_after": "case inr.inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 0 < \u2191\u2191n"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case inr.inr.inl\nn : \u2115\nn_pos : n > 0\n\u22a2 0 < \u2191\u2191n", "state_after": "no goals"}, {"tactic": "replace x_real := ENNReal.toReal_pos_iff.1 x_real", "annotated_tactic": ["replace x_real := <a>ENNReal.toReal_pos_iff</a>.1 x_real", [{"full_name": "ENNReal.toReal_pos_iff", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [167, 9], "def_end_pos": [167, 23]}]], "state_before": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x.toReal\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log"}, {"tactic": "have x_ne_zero := Ne.symm (LT.lt.ne x_real.1)", "annotated_tactic": ["have x_ne_zero := <a>Ne.symm</a> (<a>LT.lt.ne</a> x_real.1)", [{"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log"}, {"tactic": "have x_ne_top := LT.lt.ne x_real.2", "annotated_tactic": ["have x_ne_top := <a>LT.lt.ne</a> x_real.2", [{"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\nx_ne_top : x \u2260 \u22a4\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log"}, {"tactic": "simp only [log, pow_eq_zero_iff', x_ne_zero, ne_eq, false_and, \u2193reduceIte, pow_eq_top_iff,\n  x_ne_top, toReal_pow, Real.log_pow, EReal.coe_mul]", "annotated_tactic": ["simp only [<a>log</a>, <a>pow_eq_zero_iff'</a>, x_ne_zero, <a>ne_eq</a>, <a>false_and</a>, \u2193<a>reduceIte</a>, <a>pow_eq_top_iff</a>,\n      x_ne_top, <a>toReal_pow</a>, <a>Real.log_pow</a>, <a>EReal.coe_mul</a>]", [{"full_name": "ENNReal.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/ENNReal.lean", "def_pos": [47, 19], "def_end_pos": [47, 22]}, {"full_name": "pow_eq_zero_iff'", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [209, 15], "def_end_pos": [209, 31]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}, {"full_name": "reduceIte", "def_path": ".lake/packages/lean4/src/lean/Lean/Meta/Tactic/Simp/BuiltinSimprocs/Core.lean", "def_pos": [12, 33], "def_end_pos": [12, 42]}, {"full_name": "ENNReal.pow_eq_top_iff", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [267, 17], "def_end_pos": [267, 31]}, {"full_name": "ENNReal.toReal_pow", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [432, 9], "def_end_pos": [432, 19]}, {"full_name": "Real.log_pow", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 16]}, {"full_name": "EReal.coe_mul", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [168, 9], "def_end_pos": [168, 16]}]], "state_before": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\nx_ne_top : x \u2260 \u22a4\n\u22a2 (x ^ n).log = \u2191\u2191n * x.log", "state_after": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\nx_ne_top : x \u2260 \u22a4\n\u22a2 \u2191\u2191n * \u2191(Real.log x.toReal) = \u2191\u2191n * \u2191(Real.log x.toReal)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inr.inr.inr\nx : \u211d\u22650\u221e\nn : \u2115\nn_pos : n > 0\nx_real : 0 < x \u2227 x < \u22a4\nx_ne_zero : x \u2260 0\nx_ne_top : x \u2260 \u22a4\n\u22a2 \u2191\u2191n * \u2191(Real.log x.toReal) = \u2191\u2191n * \u2191(Real.log x.toReal)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Icc_union_Ico_eq_Ico", "start": [1683, 1], "end": [1686, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.dvd_iff_dvd_dvd", "start": [1407, 1], "end": [1408, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "full_name": "MDifferentiable.comp", "start": [751, 1], "end": [752, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "full_name": "Nat.case_strong_induction_on", "start": [495, 11], "end": [500, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.monomial_eq_zero", "start": [358, 1], "end": [359, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Young/SemistandardTableau.lean", "full_name": "SemistandardYoungTableau.to_fun_eq_coe", "start": [80, 1], "end": [82, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "Monotone.strictMono_iff_injective", "start": [933, 1], "end": [934, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Maps.lean", "full_name": "Ideal.map_top", "start": [120, 1], "end": [121, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "SetCoe.ext", "start": [188, 1], "end": [189, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Count.lean", "full_name": "List.count_attach", "start": [144, 1], "end": [146, 84], "traced_tactics": [{"tactic": "simp [Subtype.ext_iff]", "annotated_tactic": ["simp [<a>Subtype.ext_iff</a>]", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}]], "state_before": "\u03b1 : Type u_1\nl : List \u03b1\ninst\u271d : DecidableEq \u03b1\na x\u271d\u00b9 : { x // x \u2208 l }\nx\u271d : x\u271d\u00b9 \u2208 l.attach\n\u22a2 (x\u271d\u00b9 == a) = true \u2194 (\u2191x\u271d\u00b9 == \u2191a) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsBigOWith.sup", "start": [665, 1], "end": [667, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Complex.tan_add_int_mul_pi", "start": [1346, 1], "end": [1347, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "full_name": "LinearMap.range_zero", "start": [184, 1], "end": [185, 55], "traced_tactics": [{"tactic": "simpa only [range_eq_map] using Submodule.map_zero _", "annotated_tactic": ["simpa only [<a>range_eq_map</a>] using <a>Submodule.map_zero</a> _", [{"full_name": "LinearMap.range_eq_map", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [76, 9], "def_end_pos": [76, 21]}, {"full_name": "Submodule.map_zero", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [116, 9], "def_end_pos": [116, 17]}]], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nK : Type u_4\nK\u2082 : Type u_5\nM : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nV : Type u_9\nV\u2082 : Type u_10\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_11\ninst\u271d\u00b2 : FunLike F M M\u2082\ninst\u271d\u00b9 : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\n\u22a2 range 0 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "MetricSpace.ext", "start": [45, 1], "end": [47, 45], "traced_tactics": [{"tactic": "cases m", "annotated_tactic": ["cases m", []], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm m' : MetricSpace \u03b1\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 m = m'", "state_after": "case mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm' : MetricSpace \u03b1\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d = m'"}, {"tactic": "cases m'", "annotated_tactic": ["cases m'", []], "state_before": "case mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm' : MetricSpace \u03b1\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d = m'", "state_after": "case mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d\u00b9 = mk eq_of_dist_eq_zero\u271d"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d\u00b9 = mk eq_of_dist_eq_zero\u271d", "state_after": "case mk.mk.e_toPseudoMetricSpace\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 toPseudoMetricSpace\u271d\u00b9 = toPseudoMetricSpace\u271d"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case mk.mk.e_toPseudoMetricSpace\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 toPseudoMetricSpace\u271d\u00b9 = toPseudoMetricSpace\u271d", "state_after": "case mk.mk.e_toPseudoMetricSpace.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 PseudoMetricSpace.toDist = PseudoMetricSpace.toDist"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case mk.mk.e_toPseudoMetricSpace.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 PseudoMetricSpace.toDist = PseudoMetricSpace.toDist", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image2_image_right", "start": [219, 1], "end": [221, 12], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\n\u22a2 image2 f s (g '' t) = image2 (fun a b => f a (g b)) s t", "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f s (g '' t) \u2194 x\u271d \u2208 image2 (fun a b => f a (g b)) s t"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f s (g '' t) \u2194 x\u271d \u2208 image2 (fun a b => f a (g b)) s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Lindelof.lean", "full_name": "IsLindelof.elim_nhds_subcover'", "start": [153, 1], "end": [165, 79], "traced_tactics": [{"tactic": "have := hs.elim_countable_subcover (fun x : s \u21a6 interior (U x x.2)) (fun _ \u21a6 isOpen_interior)\n  fun x hx \u21a6\n    mem_iUnion.2 \u27e8\u27e8x, hx\u27e9, mem_interior_iff_mem_nhds.2 <| hU _ _\u27e9", "annotated_tactic": ["have := hs.elim_countable_subcover (fun x : s \u21a6 <a>interior</a> (U x x.2)) (fun _ \u21a6 <a>isOpen_interior</a>)\n    fun x hx \u21a6\n      <a>mem_iUnion</a>.2 \u27e8\u27e8x, hx\u27e9, <a>mem_interior_iff_mem_nhds</a>.2 <| hU _ _\u27e9", [{"full_name": "interior", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [111, 5], "def_end_pos": [111, 13]}, {"full_name": "isOpen_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "mem_interior_iff_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [852, 9], "def_end_pos": [852, 34]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, U \u2191x \u22ef", "state_after": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nthis : \u2203 r, r.Countable \u2227 s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, U \u2191x \u22ef"}, {"tactic": "rcases this with \u27e8r, \u27e8hr, hs\u27e9\u27e9", "annotated_tactic": ["rcases this with \u27e8r, \u27e8hr, hs\u27e9\u27e9", []], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nthis : \u2203 r, r.Countable \u2227 s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, U \u2191x \u22ef", "state_after": "case intro.intro\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, U \u2191x \u22ef"}, {"tactic": "use r, hr", "annotated_tactic": ["use r, hr", []], "state_before": "case intro.intro\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, U \u2191x \u22ef", "state_after": "case right\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 s \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef"}, {"tactic": "apply Subset.trans hs", "annotated_tactic": ["apply <a>Subset.trans</a> hs", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 21]}]], "state_before": "case right\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 s \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef", "state_after": "case right\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef"}, {"tactic": "apply iUnion\u2082_subset", "annotated_tactic": ["apply <a>iUnion\u2082_subset</a>", [{"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}]], "state_before": "case right\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef", "state_after": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2200 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\n\u22a2 \u2200 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef", "state_after": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\ni : \u2191s\nhi : i \u2208 r\n\u22a2 (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef"}, {"tactic": "apply Subset.trans interior_subset", "annotated_tactic": ["apply <a>Subset.trans</a> <a>interior_subset</a>", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 21]}, {"full_name": "interior_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 24]}]], "state_before": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\ni : \u2191s\nhi : i \u2208 r\n\u22a2 (fun x => interior (U \u2191x \u22ef)) i \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef", "state_after": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\ni : \u2191s\nhi : i \u2208 r\n\u22a2 U \u2191i \u22ef \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef"}, {"tactic": "exact subset_iUnion_of_subset i (subset_iUnion_of_subset hi (Subset.refl _))", "annotated_tactic": ["exact <a>subset_iUnion_of_subset</a> i (<a>subset_iUnion_of_subset</a> hi (<a>Subset.refl</a> _))", [{"full_name": "Set.subset_iUnion_of_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [298, 9], "def_end_pos": [298, 32]}, {"full_name": "Set.subset_iUnion_of_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [298, 9], "def_end_pos": [298, 32]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 20]}]], "state_before": "case right.h\nX : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\nhs\u271d : IsLindelof s\nU : (x : X) \u2192 x \u2208 s \u2192 Set X\nhU : \u2200 (x : X) (hx : x \u2208 s), U x hx \u2208 \ud835\udcdd x\nr : Set \u2191s\nhr : r.Countable\nhs : s \u2286 \u22c3 i \u2208 r, (fun x => interior (U \u2191x \u22ef)) i\ni : \u2191s\nhi : i \u2208 r\n\u22a2 U \u2191i \u22ef \u2286 \u22c3 x \u2208 r, U \u2191x \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exponential.lean", "full_name": "hasFDerivAt_exp_zero_of_radius_pos", "start": [77, 1], "end": [79, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Gluing.lean", "full_name": "TopCat.GlueData.isOpen_iff", "start": [105, 1], "end": [116, 32], "traced_tactics": [{"tactic": "delta CategoryTheory.GlueData.\u03b9", "annotated_tactic": ["delta <a>CategoryTheory.GlueData.\u03b9</a>", [{"full_name": "CategoryTheory.GlueData.\u03b9", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [201, 5], "def_end_pos": [201, 6]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u21d1(D.\u03b9 i) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u21d1(Multicoequalizer.\u03c0 D.diagram i) \u207b\u00b9' U)"}, {"tactic": "simp_rw [\u2190 Multicoequalizer.\u03b9_sigma\u03c0 \ud835\udda3.diagram]", "annotated_tactic": ["simp_rw [\u2190 <a>Multicoequalizer.\u03b9_sigma\u03c0</a> \ud835\udda3.<a>diagram</a>]", [{"full_name": "CategoryTheory.Limits.Multicoequalizer.\u03b9_sigma\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "def_pos": [915, 9], "def_end_pos": [915, 17]}, {"full_name": "CategoryTheory.GlueData.diagram", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [138, 5], "def_end_pos": [138, 12]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u21d1(Multicoequalizer.\u03c0 D.diagram i) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "rw [\u2190 (homeoOfIso (Multicoequalizer.isoCoequalizer \ud835\udda3.diagram).symm).isOpen_preimage]", "annotated_tactic": ["rw [\u2190 (<a>homeoOfIso</a> (<a>Multicoequalizer.isoCoequalizer</a> \ud835\udda3.<a>diagram</a>).<a>symm</a>).<a>isOpen_preimage</a>]", [{"full_name": "TopCat.homeoOfIso", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [164, 5], "def_end_pos": [164, 15]}, {"full_name": "CategoryTheory.Limits.Multicoequalizer.isoCoequalizer", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "def_pos": [902, 5], "def_end_pos": [902, 19]}, {"full_name": "CategoryTheory.GlueData.diagram", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [138, 5], "def_end_pos": [138, 12]}, {"full_name": "CategoryTheory.Iso.symm", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "Homeomorph.isOpen_preimage", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [365, 9], "def_end_pos": [365, 24]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "rw [coequalizer_isOpen_iff]", "annotated_tactic": ["rw [<a>coequalizer_isOpen_iff</a>]", [{"full_name": "TopCat.coequalizer_isOpen_iff", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "def_pos": [466, 9], "def_end_pos": [466, 31]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "dsimp only [GlueData.diagram_l, GlueData.diagram_left, GlueData.diagram_r, GlueData.diagram_right,\n  parallelPair_obj_one]", "annotated_tactic": ["dsimp only [<a>GlueData.diagram_l</a>, <a>GlueData.diagram_left</a>, <a>GlueData.diagram_r</a>, <a>GlueData.diagram_right</a>,\n    <a>parallelPair_obj_one</a>]", [{"full_name": "CategoryTheory.GlueData.diagram_l", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [150, 9], "def_end_pos": [150, 18]}, {"full_name": "CategoryTheory.GlueData.diagram_left", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [182, 9], "def_end_pos": [182, 21]}, {"full_name": "CategoryTheory.GlueData.diagram_r", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "CategoryTheory.GlueData.diagram_right", "def_path": "Mathlib/CategoryTheory/GlueData.lean", "def_pos": [187, 9], "def_end_pos": [187, 22]}, {"full_name": "CategoryTheory.Limits.parallelPair_obj_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [229, 9], "def_end_pos": [229, 29]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.diagram.right i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "rw [colimit_isOpen_iff.{_,u}]", "annotated_tactic": ["rw [<a>colimit_isOpen_iff</a>.{_,u}]", [{"full_name": "TopCat.colimit_isOpen_iff", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "def_pos": [459, 9], "def_end_pos": [459, 27]}]], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 IsOpen\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "D : GlueData\nU : Set \u2191D.glued\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))) \u2194\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "case mp\nD : GlueData\nU : Set \u2191D.glued\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))) \u2192\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)\n\ncase mpr\nD : GlueData\nU : Set \u2191D.glued\n\u22a2 (\u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)) \u2192\n    \u2200 (j : Discrete D.J),\n      IsOpen\n        (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))"}, {"tactic": "intro h j", "annotated_tactic": ["intro h j", []], "state_before": "case mp\nD : GlueData\nU : Set \u2191D.glued\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))) \u2192\n    \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)", "state_after": "case mp\nD : GlueData\nU : Set \u2191D.glued\nh :\n  \u2200 (j : Discrete D.J),\n    IsOpen\n      (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n        (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n          (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))\nj : D.J\n\u22a2 IsOpen (\u21d1(Sigma.\u03b9 D.U j \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)"}, {"tactic": "exact h \u27e8j\u27e9", 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(\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n            (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))", "state_after": "case mpr\nD : GlueData\nU : Set \u2191D.glued\nh : \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)\nj : Discrete D.J\n\u22a2 IsOpen\n    (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))"}, {"tactic": "cases j", "annotated_tactic": ["cases j", []], "state_before": "case mpr\nD : GlueData\nU : Set \u2191D.glued\nh : \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)\nj : Discrete D.J\n\u22a2 IsOpen\n    (\u21d1(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))", "state_after": "case mpr.mk\nD : GlueData\nU : Set \u2191D.glued\nh : \u2200 (i : D.J), IsOpen (\u21d1(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 D.diagram) \u207b\u00b9' U)\nas\u271d : D.J\n\u22a2 IsOpen\n    (\u21d1(colimit.\u03b9 (Discrete.functor D.U) { as := as\u271d }) \u207b\u00b9'\n      (\u21d1(colimit.\u03b9 (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) WalkingParallelPair.one) \u207b\u00b9'\n        (\u21d1(homeoOfIso (Multicoequalizer.isoCoequalizer D.diagram).symm) \u207b\u00b9' U)))"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mpr.mk\nD : GlueData\nU : Set \u2191D.glued\nh : \u2200 (i : D.J), IsOpen 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"traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Erased.lean", "full_name": "Erased.out_mk", "start": [56, 1], "end": [59, 41], "traced_tactics": [{"tactic": "let h := (mk a).2", "annotated_tactic": ["let h := (<a>mk</a> a).2", [{"full_name": "Erased.mk", "def_path": "Mathlib/Data/Erased.lean", "def_pos": [33, 5], "def_end_pos": [33, 7]}]], "state_before": "\u03b1 : Sort u_1\na : \u03b1\n\u22a2 (mk a).out = a", "state_after": "\u03b1 : Sort u_1\na : \u03b1\nh : \u2203 a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n\u22a2 (mk a).out = a"}, {"tactic": "show Classical.choose h = a", "annotated_tactic": ["show <a>Classical.choose</a> h = a", [{"full_name": "Classical.choose", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [25, 19], "def_end_pos": [25, 25]}]], "state_before": "\u03b1 : Sort u_1\na : \u03b1\nh : \u2203 a_1, (fun b => a_1 = b) = 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[\u2190 eq_nil v]", "annotated_tactic": ["rw [\u2190 <a>eq_nil</a> v]", [{"full_name": "Vector3.eq_nil", "def_path": "Mathlib/Data/Vector3.lean", "def_pos": [104, 9], "def_end_pos": [104, 15]}]], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f []", "state_after": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f v"}, {"tactic": "exact fv", "annotated_tactic": ["exact fv", []], "state_before": "\u03b1 : Type u_1\nm n : \u2115\nf : Vector3 \u03b1 0 \u2192 Prop\nx\u271d : Exists f\nv : Vector3 \u03b1 0\nfv : f v\n\u22a2 f v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean", "full_name": "TangentBundle.trivializationAt_continuousLinearMapAt", "start": [308, 1], "end": [312, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "negSucc_zsmul", "start": [1045, 1], "end": [1048, 36], "traced_tactics": [{"tactic": "rw [\u2190 natCast_zsmul]", "annotated_tactic": ["rw [\u2190 <a>natCast_zsmul</a>]", [{"full_name": "natCast_zsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1020, 41], "def_end_pos": [1020, 54]}]], "state_before": "G\u271d : Type u_1\ninst\u271d\u00b9 : DivInvMonoid G\u271d\na\u271d b : G\u271d\nG : Type u_2\ninst\u271d : SubNegMonoid G\na : G\nn : \u2115\n\u22a2 Int.negSucc n \u2022 a = -((n + 1) \u2022 a)", "state_after": "G\u271d : Type u_1\ninst\u271d\u00b9 : DivInvMonoid G\u271d\na\u271d b : G\u271d\nG : Type u_2\ninst\u271d : SubNegMonoid G\na : G\nn : \u2115\n\u22a2 Int.negSucc n \u2022 a = -(\u2191(n + 1) \u2022 a)"}, {"tactic": "exact SubNegMonoid.zsmul_neg' n a", "annotated_tactic": ["exact <a>SubNegMonoid.zsmul_neg'</a> n a", [{"full_name": "SubNegMonoid.zsmul_neg'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [989, 13], "def_end_pos": [989, 23]}]], "state_before": "G\u271d : Type u_1\ninst\u271d\u00b9 : DivInvMonoid G\u271d\na\u271d b : G\u271d\nG : Type u_2\ninst\u271d : SubNegMonoid G\na : G\nn : \u2115\n\u22a2 Int.negSucc n \u2022 a = -(\u2191(n + 1) \u2022 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/ChosenFiniteProducts/FunctorCategory.lean", "full_name": "CategoryTheory.Functor.Monoidal.tensorHom_app_snd", "start": [104, 1], "end": [109, 6], "traced_tactics": [{"tactic": "change (f \u2297 g).app j \u226b (snd F\u2081' F\u2082').app j = _", "annotated_tactic": ["change (f \u2297 g).<a>app</a> j \u226b (<a>snd</a> F\u2081' F\u2082').<a>app</a> j = _", [{"full_name": "CategoryTheory.NatTrans.app", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [50, 3], "def_end_pos": [50, 6]}, {"full_name": "CategoryTheory.ChosenFiniteProducts.snd", "def_path": "Mathlib/CategoryTheory/ChosenFiniteProducts.lean", "def_pos": [94, 5], "def_end_pos": [94, 8]}, {"full_name": "CategoryTheory.NatTrans.app", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [50, 3], "def_end_pos": [50, 6]}]], "state_before": "J : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} J\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : ChosenFiniteProducts C\nF\u2081 F\u2081' F\u2082 F\u2082' : J \u2964 C\nf : F\u2081 \u27f6 F\u2081'\ng : F\u2082 \u27f6 F\u2082'\nj : J\n\u22a2 (f \u2297 g).app j \u226b snd (F\u2081'.obj j) (F\u2082'.obj j) = snd (F\u2081.obj j) (F\u2082.obj j) \u226b g.app j", "state_after": "J : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} J\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : ChosenFiniteProducts C\nF\u2081 F\u2081' F\u2082 F\u2082' : J \u2964 C\nf : F\u2081 \u27f6 F\u2081'\ng : F\u2082 \u27f6 F\u2082'\nj : J\n\u22a2 (f \u2297 g).app j \u226b (snd F\u2081' F\u2082').app j = snd (F\u2081.obj j) (F\u2082.obj j) \u226b g.app j"}, {"tactic": "rw [\u2190 NatTrans.comp_app, tensorHom_snd, NatTrans.comp_app]", "annotated_tactic": ["rw [\u2190 <a>NatTrans.comp_app</a>, <a>tensorHom_snd</a>, <a>NatTrans.comp_app</a>]", [{"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "CategoryTheory.ChosenFiniteProducts.tensorHom_snd", "def_path": "Mathlib/CategoryTheory/ChosenFiniteProducts.lean", "def_pos": [118, 7], "def_end_pos": [118, 20]}, {"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}]], "state_before": "J : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} J\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : ChosenFiniteProducts C\nF\u2081 F\u2081' F\u2082 F\u2082' : J \u2964 C\nf : F\u2081 \u27f6 F\u2081'\ng : F\u2082 \u27f6 F\u2082'\nj : J\n\u22a2 (f \u2297 g).app j \u226b (snd F\u2081' F\u2082').app j = snd (F\u2081.obj j) (F\u2082.obj j) \u226b g.app j", "state_after": "J : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} J\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : ChosenFiniteProducts C\nF\u2081 F\u2081' F\u2082 F\u2082' : J \u2964 C\nf : F\u2081 \u27f6 F\u2081'\ng : F\u2082 \u27f6 F\u2082'\nj : J\n\u22a2 (snd F\u2081 F\u2082).app j \u226b g.app j = snd (F\u2081.obj j) (F\u2082.obj j) \u226b g.app j"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "J : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_1} J\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : ChosenFiniteProducts C\nF\u2081 F\u2081' F\u2082 F\u2082' : J \u2964 C\nf : F\u2081 \u27f6 F\u2081'\ng : F\u2082 \u27f6 F\u2082'\nj : J\n\u22a2 (snd F\u2081 F\u2082).app j \u226b g.app j = snd (F\u2081.obj j) (F\u2082.obj j) \u226b g.app j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.prod.lift_fst_snd", "start": [735, 1], "end": [736, 63], "traced_tactics": [{"tactic": "ext <;> simp", "annotated_tactic": ["ext <;> simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d X Y : C\ninst\u271d : HasBinaryProduct X Y\n\u22a2 lift fst snd = \ud835\udfd9 (X \u2a2f Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/ToDFinsupp.lean", "full_name": "DFinsupp.toFinsupp_single", "start": [123, 1], "end": [126, 53], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Zero M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\ni : \u03b9\nm : M\n\u22a2 (single i m).toFinsupp = Finsupp.single i m", "state_after": "case h\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Zero M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\ni : \u03b9\nm : M\na\u271d : \u03b9\n\u22a2 (single i m).toFinsupp a\u271d = (Finsupp.single i m) a\u271d"}, {"tactic": "simp [Finsupp.single_apply, DFinsupp.single_apply]", "annotated_tactic": ["simp [<a>Finsupp.single_apply</a>, <a>DFinsupp.single_apply</a>]", [{"full_name": "Finsupp.single_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [287, 9], "def_end_pos": [287, 21]}, {"full_name": "DFinsupp.single_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [625, 9], "def_end_pos": [625, 21]}]], "state_before": "case h\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Zero M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\ni : \u03b9\nm : M\na\u271d : \u03b9\n\u22a2 (single i m).toFinsupp a\u271d = (Finsupp.single i m) a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_inf_distrib_left", "start": [493, 1], "end": [494, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Pointwise.lean", "full_name": "AddSubmonoid.mem_one", "start": [508, 1], "end": [509, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Deriv.lean", "full_name": "StrictConcaveOn.lt_slope_of_hasDerivWithinAt_Iio", "start": [825, 1], "end": [829, 76], "traced_tactics": [{"tactic": "simpa only [Pi.neg_def, slope_neg, neg_neg] using\n  neg_lt_neg (hfc.neg.slope_lt_of_hasDerivWithinAt_Iio hx hy hxy hf'.neg)", "annotated_tactic": ["simpa only [<a>Pi.neg_def</a>, <a>slope_neg</a>, <a>neg_neg</a>] using\n    <a>neg_lt_neg</a> (hfc.neg.slope_lt_of_hasDerivWithinAt_Iio hx hy hxy hf'.neg)", [{"full_name": "Pi.neg_def", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [172, 3], "def_end_pos": [172, 14]}, {"full_name": "slope_neg", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "def_pos": [96, 15], "def_end_pos": [96, 24]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "neg_lt_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1203, 32], "def_end_pos": [1203, 42]}]], "state_before": "S : Set \u211d\nf : \u211d \u2192 \u211d\nx y f' : \u211d\nhfc : StrictConcaveOn \u211d S f\nhx : x \u2208 S\nhy : y \u2208 S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n\u22a2 f' < slope f x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "lt_compl_self", "start": [847, 9], "end": [848, 30], "traced_tactics": [{"tactic": "rw [lt_iff_le_and_ne]", "annotated_tactic": ["rw [<a>lt_iff_le_and_ne</a>]", [{"full_name": "lt_iff_le_and_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\na b c : \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 a < a\u1d9c \u2194 a = \u22a5", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\na b c : \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 a \u2264 a\u1d9c \u2227 a \u2260 a\u1d9c \u2194 a = \u22a5"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\na b c : \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 a \u2264 a\u1d9c \u2227 a \u2260 a\u1d9c \u2194 a = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nadd_le_nadd", "start": [453, 1], "end": [454, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.scalar_commute", "start": [1307, 1], "end": [1308, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Analytic.lean", "full_name": "HasFiniteFPowerSeriesOnBall.hasFDerivAt", "start": [192, 1], "end": [195, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Subring.lean", "full_name": "Ring.image_closure", "start": [205, 1], "end": [222, 15], "traced_tactics": [{"tactic": "refine le_antisymm ?_ (closure_subset (RingHom.isSubring_image _ closure.isSubring) <|\n  Set.image_subset _ subset_closure)", "annotated_tactic": ["refine <a>le_antisymm</a> ?_ (<a>closure_subset</a> (<a>RingHom.isSubring_image</a> _ <a>closure.isSubring</a>) <|\n    <a>Set.image_subset</a> _ <a>subset_closure</a>)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Ring.closure_subset", "def_path": "Mathlib/Deprecated/Subring.lean", "def_pos": [192, 9], "def_end_pos": [192, 23]}, {"full_name": "RingHom.isSubring_image", "def_path": "Mathlib/Deprecated/Subring.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}, {"full_name": "Ring.closure.isSubring", "def_path": "Mathlib/Deprecated/Subring.lean", "def_pos": [168, 9], "def_end_pos": [168, 26]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Ring.subset_closure", "def_path": "Mathlib/Deprecated/Subring.lean", "def_pos": [188, 9], "def_end_pos": [188, 23]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\n\u22a2 \u21d1f '' closure s = closure (\u21d1f '' s)", "state_after": "R : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\n\u22a2 \u21d1f '' closure s \u2264 closure (\u21d1f '' s)"}, {"tactic": "rintro _ \u27e8x, hx, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, hx, rfl\u27e9", []], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\n\u22a2 \u21d1f '' closure s \u2264 closure (\u21d1f '' s)", "state_after": "case intro.intro\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 f x \u2208 closure (\u21d1f '' s)"}, {"tactic": "apply AddGroup.InClosure.recOn (motive := fun {x} _ \u21a6 f x \u2208 closure (f '' s)) hx _ <;> intros", "annotated_tactic": ["apply <a>AddGroup.InClosure.recOn</a> (motive := fun {x} _ \u21a6 f x \u2208 <a>closure</a> (f '' s)) hx _ <;> intros", [{"full_name": "AddGroup.InClosure.recOn", "def_path": "Mathlib/Deprecated/Subgroup.lean", "def_pos": [490, 11], "def_end_pos": [490, 20]}, {"full_name": "Ring.closure", "def_path": "Mathlib/Deprecated/Subring.lean", "def_pos": [99, 5], "def_end_pos": [99, 12]}]], "state_before": "case intro.intro\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 f x \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 f 0 \u2208 closure (\u21d1f '' s)\n\ncase intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 f (-a\u271d\u00b9) \u2208 closure (\u21d1f '' s)\n\ncase intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f (a\u271d\u00b2 + b\u271d) \u2208 closure (\u21d1f '' s)\n\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)"}, {"tactic": "rw [f.map_zero]", "annotated_tactic": ["rw [f.map_zero]", []], "state_before": "case intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 f 0 \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 0 \u2208 closure (\u21d1f '' s)"}, {"tactic": "apply closure.isSubring.zero_mem", "annotated_tactic": ["apply closure.isSubring.zero_mem", []], "state_before": "case intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 0 \u2208 closure (\u21d1f '' s)", "state_after": "no goals"}, {"tactic": "rw [f.map_neg]", "annotated_tactic": ["rw [f.map_neg]", []], "state_before": "case intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 f (-a\u271d\u00b9) \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 -f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)"}, {"tactic": "apply closure.isSubring.neg_mem", "annotated_tactic": ["apply closure.isSubring.neg_mem", []], "state_before": "case intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 -f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)", "state_after": "no goals"}, {"tactic": "rw [f.map_add]", "annotated_tactic": ["rw [f.map_add]", []], "state_before": "case intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f (a\u271d\u00b2 + b\u271d) \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b2 + f b\u271d \u2208 closure (\u21d1f '' s)"}, {"tactic": "apply closure.isSubring.add_mem", "annotated_tactic": ["apply closure.isSubring.add_mem", []], "state_before": "case intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b2 + f b\u271d \u2208 closure (\u21d1f '' s)", "state_after": "case intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\n\ncase intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f b\u271d \u2208 closure (\u21d1f '' s)"}, {"tactic": "assumption'", "annotated_tactic": ["assumption'", []], "state_before": "case intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\n\ncase intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 closure (\u21d1f '' s)\na_ih\u271d : f b\u271d \u2208 closure (\u21d1f '' s)\n\u22a2 f b\u271d \u2208 closure (\u21d1f '' s)", "state_after": "no goals"}, {"tactic": "apply AddGroup.mem_closure", "annotated_tactic": ["apply <a>AddGroup.mem_closure</a>", [{"full_name": "AddGroup.mem_closure", "def_path": "Mathlib/Deprecated/Subgroup.lean", "def_pos": [525, 3], "def_end_pos": [525, 14]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 closure (\u21d1f '' s)", "state_after": "case a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 Monoid.Closure (\u21d1f '' s)"}, {"tactic": "rw [\u2190 Monoid.image_closure f.to_isMonoidHom]", "annotated_tactic": ["rw [\u2190 <a>Monoid.image_closure</a> f.to_isMonoidHom]", [{"full_name": "Monoid.image_closure", "def_path": "Mathlib/Deprecated/Submonoid.lean", "def_pos": [349, 9], "def_end_pos": [349, 22]}]], "state_before": "case a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 Monoid.Closure (\u21d1f '' s)", "state_after": "case a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 \u21d1f '' Monoid.Closure s"}, {"tactic": "apply Set.mem_image_of_mem", "annotated_tactic": ["apply <a>Set.mem_image_of_mem</a>", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "case a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 f a\u271d\u00b9 \u2208 \u21d1f '' Monoid.Closure s", "state_after": "case a.h\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 a\u271d\u00b9 \u2208 Monoid.Closure s"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case a.h\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 a\u271d\u00b9 \u2208 Monoid.Closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "full_name": "CategoryTheory.Subobject.inf_factors", "start": [417, 1], "end": [423, 50], "traced_tactics": [{"tactic": "revert X Y", "annotated_tactic": ["revert X Y", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nX Y : Subobject B\nf : A \u27f6 B\n\u22a2 X.Factors f \u2227 Y.Factors f \u2192 (X \u2293 Y).Factors f", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nf : A \u27f6 B\n\u22a2 \u2200 {X Y : Subobject B}, X.Factors f \u2227 Y.Factors f \u2192 (X \u2293 Y).Factors f"}, {"tactic": "apply Quotient.ind\u2082'", "annotated_tactic": ["apply <a>Quotient.ind\u2082'</a>", [{"full_name": "Quotient.ind\u2082'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [668, 19], "def_end_pos": [668, 24]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nf : A \u27f6 B\n\u22a2 \u2200 {X Y : Subobject B}, X.Factors f \u2227 Y.Factors f \u2192 (X \u2293 Y).Factors f", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nf : A \u27f6 B\n\u22a2 \u2200 (a\u2081 a\u2082 : MonoOver B),\n    Factors (Quotient.mk'' a\u2081) f \u2227 Factors (Quotient.mk'' a\u2082) f \u2192 (Quotient.mk'' a\u2081 \u2293 Quotient.mk'' a\u2082).Factors f"}, {"tactic": "rintro X Y \u27e8\u27e8g\u2081, rfl\u27e9, \u27e8g\u2082, hg\u2082\u27e9\u27e9", "annotated_tactic": ["rintro X Y \u27e8\u27e8g\u2081, rfl\u27e9, \u27e8g\u2082, hg\u2082\u27e9\u27e9", []], "state_before": "case h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nf : A \u27f6 B\n\u22a2 \u2200 (a\u2081 a\u2082 : MonoOver B),\n    Factors (Quotient.mk'' a\u2081) f \u2227 Factors (Quotient.mk'' a\u2082) f \u2192 (Quotient.mk'' a\u2081 \u2293 Quotient.mk'' a\u2082).Factors f", "state_after": "case h.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nX Y : MonoOver B\ng\u2081 : A \u27f6 X.obj.left\ng\u2082 : A \u27f6 Y.obj.left\nhg\u2082 : g\u2082 \u226b Y.arrow = g\u2081 \u226b X.arrow\n\u22a2 (Quotient.mk'' X \u2293 Quotient.mk'' Y).Factors (g\u2081 \u226b X.arrow)"}, {"tactic": "exact \u27e8_, pullback.lift_snd_assoc _ _ hg\u2082 _\u27e9", "annotated_tactic": ["exact \u27e8_, <a>pullback.lift_snd_assoc</a> _ _ hg\u2082 _\u27e9", [{"full_name": "CategoryTheory.Limits.pullback.lift_snd_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1210, 3], "def_end_pos": [1210, 10]}]], "state_before": "case h.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nA B : C\nX Y : MonoOver B\ng\u2081 : A \u27f6 X.obj.left\ng\u2082 : A \u27f6 Y.obj.left\nhg\u2082 : g\u2082 \u226b Y.arrow = g\u2081 \u226b X.arrow\n\u22a2 (Quotient.mk'' X \u2293 Quotient.mk'' Y).Factors (g\u2081 \u226b X.arrow)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Parity.lean", "full_name": "Nat.even_xor_odd'", "start": [285, 1], "end": [288, 59], "traced_tactics": [{"tactic": "obtain \u27e8k, rfl\u27e9 | \u27e8k, rfl\u27e9 := even_or_odd n <;> use k", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 | \u27e8k, rfl\u27e9 := <a>even_or_odd</a> n <;> use k", [{"full_name": "Nat.even_or_odd", "def_path": "Mathlib/Algebra/Ring/Parity.lean", "def_pos": [278, 7], "def_end_pos": [278, 18]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n\u271d n : \u2115\n\u22a2 \u2203 k, Xor' (n = 2 * k) (n = 2 * k + 1)", "state_after": "case h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n k : \u2115\n\u22a2 Xor' (k + k = 2 * k) (k + k = 2 * k + 1)\n\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n k : \u2115\n\u22a2 Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)"}, {"tactic": "simpa only [\u2190 two_mul, eq_self_iff_true, xor_true] using (succ_ne_self (2 * k)).symm", "annotated_tactic": ["simpa only [\u2190 <a>two_mul</a>, <a>eq_self_iff_true</a>, <a>xor_true</a>] using (<a>succ_ne_self</a> (2 * k)).<a>symm</a>", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [179, 9], "def_end_pos": [179, 16]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "xor_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [302, 17], "def_end_pos": [302, 25]}, {"full_name": "Nat.succ_ne_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [612, 9], "def_end_pos": [612, 21]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n k : \u2115\n\u22a2 Xor' (k + k = 2 * k) (k + k = 2 * k + 1)", "state_after": "no goals"}, {"tactic": "simpa only [xor_true, xor_comm] using (succ_ne_self _)", "annotated_tactic": ["simpa only [<a>xor_true</a>, <a>xor_comm</a>] using (<a>succ_ne_self</a> _)", [{"full_name": "xor_true", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [302, 17], "def_end_pos": [302, 25]}, {"full_name": "xor_comm", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [309, 9], "def_end_pos": [309, 17]}, {"full_name": "Nat.succ_ne_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [612, 9], "def_end_pos": [612, 21]}]], "state_before": "case h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\nm n k : \u2115\n\u22a2 Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Engel.lean", "full_name": "LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer", "start": [192, 1], "end": [216, 89], "traced_tactics": [{"tactic": "obtain \u27e8x, hx\u2081, hx\u2082\u27e9 := SetLike.exists_of_lt hK\u2082", "annotated_tactic": ["obtain \u27e8x, hx\u2081, hx\u2082\u27e9 := <a>SetLike.exists_of_lt</a> hK\u2082", [{"full_name": "SetLike.exists_of_lt", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 21]}]], "state_before": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'"}, {"tactic": "let K' : LieSubalgebra R L :=\n  { (R \u2219 x) \u2294 (K : Submodule R L) with\n    lie_mem' := fun {y z} => LieSubalgebra.lie_mem_sup_of_mem_normalizer hx\u2081 }", "annotated_tactic": ["let K' : <a>LieSubalgebra</a> R L :=\n    { (R \u2219 x) \u2294 (K : <a>Submodule</a> R L) with\n      lie_mem' := fun {y z} => <a>LieSubalgebra.lie_mem_sup_of_mem_normalizer</a> hx\u2081 }", [{"full_name": "LieSubalgebra", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [41, 11], "def_end_pos": [41, 24]}, {"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [37, 11], "def_end_pos": [37, 20]}, {"full_name": "LieSubalgebra.lie_mem_sup_of_mem_normalizer", "def_path": "Mathlib/Algebra/Lie/Normalizer.lean", "def_pos": [158, 9], "def_end_pos": [158, 38]}]], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'"}, {"tactic": "have hxK' : x \u2208 K' := Submodule.mem_sup_left (Submodule.subset_span (Set.mem_singleton _))", "annotated_tactic": ["have hxK' : x \u2208 K' := <a>Submodule.mem_sup_left</a> (<a>Submodule.subset_span</a> (<a>Set.mem_singleton</a> _))", [{"full_name": "Submodule.mem_sup_left", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [272, 9], "def_end_pos": [272, 21]}, {"full_name": "Submodule.subset_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [78, 9], "def_end_pos": [78, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'"}, {"tactic": "have hKK' : K \u2264 K' := (LieSubalgebra.coe_submodule_le_coe_submodule K K').mp le_sup_right", "annotated_tactic": ["have hKK' : K \u2264 K' := (<a>LieSubalgebra.coe_submodule_le_coe_submodule</a> K K').<a>mp</a> <a>le_sup_right</a>", [{"full_name": "LieSubalgebra.coe_submodule_le_coe_submodule", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [419, 9], "def_end_pos": [419, 39]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'"}, {"tactic": "have hK' : K' \u2264 K.normalizer := by\n  rw [\u2190 LieSubalgebra.coe_submodule_le_coe_submodule]\n  exact sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx\u2081) hK\u2082.le", "annotated_tactic": ["have hK' : K' \u2264 K.normalizer := by\n    rw [\u2190 <a>LieSubalgebra.coe_submodule_le_coe_submodule</a>]\n    exact <a>sup_le</a> ((<a>Submodule.span_singleton_le_iff_mem</a> _ _).<a>mpr</a> hx\u2081) hK\u2082.le", [{"full_name": "LieSubalgebra.coe_submodule_le_coe_submodule", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [419, 9], "def_end_pos": [419, 39]}, {"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [143, 9], "def_end_pos": [143, 15]}, {"full_name": "Submodule.span_singleton_le_iff_mem", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [624, 9], "def_end_pos": [624, 34]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'"}, {"tactic": "refine \u27e8K', ?_, lt_iff_le_and_ne.mpr \u27e8hKK', fun contra => hx\u2082 (contra.symm \u25b8 hxK')\u27e9\u27e9", "annotated_tactic": ["refine \u27e8K', ?_, lt_iff_le_and_ne.mpr \u27e8hKK', fun contra => hx\u2082 (contra.symm \u25b8 hxK')\u27e9\u27e9", []], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\n\u22a2 \u2203 K', IsEngelian R \u21a5K' \u2227 K < K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\n\u22a2 IsEngelian R \u21a5K'"}, {"tactic": "intro M _i1 _i2 _i3 _i4 h", "annotated_tactic": ["intro M _i1 _i2 _i3 _i4 h", []], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\n\u22a2 IsEngelian R \u21a5K'", "state_after": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M"}, {"tactic": "obtain \u27e8I, hI\u2081 : (I : LieSubalgebra R K') = LieSubalgebra.ofLe hKK'\u27e9 :=\n  LieSubalgebra.exists_nested_lieIdeal_ofLe_normalizer hKK' hK'", "annotated_tactic": ["obtain \u27e8I, hI\u2081 : (I : <a>LieSubalgebra</a> R K') = <a>LieSubalgebra.ofLe</a> hKK'\u27e9 :=\n    <a>LieSubalgebra.exists_nested_lieIdeal_ofLe_normalizer</a> hKK' hK'", [{"full_name": "LieSubalgebra", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [41, 11], "def_end_pos": [41, 24]}, {"full_name": "LieSubalgebra.ofLe", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [612, 5], "def_end_pos": [612, 9]}, {"full_name": "LieSubalgebra.exists_nested_lieIdeal_ofLe_normalizer", "def_path": "Mathlib/Algebra/Lie/Normalizer.lean", "def_pos": [180, 9], "def_end_pos": [180, 47]}]], "state_before": "case intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M", "state_after": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M"}, {"tactic": "have hI\u2082 : (R \u2219 (\u27e8x, hxK'\u27e9 : K')) \u2294 (LieSubmodule.toSubmodule I) = \u22a4 := by\n  rw [\u2190 LieIdeal.coe_to_lieSubalgebra_to_submodule R K' I, hI\u2081]\n  apply Submodule.map_injective_of_injective (K' : Submodule R L).injective_subtype\n  simp", "annotated_tactic": ["have hI\u2082 : (R \u2219 (\u27e8x, hxK'\u27e9 : K')) \u2294 (LieSubmodule.toSubmodule I) = \u22a4 := by\n    rw [\u2190 <a>LieIdeal.coe_to_lieSubalgebra_to_submodule</a> R K' I, hI\u2081]\n    apply <a>Submodule.map_injective_of_injective</a> (K' : <a>Submodule</a> R L).<a>injective_subtype</a>\n    simp", [{"full_name": "LieIdeal.coe_to_lieSubalgebra_to_submodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [257, 9], "def_end_pos": [257, 51]}, {"full_name": "Submodule.map_injective_of_injective", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [357, 9], "def_end_pos": [357, 35]}, {"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [37, 11], "def_end_pos": [37, 20]}, {"full_name": "Submodule.injective_subtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [88, 9], "def_end_pos": [88, 26]}]], "state_before": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M", "state_after": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M"}, {"tactic": "have e : K \u2243\u2097\u2045R\u2046 I :=\n  (LieSubalgebra.equivOfLe hKK').trans\n    (LieEquiv.ofEq _ _ ((LieSubalgebra.coe_set_eq _ _).mpr hI\u2081.symm))", "annotated_tactic": ["have e : K \u2243\u2097\u2045R\u2046 I :=\n    (<a>LieSubalgebra.equivOfLe</a> hKK').<a>trans</a>\n      (<a>LieEquiv.ofEq</a> _ _ ((<a>LieSubalgebra.coe_set_eq</a> _ _).<a>mpr</a> hI\u2081.symm))", [{"full_name": "LieSubalgebra.equivOfLe", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [639, 19], "def_end_pos": [639, 28]}, {"full_name": "LieEquiv.trans", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [645, 5], "def_end_pos": [645, 10]}, {"full_name": "LieEquiv.ofEq", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [786, 5], "def_end_pos": [786, 9]}, {"full_name": "LieSubalgebra.coe_set_eq", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M", "state_after": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\ne : \u21a5K \u2243\u2097\u2045R\u2046 \u21a5\u2191I\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M"}, {"tactic": "have hI\u2083 : LieAlgebra.IsEngelian R I := e.isEngelian_iff.mp hK\u2081", "annotated_tactic": ["have hI\u2083 : <a>LieAlgebra.IsEngelian</a> R I := e.isEngelian_iff.mp hK\u2081", [{"full_name": "LieAlgebra.IsEngelian", "def_path": "Mathlib/Algebra/Lie/Engel.lean", "def_pos": [158, 5], "def_end_pos": [158, 26]}]], "state_before": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\ne : \u21a5K \u2243\u2097\u2045R\u2046 \u21a5\u2191I\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M", "state_after": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\ne : \u21a5K \u2243\u2097\u2045R\u2046 \u21a5\u2191I\nhI\u2083 : IsEngelian R \u21a5\u2191I\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M"}, {"tactic": "exact LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop hI\u2082 (h _) (hI\u2083 _ fun x => h x)", "annotated_tactic": ["exact <a>LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop</a> hI\u2082 (h _) (hI\u2083 _ fun x => h x)", [{"full_name": "LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop", "def_path": "Mathlib/Algebra/Lie/Engel.lean", "def_pos": [128, 9], "def_end_pos": [128, 45]}]], "state_before": "case intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4\ne : \u21a5K \u2243\u2097\u2045R\u2046 \u21a5\u2191I\nhI\u2083 : IsEngelian R \u21a5\u2191I\n\u22a2 LieModule.IsNilpotent R (\u21a5K') M", "state_after": "no goals"}, {"tactic": "rw [\u2190 LieSubalgebra.coe_submodule_le_coe_submodule]", "annotated_tactic": ["rw [\u2190 <a>LieSubalgebra.coe_submodule_le_coe_submodule</a>]", [{"full_name": "LieSubalgebra.coe_submodule_le_coe_submodule", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [419, 9], "def_end_pos": [419, 39]}]], "state_before": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 K' \u2264 K.normalizer", "state_after": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 K'.toSubmodule \u2264 K.normalizer.toSubmodule"}, {"tactic": "exact sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx\u2081) hK\u2082.le", "annotated_tactic": ["exact <a>sup_le</a> ((<a>Submodule.span_singleton_le_iff_mem</a> _ _).<a>mpr</a> hx\u2081) hK\u2082.le", [{"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [143, 9], "def_end_pos": [143, 15]}, {"full_name": "Submodule.span_singleton_le_iff_mem", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [624, 9], "def_end_pos": [624, 34]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 K'.toSubmodule \u2264 K.normalizer.toSubmodule", "state_after": "no goals"}, {"tactic": "rw [\u2190 LieIdeal.coe_to_lieSubalgebra_to_submodule R K' I, hI\u2081]", "annotated_tactic": ["rw [\u2190 <a>LieIdeal.coe_to_lieSubalgebra_to_submodule</a> R K' I, hI\u2081]", [{"full_name": "LieIdeal.coe_to_lieSubalgebra_to_submodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [257, 9], "def_end_pos": [257, 51]}]], "state_before": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 \u2191I = \u22a4", "state_after": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule = \u22a4"}, {"tactic": "apply Submodule.map_injective_of_injective (K' : Submodule R L).injective_subtype", "annotated_tactic": ["apply <a>Submodule.map_injective_of_injective</a> (K' : <a>Submodule</a> R L).<a>injective_subtype</a>", [{"full_name": "Submodule.map_injective_of_injective", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [357, 9], "def_end_pos": [357, 35]}, {"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [37, 11], "def_end_pos": [37, 20]}, {"full_name": "Submodule.injective_subtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [88, 9], "def_end_pos": [88, 26]}]], "state_before": "R : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule = \u22a4", "state_after": "case a\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.map K'.subtype (Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule) =\n    Submodule.map K'.subtype \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case a\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R \u21a5K\nhK\u2082 : K < K.normalizer\nx : L\nhx\u2081 : x \u2208 K.normalizer\nhx\u2082 : x \u2209 K\nK' : LieSubalgebra R L :=\n  let __src := Submodule.span R {x} \u2294 K.toSubmodule;\n  { toSubmodule := __src, lie_mem' := \u22ef }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 K.normalizer\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule (\u21a5K') M\n_i4 : LieModule R (\u21a5K') M\nh : \u2200 (x : \u21a5K'), _root_.IsNilpotent ((toEnd R (\u21a5K') M) x)\nI : LieIdeal R \u21a5K'\nhI\u2081 : lieIdealSubalgebra R (\u21a5K') I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.map K'.subtype (Submodule.span R {\u27e8x, hxK'\u27e9} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule) =\n    Submodule.map K'.subtype \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "full_name": "mul_lt_mul_of_pos_right", "start": [220, 1], "end": [221, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "eq_zero_of_mul_eq_self_left", "start": [273, 1], "end": [274, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.length_edges", "start": [798, 1], "end": [798, 95], "traced_tactics": [{"tactic": "simp [edges]", "annotated_tactic": ["simp [<a>edges</a>]", [{"full_name": "SimpleGraph.Walk.edges", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [543, 5], "def_end_pos": [543, 10]}]], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : G.Walk u v\n\u22a2 p.edges.length = p.length", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "csSup_pair", "start": [679, 1], "end": [680, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Dilation.lean", "full_name": "Dilation.comp_apply", "start": [341, 1], "end": [342, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Products.lean", "full_name": "CategoryTheory.Limits.Sigma.map_id", "start": [446, 1], "end": [448, 12], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b2 : Type w\n\u03b1 : Type w\u2082\n\u03b3 : Type w\u2083\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : \u03b1 \u2192 C\ninst\u271d : HasCoproduct f\n\u22a2 (map fun a => \ud835\udfd9 (f a)) = \ud835\udfd9 (\u2210 f)", "state_after": "case h\n\u03b2 : Type w\n\u03b1 : Type w\u2082\n\u03b3 : Type w\u2083\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : \u03b1 \u2192 C\ninst\u271d : HasCoproduct f\nb\u271d : \u03b1\n\u22a2 (\u03b9 f b\u271d \u226b map fun a => \ud835\udfd9 (f a)) = \u03b9 f b\u271d \u226b \ud835\udfd9 (\u2210 f)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03b2 : Type w\n\u03b1 : Type w\u2082\n\u03b3 : Type w\u2083\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : \u03b1 \u2192 C\ninst\u271d : HasCoproduct f\nb\u271d : \u03b1\n\u22a2 (\u03b9 f b\u271d \u226b map fun a => \ud835\udfd9 (f a)) = \u03b9 f b\u271d \u226b \ud835\udfd9 (\u2210 f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Codisjoint.sup_left", "start": [355, 1], "end": [356, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Polynomial.lean", "full_name": "Polynomial.exists_forall_norm_le", "start": [144, 1], "end": [148, 73], "traced_tactics": [{"tactic": "rw [eq_C_of_degree_le_zero (le_of_not_gt hp0)]", "annotated_tactic": ["rw [<a>eq_C_of_degree_le_zero</a> (<a>le_of_not_gt</a> hp0)]", [{"full_name": "Polynomial.eq_C_of_degree_le_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [640, 9], "def_end_pos": [640, 31]}, {"full_name": "le_of_not_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [330, 9], "def_end_pos": [330, 21]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : NormedRing R\ninst\u271d\u00b9 : IsAbsoluteValue norm\ninst\u271d : ProperSpace R\np : R[X]\nhp0 : \u00ac0 < p.degree\n\u22a2 \u2200 (y : R), \u2016eval (p.coeff 0) p\u2016 \u2264 \u2016eval y p\u2016", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : NormedRing R\ninst\u271d\u00b9 : IsAbsoluteValue norm\ninst\u271d : ProperSpace R\np : R[X]\nhp0 : \u00ac0 < p.degree\n\u22a2 \u2200 (y : R), \u2016eval ((C (p.coeff 0)).coeff 0) (C (p.coeff 0))\u2016 \u2264 \u2016eval y (C (p.coeff 0))\u2016"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : NormedRing R\ninst\u271d\u00b9 : IsAbsoluteValue norm\ninst\u271d : ProperSpace R\np : R[X]\nhp0 : \u00ac0 < p.degree\n\u22a2 \u2200 (y : R), \u2016eval ((C (p.coeff 0)).coeff 0) (C (p.coeff 0))\u2016 \u2264 \u2016eval y (C (p.coeff 0))\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "full_name": "Pi.mulSingle_comm", "start": [395, 1], "end": [397, 34], "traced_tactics": [{"tactic": "simp [mulSingle_apply, eq_comm]", "annotated_tactic": ["simp [<a>mulSingle_apply</a>, <a>eq_comm</a>]", [{"full_name": "Pi.mulSingle_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [386, 9], "def_end_pos": [386, 24]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f i\ni\u271d : I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : (i : I) \u2192 One (f i)\ninst\u271d\u00b2 : (i : I) \u2192 One (g i)\ninst\u271d\u00b9 : (i : I) \u2192 One (h i)\ninst\u271d : One \u03b2\ni : I\nx : \u03b2\ni' : I\n\u22a2 mulSingle i x i' = mulSingle i' x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order.lean", "full_name": "TopologicalSpace.isOpen_generateFrom_of_mem", "start": [73, 1], "end": [75, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Strong.lean", "full_name": "aux_add", "start": [159, 1], "end": [163, 61], "traced_tactics": [{"tactic": "simpa [neg_div] using aux_sub (E := E) (m := -m) ha hb hab", "annotated_tactic": ["simpa [<a>neg_div</a>] using <a>aux_sub</a> (E := E) (m := -m) ha hb hab", [{"full_name": "neg_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [117, 9], "def_end_pos": [117, 16]}, {"full_name": "_private.Mathlib.Analysis.Convex.Strong.0.aux_sub", "def_path": "Mathlib/Analysis/Convex/Strong.lean", "def_pos": [150, 15], "def_end_pos": [150, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 \u03c8 : \u211d \u2192 \u211d\ns : Set E\na b m : \u211d\nx y : E\nf : E \u2192 \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a * (f x + m / 2 * \u2016x\u2016 ^ 2) + b * (f y + m / 2 * \u2016y\u2016 ^ 2) - m / 2 * \u2016a \u2022 x + b \u2022 y\u2016 ^ 2 =\n    a * f x + b * f y + m / 2 * a * b * \u2016x - y\u2016 ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "full_name": "OrderRingHom.cancel_left", "start": [314, 1], "end": [316, 81], "traced_tactics": [{"tactic": "rw [\u2190 comp_apply, h, comp_apply]", "annotated_tactic": ["rw [\u2190 <a>comp_apply</a>, h, <a>comp_apply</a>]", [{"full_name": "OrderRingHom.comp_apply", "def_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "def_pos": [288, 9], "def_end_pos": [288, 19]}, {"full_name": "OrderRingHom.comp_apply", "def_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "def_pos": [288, 9], "def_end_pos": [288, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : NonAssocSemiring \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : NonAssocSemiring \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : NonAssocSemiring \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : NonAssocSemiring \u03b4\ninst\u271d : Preorder \u03b4\nf : \u03b2 \u2192+*o \u03b3\ng\u2081 g\u2082 : \u03b1 \u2192+*o \u03b2\nhf : Injective \u21d1f\nh : f.comp g\u2081 = f.comp g\u2082\na : \u03b1\n\u22a2 f (g\u2081 a) = f (g\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.image_subtype", "start": [1093, 1], "end": [1099, 26], "traced_tactics": [{"tactic": "rw [\u2190 Subtype.range_coe (s := s)] at hf_inj", "annotated_tactic": ["rw [\u2190 <a>Subtype.range_coe</a> (s := s)] at hf_inj", [{"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun (x : \u2191s) => \u2191x\nhf_inj : Disjoint (span R s) (ker f)\n\u22a2 LinearIndependent R fun (x : \u2191(\u21d1f '' s)) => \u2191x", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun (x : \u2191s) => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 LinearIndependent R fun (x : \u2191(\u21d1f '' s)) => \u2191x"}, {"tactic": "refine (hs.map hf_inj).to_subtype_range' ?_", "annotated_tactic": ["refine (hs.map hf_inj).<a>to_subtype_range'</a> ?_", [{"full_name": "LinearIndependent.to_subtype_range'", "def_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "def_pos": [588, 9], "def_end_pos": [588, 44]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun (x : \u2191s) => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 LinearIndependent R fun (x : \u2191(\u21d1f '' s)) => \u2191x", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun (x : \u2191s) => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 Set.range (\u21d1f \u2218 fun x => \u2191x) = \u21d1f '' s"}, {"tactic": "simp [Set.range_comp f]", "annotated_tactic": ["simp [<a>Set.range_comp</a> f]", [{"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [731, 9], "def_end_pos": [731, 19]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun (x : \u2191s) => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 Set.range (\u21d1f \u2218 fun x => \u2191x) = \u21d1f '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "full_name": "OrdinalApprox.lfpApprox_mem_fixedPoints_of_eq", "start": [148, 1], "end": [161, 35], "traced_tactics": [{"tactic": "have lfpApprox_mem_fixedPoint :\n    lfpApprox f x a \u2208 fixedPoints f := by\n  rw [mem_fixedPoints_iff, \u2190 lfpApprox_add_one f x h_init]\n  exact Monotone.eq_of_le_of_le (lfpApprox_monotone f x)\n    h_fab (SuccOrder.le_succ a) (SuccOrder.succ_le_of_lt h_ab)", "annotated_tactic": ["have lfpApprox_mem_fixedPoint :\n      <a>lfpApprox</a> f x a \u2208 <a>fixedPoints</a> f := by\n    rw [<a>mem_fixedPoints_iff</a>, \u2190 <a>lfpApprox_add_one</a> f x h_init]\n    exact <a>Monotone.eq_of_le_of_le</a> (<a>lfpApprox_monotone</a> f x)\n      h_fab (<a>SuccOrder.le_succ</a> a) (<a>SuccOrder.succ_le_of_lt</a> h_ab)", [{"full_name": "OrdinalApprox.lfpApprox", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [72, 5], "def_end_pos": [72, 14]}, {"full_name": "Function.fixedPoints", "def_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "def_pos": [132, 5], "def_end_pos": [132, 16]}, {"full_name": "Function.mem_fixedPoints_iff", "def_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 28]}, {"full_name": "OrdinalApprox.lfpApprox_add_one", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [92, 9], "def_end_pos": [92, 26]}, {"full_name": "Monotone.eq_of_le_of_le", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [942, 9], "def_end_pos": [942, 32]}, {"full_name": "OrdinalApprox.lfpApprox_monotone", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "SuccOrder.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [68, 3], "def_end_pos": [68, 10]}, {"full_name": "SuccOrder.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\n\u22a2 lfpApprox f x c \u2208 fixedPoints \u21d1f", "state_after": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x c \u2208 fixedPoints \u21d1f"}, {"tactic": "rw [lfpApprox_eq_of_mem_fixedPoints f x h_init]", "annotated_tactic": ["rw [<a>lfpApprox_eq_of_mem_fixedPoints</a> f x h_init]", [{"full_name": "OrdinalApprox.lfpApprox_eq_of_mem_fixedPoints", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [116, 9], "def_end_pos": [116, 40]}]], "state_before": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x c \u2208 fixedPoints \u21d1f", "state_after": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x ?m.12529 \u2208 fixedPoints \u21d1f\n\ncase h_ab\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 ?m.12529 \u2264 c\n\ncase h\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x ?m.12529 \u2208 fixedPoints \u21d1f\n\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 Ordinal.{u}"}, {"tactic": "rw [mem_fixedPoints_iff, \u2190 lfpApprox_add_one f x h_init]", "annotated_tactic": ["rw [<a>mem_fixedPoints_iff</a>, \u2190 <a>lfpApprox_add_one</a> f x h_init]", [{"full_name": "Function.mem_fixedPoints_iff", "def_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 28]}, {"full_name": "OrdinalApprox.lfpApprox_add_one", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [92, 9], "def_end_pos": [92, 26]}]], "state_before": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\n\u22a2 lfpApprox f x a \u2208 fixedPoints \u21d1f", "state_after": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\n\u22a2 lfpApprox f x (a + 1) = lfpApprox f x a"}, {"tactic": "exact Monotone.eq_of_le_of_le (lfpApprox_monotone f x)\n  h_fab (SuccOrder.le_succ a) (SuccOrder.succ_le_of_lt h_ab)", "annotated_tactic": ["exact <a>Monotone.eq_of_le_of_le</a> (<a>lfpApprox_monotone</a> f x)\n      h_fab (<a>SuccOrder.le_succ</a> a) (<a>SuccOrder.succ_le_of_lt</a> h_ab)", [{"full_name": "Monotone.eq_of_le_of_le", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [942, 9], "def_end_pos": [942, 32]}, {"full_name": "OrdinalApprox.lfpApprox_monotone", "def_path": "Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "SuccOrder.le_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [68, 3], "def_end_pos": [68, 10]}, {"full_name": "SuccOrder.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\n\u22a2 lfpApprox f x (a + 1) = lfpApprox f x a", "state_after": "no goals"}, {"tactic": "exact lfpApprox_mem_fixedPoint", "annotated_tactic": ["exact lfpApprox_mem_fixedPoint", []], "state_before": "\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x ?m.12529 \u2208 fixedPoints \u21d1f", "state_after": "no goals"}, {"tactic": "exact h_ac", "annotated_tactic": ["exact h_ac", []], "state_before": "case h_ab\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 a \u2264 c", "state_after": "no goals"}, {"tactic": "exact lfpApprox_mem_fixedPoint", "annotated_tactic": ["exact lfpApprox_mem_fixedPoint", []], "state_before": "case h\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nf : \u03b1 \u2192o \u03b1\nx : \u03b1\na b c : Ordinal.{u}\nh_init : x \u2264 f x\nh_ab : a < b\nh_ac : a \u2264 c\nh_fab : lfpApprox f x a = lfpApprox f x b\nlfpApprox_mem_fixedPoint : lfpApprox f x a \u2208 fixedPoints \u21d1f\n\u22a2 lfpApprox f x a \u2208 fixedPoints \u21d1f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "GradedMonoid.fst_one", "start": [165, 9], "end": [165, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "Substring.Valid.prevn", "start": [992, 1], "end": [996, 57], "traced_tactics": [{"tactic": "let \u27e8l, m, r, h\u27e9 := h.validFor", "annotated_tactic": ["let \u27e8l, m, r, h\u27e9 := h.validFor", []], "state_before": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081.reverse ++ m\u2082\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }", "state_after": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081.reverse ++ m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }"}, {"tactic": "simp only [h.toString] at e", "annotated_tactic": ["simp only [h.toString] at e", []], "state_before": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081.reverse ++ m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }", "state_after": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081.reverse ++ m\u2082\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }"}, {"tactic": "subst e", "annotated_tactic": ["subst e", []], "state_before": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081.reverse ++ m\u2082\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }", "state_after": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl r : List Char\nh : ValidFor l (m\u2081.reverse ++ m\u2082) r x\u271d\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }"}, {"tactic": "simp [h.prevn]", "annotated_tactic": ["simp [h.prevn]", []], "state_before": "m\u2082 m\u2081 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl r : List Char\nh : ValidFor l (m\u2081.reverse ++ m\u2082) r x\u271d\n\u22a2 \u2200 (n : Nat), x\u271d.prevn n { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len (List.drop n m\u2081) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.re", "start": [41, 11], "end": [43, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.transpose_one", "start": [2087, 1], "end": [2088, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.subset_sUnion_of_mem", "start": [1009, 1], "end": [1010, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_sub", "start": [2561, 1], "end": [2566, 51], "traced_tactics": [{"tactic": "revert s", "annotated_tactic": ["revert s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 count a (s - t) = count a s - count a t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t"}, {"tactic": "refine Multiset.induction_on t (by simp) fun b t IH s => ?_", "annotated_tactic": ["refine <a>Multiset.induction_on</a> t (by simp) fun b t IH s => ?_", [{"full_name": "Multiset.induction_on", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s - b ::\u2098 t) = count a s - count a (b ::\u2098 t)"}, {"tactic": "rw [sub_cons, IH]", "annotated_tactic": ["rw [<a>sub_cons</a>, IH]", [{"full_name": "Multiset.sub_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1710, 9], "def_end_pos": [1710, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s - b ::\u2098 t) = count a s - count a (b ::\u2098 t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s.erase b) - count a t = count a s - count a (b ::\u2098 t)"}, {"tactic": "rcases Decidable.eq_or_ne a b with rfl | ab", "annotated_tactic": ["rcases <a>Decidable.eq_or_ne</a> a b with rfl | ab", [{"full_name": "Decidable.eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s.erase b) - count a t = count a s - count a (b ::\u2098 t)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d t : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s.erase a) - count a t = count a s - count a (a ::\u2098 t)\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\nab : a \u2260 b\n\u22a2 count a (s.erase b) - count a t = count a s - count a (b ::\u2098 t)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), count a (s - 0) = count a s - count a 0", "state_after": "no goals"}, {"tactic": "rw [count_erase_self, count_cons_self, Nat.sub_sub, add_comm]", "annotated_tactic": ["rw [<a>count_erase_self</a>, <a>count_cons_self</a>, <a>Nat.sub_sub</a>, <a>add_comm</a>]", [{"full_name": "Multiset.count_erase_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2548, 9], "def_end_pos": [2548, 25]}, {"full_name": "Multiset.count_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2455, 9], "def_end_pos": [2455, 24]}, {"full_name": "Nat.sub_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 26]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d t : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s.erase a) - count a t = count a s - count a (a ::\u2098 t)", "state_after": "no goals"}, {"tactic": "rw [count_erase_of_ne ab, count_cons_of_ne ab]", "annotated_tactic": ["rw [<a>count_erase_of_ne</a> ab, <a>count_cons_of_ne</a> ab]", [{"full_name": "Multiset.count_erase_of_ne", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2554, 9], "def_end_pos": [2554, 26]}, {"full_name": "Multiset.count_cons_of_ne", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2460, 9], "def_end_pos": [2460, 25]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\nab : a \u2260 b\n\u22a2 count a (s.erase b) - count a t = count a s - count a (b ::\u2098 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "full_name": "OneHom.mk_coe", "start": [654, 1], "end": [655, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "IsGroupHom.to_isMonoidHom", "start": [283, 1], "end": [284, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/CompleteLattice.lean", "full_name": "CompleteLatticeHom.ext", "start": [689, 1], "end": [690, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.relabel_moveRight'", "start": [1238, 1], "end": [1240, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "List.prod_map_erase", "start": [408, 1], "end": [415, 35], "traced_tactics": [{"tactic": "obtain rfl | \u27e8ne, h\u27e9 := List.eq_or_ne_mem_of_mem h", "annotated_tactic": ["obtain rfl | \u27e8ne, h\u27e9 := <a>List.eq_or_ne_mem_of_mem</a> h", [{"full_name": "List.eq_or_ne_mem_of_mem", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [327, 9], "def_end_pos": [327, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b9 : CommMonoid M\na\u271d : M\nl\u271d l\u2081 l\u2082 : List M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 M\na b : \u03b1\nl : List \u03b1\nh : a \u2208 b :: l\n\u22a2 f a * (map f ((b :: l).erase a)).prod = (map f (b :: l)).prod", "state_after": "case inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b9 : CommMonoid M\na\u271d : M\nl\u271d l\u2081 l\u2082 : List M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nl : List \u03b1\nh : a \u2208 a :: 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"differentiableAt_id'", "start": [1101, 1], "end": [1102, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNReal/Basic.lean", "full_name": "NNReal.abs_eq", "start": [1049, 1], "end": [1050, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Rank.lean", "full_name": "LieAlgebra.rank_le_finrank_engel", "start": [197, 1], "end": [200, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "full_name": "Polynomial.coeff_eq_zero_of_lt_natTrailingDegree", "start": [261, 1], "end": [268, 33], "traced_tactics": [{"tactic": "apply coeff_eq_zero_of_lt_trailingDegree", "annotated_tactic": ["apply <a>coeff_eq_zero_of_lt_trailingDegree</a>", [{"full_name": "Polynomial.coeff_eq_zero_of_lt_trailingDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [257, 9], "def_end_pos": [257, 43]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\n\u22a2 p.coeff n = 0", "state_after": "case h\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\n\u22a2 \u2191n < p.trailingDegree"}, {"tactic": "by_cases hp : p = 0", "annotated_tactic": ["by_cases hp : p = 0", []], "state_before": "case h\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\n\u22a2 \u2191n < p.trailingDegree", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : p = 0\n\u22a2 \u2191n < p.trailingDegree\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : \u00acp = 0\n\u22a2 \u2191n < p.trailingDegree"}, {"tactic": "rw [hp, trailingDegree_zero]", "annotated_tactic": ["rw [hp, <a>trailingDegree_zero</a>]", [{"full_name": "Polynomial.trailingDegree_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [84, 9], "def_end_pos": [84, 28]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : p = 0\n\u22a2 \u2191n < p.trailingDegree", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : p = 0\n\u22a2 \u2191n < \u22a4"}, {"tactic": "exact WithTop.coe_lt_top n", "annotated_tactic": ["exact <a>WithTop.coe_lt_top</a> n", [{"full_name": "WithTop.coe_lt_top", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 19]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : p = 0\n\u22a2 \u2191n < \u22a4", "state_after": "no goals"}, {"tactic": "rw [trailingDegree_eq_natTrailingDegree hp]", "annotated_tactic": ["rw [<a>trailingDegree_eq_natTrailingDegree</a> hp]", [{"full_name": "Polynomial.trailingDegree_eq_natTrailingDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [103, 9], "def_end_pos": [103, 44]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : \u00acp = 0\n\u22a2 \u2191n < p.trailingDegree", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : \u00acp = 0\n\u22a2 \u2191n < \u2191p.natTrailingDegree"}, {"tactic": "exact WithTop.coe_lt_coe.2 h", "annotated_tactic": ["exact <a>WithTop.coe_lt_coe</a>.2 h", [{"full_name": "WithTop.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [1153, 9], "def_end_pos": [1153, 19]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn : \u2115\nh : n < p.natTrailingDegree\nhp : \u00acp = 0\n\u22a2 \u2191n < \u2191p.natTrailingDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "full_name": "AddCircle.norm_coe_eq_abs_iff", "start": [142, 1], "end": [164, 15], "traced_tactics": [{"tactic": "refine \u27e8fun hx => hx \u25b8 norm_le_half_period p hp, fun hx => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun hx => hx \u25b8 <a>norm_le_half_period</a> p hp, fun hx => ?_\u27e9", [{"full_name": "AddCircle.norm_le_half_period", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [127, 9], "def_end_pos": [127, 28]}]], "state_before": "p x : \u211d\nhp : p \u2260 0\n\u22a2 \u2016\u2191x\u2016 = |x| \u2194 |x| \u2264 |p| / 2", "state_after": "p x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "clear hx", "annotated_tactic": ["clear hx", []], "state_before": "p x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\n\u22a2 \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|", "state_after": "p x : \u211d\nhp : p \u2260 0\n\u22a2 \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|"}, {"tactic": "intro p hp hx", "annotated_tactic": ["intro p hp hx", []], "state_before": "p x : \u211d\nhp : p \u2260 0\n\u22a2 \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|", "state_after": "p\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "rcases eq_or_ne x (p / (2 : \u211d)) with (rfl | hx')", "annotated_tactic": ["rcases <a>eq_or_ne</a> x (p / (2 : \u211d)) with (rfl | hx')", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "p\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "case inl\np\u271d : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |p / 2| \u2264 p / 2\n\u22a2 \u2016\u2191(p / 2)\u2016 = |p / 2|\n\ncase inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "suffices round (p\u207b\u00b9 * x) = 0 by simp [norm_eq, this]", "annotated_tactic": ["suffices <a>round</a> (p\u207b\u00b9 * x) = 0 by simp [<a>norm_eq</a>, this]", [{"full_name": "round", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1448, 5], "def_end_pos": [1448, 10]}, {"full_name": "AddCircle.norm_eq", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [86, 9], "def_end_pos": [86, 16]}]], "state_before": "case inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "case inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 round (p\u207b\u00b9 * x) = 0"}, {"tactic": "rw [round_eq_zero_iff]", "annotated_tactic": ["rw [<a>round_eq_zero_iff</a>]", [{"full_name": "round_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1598, 9], "def_end_pos": [1598, 26]}]], "state_before": "case inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 round (p\u207b\u00b9 * x) = 0", "state_after": "case inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)"}, {"tactic": "obtain \u27e8hx\u2081, hx\u2082\u27e9 := abs_le.mp hx", "annotated_tactic": ["obtain \u27e8hx\u2081, hx\u2082\u27e9 := abs_le.mp hx", []], "state_before": "case inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)", "state_after": "case inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x \u2264 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)"}, {"tactic": "replace hx\u2082 := Ne.lt_of_le hx' hx\u2082", "annotated_tactic": ["replace hx\u2082 := <a>Ne.lt_of_le</a> hx' hx\u2082", [{"full_name": "Ne.lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}]], "state_before": "case inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x \u2264 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)", "state_after": "case inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)", "state_after": "case inr.intro.left\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 -(1 / 2) \u2264 p\u207b\u00b9 * x\n\ncase inr.intro.right\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x < 1 / 2"}, {"tactic": "rcases hp.symm.lt_or_lt with (hp | hp)", "annotated_tactic": ["rcases hp.symm.lt_or_lt with (hp | hp)", []], "state_before": "p x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "case inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|\n\ncase inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "rw [abs_eq_self.mpr hp.le] at hx", "annotated_tactic": ["rw [abs_eq_self.mpr hp.le] at hx", []], "state_before": "case inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "case inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "exact this p hp hx", "annotated_tactic": ["exact this p hp hx", []], "state_before": "case inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "no goals"}, {"tactic": "rw [abs_eq_neg_self.mpr hp.le] at hx", "annotated_tactic": ["rw [abs_eq_neg_self.mpr hp.le] at hx", []], "state_before": "case inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "case inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 -p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|"}, {"tactic": "exact this (-p) (neg_pos.mpr hp) hx", "annotated_tactic": ["exact this (-p) (neg_pos.mpr hp) hx", []], "state_before": "case inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 -p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "no goals"}, {"tactic": "simp [abs_div, abs_two]", "annotated_tactic": ["simp [<a>abs_div</a>, <a>abs_two</a>]", [{"full_name": "abs_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 16]}, {"full_name": "abs_two", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [45, 7], "def_end_pos": [45, 14]}]], "state_before": "case inl\np\u271d : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |p / 2| \u2264 p / 2\n\u22a2 \u2016\u2191(p / 2)\u2016 = |p / 2|", "state_after": "no goals"}, {"tactic": "simp [norm_eq, this]", "annotated_tactic": ["simp [<a>norm_eq</a>, this]", [{"full_name": "AddCircle.norm_eq", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [86, 9], "def_end_pos": [86, 16]}]], "state_before": "p\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nthis : round (p\u207b\u00b9 * x) = 0\n\u22a2 \u2016\u2191x\u2016 = |x|", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mul_le_mul_left hp, \u2190 mul_assoc, mul_inv_cancel hp.ne.symm, one_mul, mul_neg, \u2190\n  mul_div_assoc, mul_one]", "annotated_tactic": ["rwa [\u2190 <a>mul_le_mul_left</a> hp, \u2190 <a>mul_assoc</a>, <a>mul_inv_cancel</a> hp.ne.symm, <a>one_mul</a>, <a>mul_neg</a>, \u2190\n      <a>mul_div_assoc</a>, <a>mul_one</a>]", [{"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [265, 9], "def_end_pos": [265, 24]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [223, 15], "def_end_pos": [223, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case inr.intro.left\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 -(1 / 2) \u2264 p\u207b\u00b9 * x", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mul_lt_mul_left hp, \u2190 mul_assoc, mul_inv_cancel hp.ne.symm, one_mul, \u2190 mul_div_assoc,\n  mul_one]", "annotated_tactic": ["rwa [\u2190 <a>mul_lt_mul_left</a> hp, \u2190 <a>mul_assoc</a>, <a>mul_inv_cancel</a> hp.ne.symm, <a>one_mul</a>, \u2190 <a>mul_div_assoc</a>,\n      <a>mul_one</a>]", [{"full_name": "mul_lt_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [223, 15], "def_end_pos": [223, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case inr.intro.right\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x < 1 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.norm_compMeasurePreserving", "start": [1022, 1], "end": [1024, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Choose/Multinomial.lean", "full_name": "Nat.multinomial_insert", "start": [70, 1], "end": [72, 49], "traced_tactics": [{"tactic": "rw [\u2190 cons_eq_insert _ _ ha, multinomial_cons]", "annotated_tactic": ["rw [\u2190 <a>cons_eq_insert</a> _ _ ha, <a>multinomial_cons</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1137, 9], "def_end_pos": [1137, 23]}, {"full_name": "Nat.multinomial_cons", "def_path": "Mathlib/Data/Nat/Choose/Multinomial.lean", "def_pos": [62, 7], "def_end_pos": [62, 23]}]], "state_before": "\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nha : a \u2209 s\nf : \u03b1 \u2192 \u2115\n\u22a2 multinomial (insert a s) f = (f a + \u2211 i \u2208 s, f i).choose (f a) * multinomial s f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Prefunctor.mapPath_comp", "start": [227, 1], "end": [230, 56], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "V : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nW : Type u\u2082\ninst\u271d : Quiver W\nF : V \u2964q W\na b : V\np : Path a b\nc b\u271d : V\nq : Path b b\u271d\ne : b\u271d \u27f6 c\n\u22a2 F.mapPath (p.comp (q.cons e)) = (F.mapPath p).comp (F.mapPath (q.cons e))", "state_after": "V : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nW : Type u\u2082\ninst\u271d : Quiver W\nF : V \u2964q W\na b : V\np : Path a b\nc b\u271d : V\nq : Path b b\u271d\ne : b\u271d \u27f6 c\n\u22a2 (F.mapPath (p.comp q)).cons (F.map e) = ((F.mapPath p).comp (F.mapPath q)).cons (F.map e)"}, {"tactic": "rw [mapPath_comp p q]", "annotated_tactic": ["rw [mapPath_comp p q]", []], "state_before": "V : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nW : Type u\u2082\ninst\u271d : Quiver W\nF : V \u2964q W\na b : V\np : Path a b\nc b\u271d : V\nq : Path b b\u271d\ne : b\u271d \u27f6 c\n\u22a2 (F.mapPath (p.comp q)).cons (F.map e) = ((F.mapPath p).comp (F.mapPath q)).cons (F.map e)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.AbsolutelyContinuous.isOpenPosMeasure", "start": [76, 11], "end": [77, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean", "full_name": "Subsemiring.smul_mem_pointwise_smul_iff", "start": [106, 1], "end": [107, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/CommMon_.lean", "full_name": "CommMon_.id'", "start": [71, 1], "end": [72, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Martingale/Centering.lean", "full_name": "MeasureTheory.predictablePart_zero", "start": [50, 1], "end": [51, 65], "traced_tactics": [{"tactic": "simp_rw [predictablePart, Finset.range_zero, Finset.sum_empty]", "annotated_tactic": ["simp_rw [<a>predictablePart</a>, <a>Finset.range_zero</a>, <a>Finset.sum_empty</a>]", [{"full_name": "MeasureTheory.predictablePart", "def_path": "Mathlib/Probability/Martingale/Centering.lean", "def_pos": [45, 19], "def_end_pos": [45, 34]}, {"full_name": "Finset.range_zero", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2945, 9], "def_end_pos": [2945, 19]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 predictablePart f \u2131 \u03bc 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Piecewise.lean", "full_name": "Finset.piecewise_congr", "start": [61, 1], "end": [63, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.takeWhile_cons_of_pos", "start": [3006, 1], "end": [3008, 27], "traced_tactics": [{"tactic": "simp [takeWhile_cons, h]", "annotated_tactic": ["simp [<a>takeWhile_cons</a>, h]", [{"full_name": "List.takeWhile_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1718, 9], "def_end_pos": [1718, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nx : \u03b1\nh : p x = true\n\u22a2 takeWhile p (x :: l) = x :: takeWhile p l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Choose/Central.lean", "full_name": "Nat.two_le_centralBinom", "start": [63, 1], "end": [67, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Zlattice/Basic.lean", "full_name": "Zspan.repr_ceil_apply", "start": [109, 1], "end": [112, 87], "traced_tactics": [{"tactic": "classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n  Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n  Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]", "annotated_tactic": ["classical simp only [<a>ceil</a>, <a>zsmul_eq_smul_cast</a> K, b.repr.map_smul, <a>Finsupp.single_apply</a>,\n    <a>Finset.sum_apply'</a>, <a>Basis.repr_self</a>, <a>Finsupp.smul_single'</a>, <a>mul_one</a>, <a>Finset.sum_ite_eq'</a>, <a>coe_sum</a>,\n    <a>Finset.mem_univ</a>, <a>if_true</a>, <a>coe_smul_of_tower</a>, <a>Basis.restrictScalars_apply</a>, <a>map_sum</a>]", [{"full_name": "Zspan.ceil", "def_path": "Mathlib/Algebra/Module/Zlattice/Basic.lean", "def_pos": [98, 5], "def_end_pos": [98, 9]}, {"full_name": "zsmul_eq_smul_cast", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [404, 9], "def_end_pos": [404, 27]}, {"full_name": "Finsupp.single_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [287, 9], "def_end_pos": [287, 21]}, {"full_name": "Finset.sum_apply'", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [657, 9], "def_end_pos": [657, 26]}, {"full_name": "Basis.repr_self", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [145, 9], "def_end_pos": [145, 18]}, {"full_name": "Finsupp.smul_single'", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [1579, 9], "def_end_pos": [1579, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "Finset.sum_ite_eq'", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1330, 3], "def_end_pos": [1330, 14]}, {"full_name": "Submodule.coe_sum", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [94, 9], "def_end_pos": [94, 16]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "if_true", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [24, 17], "def_end_pos": [24, 24]}, {"full_name": "Submodule.coe_smul_of_tower", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [309, 9], "def_end_pos": [309, 26]}, {"full_name": "Basis.restrictScalars_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 36]}, {"full_name": "map_sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [295, 3], "def_end_pos": [295, 14]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm : E\ni : \u03b9\n\u22a2 (b.repr \u2191(ceil b m)) i = \u2191\u2308(b.repr m) i\u2309", "state_after": "no goals"}, {"tactic": "simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\nFinset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\nFinset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]", "annotated_tactic": ["simp only [<a>ceil</a>, <a>zsmul_eq_smul_cast</a> 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u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\n\u22a2 (t i\u2080 \u2229 \u22c2 i, t i).Nonempty", "state_after": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 t i\u2080 \u2229 \u22c2 i, t i \u2260 \u2205"}, {"tactic": "apply mt ((htc i\u2080).elim_directed_family_closed t htcl)", "annotated_tactic": ["apply <a>mt</a> ((htc i\u2080).<a>elim_directed_family_closed</a> t htcl)", [{"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "IsCompact.elim_directed_family_closed", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [242, 9], "def_end_pos": [242, 46]}]], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 t i\u2080 \u2229 \u22c2 i, t i \u2260 \u2205", "state_after": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 \u00ac(Directed (fun x x_1 => x \u2287 x_1) t \u2192 \u2203 i, t i\u2080 \u2229 t i = \u2205)"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 \u00ac(Directed (fun x x_1 => x \u2287 x_1) t \u2192 \u2203 i, t i\u2080 \u2229 t i = \u2205)", "state_after": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) t \u2227 \u2200 (i : \u03b9), (t i\u2080 \u2229 t i).Nonempty"}, {"tactic": "simp only [\u2190 nonempty_iff_ne_empty] at htn \u22a2", "annotated_tactic": ["simp only [\u2190 <a>nonempty_iff_ne_empty</a>] at htn \u22a2", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [580, 9], "def_end_pos": [580, 30]}]], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), t i \u2260 \u2205\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) t \u2227 \u2200 (i : \u03b9), (t i\u2080 \u2229 t i).Nonempty", "state_after": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) t \u2227 \u2200 (i : \u03b9), (t i\u2080 \u2229 t i).Nonempty"}, {"tactic": "refine \u27e8htd, fun i => ?_\u27e9", "annotated_tactic": ["refine \u27e8htd, fun i => ?_\u27e9", []], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) t \u2227 \u2200 (i : \u03b9), (t i\u2080 \u2229 t i).Nonempty", "state_after": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\ni : \u03b9\n\u22a2 (t i\u2080 \u2229 t i).Nonempty"}, {"tactic": "rcases htd i\u2080 i with \u27e8j, hji\u2080, hji\u27e9", "annotated_tactic": ["rcases htd i\u2080 i with \u27e8j, hji\u2080, hji\u27e9", []], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\ni : \u03b9\n\u22a2 (t i\u2080 \u2229 t i).Nonempty", "state_after": "case intro.intro\nX : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\ni j : \u03b9\nhji\u2080 : t i\u2080 \u2287 t j\nhji : t i \u2287 t j\n\u22a2 (t i\u2080 \u2229 t i).Nonempty"}, {"tactic": "exact (htn j).mono (subset_inter hji\u2080 hji)", "annotated_tactic": ["exact (htn j).<a>mono</a> (<a>subset_inter</a> hji\u2080 hji)", [{"full_name": "Set.Nonempty.mono", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [446, 9], "def_end_pos": [446, 22]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [937, 9], "def_end_pos": [937, 21]}]], "state_before": "case intro.intro\nX : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\ni j : \u03b9\nhji\u2080 : t i\u2080 \u2287 t j\nhji : t i \u2287 t j\n\u22a2 (t i\u2080 \u2229 t i).Nonempty", "state_after": "no goals"}, {"tactic": "rwa [inter_eq_right.mpr (iInter_subset _ i\u2080)] at this", "annotated_tactic": ["rwa [inter_eq_right.mpr (<a>iInter_subset</a> _ i\u2080)] at this", [{"full_name": "Set.iInter_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [282, 9], "def_end_pos": [282, 22]}]], "state_before": "X : Type u\nY : Type v\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nt : \u03b9 \u2192 Set X\nhtd : Directed (fun x x_1 => x \u2287 x_1) t\nhtn : \u2200 (i : \u03b9), (t i).Nonempty\nhtc : \u2200 (i : \u03b9), IsCompact (t i)\nhtcl : \u2200 (i : \u03b9), IsClosed (t i)\ni\u2080 : \u03b9 := h\u03b9.some\nthis : (t i\u2080 \u2229 \u22c2 i, t i).Nonempty\n\u22a2 (\u22c2 i, t i).Nonempty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "full_name": "Function.IsFixedPt.apply", "start": [94, 11], "end": [94, 90], "traced_tactics": [{"tactic": "convert hx", "annotated_tactic": ["convert hx", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf fa g : \u03b1 \u2192 \u03b1\nx\u271d y : \u03b1\nfb : \u03b2 \u2192 \u03b2\nm n k : \u2115\ne : Perm \u03b1\nx : \u03b1\nhx : IsFixedPt f x\n\u22a2 IsFixedPt f (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part", "start": [1113, 1], "end": [1140, 42], "traced_tactics": [{"tactic": "let f\u2081 := hf.toL1 f", "annotated_tactic": ["let f\u2081 := hf.toL1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal"}, {"tactic": "have eq\u2081 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  rw [Real.nnnorm_of_nonneg (le_max_right _ _)]\n  rw [Real.coe_toNNReal', NNReal.coe_mk]", "annotated_tactic": ["have eq\u2081 : <a>ENNReal.toReal</a> (\u222b\u207b a, <a>ENNReal.ofReal</a> (f a) \u2202\u03bc) = \u2016<a>Lp.posPart</a> f\u2081\u2016 := by\n    rw [<a>L1.norm_def</a>]\n    congr 1\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [<a>Lp.coeFn_posPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n    rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]\n    congr 1\n    apply <a>NNReal.eq</a>\n    rw [<a>Real.nnnorm_of_nonneg</a> (<a>le_max_right</a> _ _)]\n    rw [<a>Real.coe_toNNReal'</a>, <a>NNReal.coe_mk</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "MeasureTheory.Lp.posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1302, 5], "def_end_pos": [1302, 12]}, {"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1444, 9], "def_end_pos": [1444, 17]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [328, 9], "def_end_pos": [328, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1316, 9], "def_end_pos": [1316, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "NNReal.eq", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [97, 26], "def_end_pos": [97, 28]}, {"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1476, 9], "def_end_pos": [1476, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 34]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal"}, {"tactic": "have eq\u2082 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]\n  rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero]", "annotated_tactic": ["have eq\u2082 : <a>ENNReal.toReal</a> (\u222b\u207b a, <a>ENNReal.ofReal</a> (-f a) \u2202\u03bc) = \u2016<a>Lp.negPart</a> f\u2081\u2016 := by\n    rw [<a>L1.norm_def</a>]\n    congr 1\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [<a>Lp.coeFn_negPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n    rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]\n    congr 1\n    apply <a>NNReal.eq</a>\n    simp only [<a>Real.coe_toNNReal'</a>, <a>coe_nnnorm</a>, <a>nnnorm_neg</a>]\n    rw [<a>Real.norm_of_nonpos</a> (<a>min_le_right</a> _ _), \u2190 <a>max_neg_neg</a>, <a>neg_zero</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [194, 15], "def_end_pos": [194, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "MeasureTheory.Lp.negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1307, 5], "def_end_pos": [1307, 12]}, {"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1444, 9], "def_end_pos": [1444, 17]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [328, 9], "def_end_pos": [328, 27]}, {"full_name": "MeasureTheory.Lp.coeFn_negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 22]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "NNReal.eq", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [97, 26], "def_end_pos": [97, 28]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [764, 41], "def_end_pos": [764, 51]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [811, 30], "def_end_pos": [811, 40]}, {"full_name": "Real.norm_of_nonpos", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [48, 15], "def_end_pos": [48, 26]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal"}, {"tactic": "rw [eq\u2081, eq\u2082, integral, dif_pos, dif_pos]", "annotated_tactic": ["rw [eq\u2081, eq\u2082, <a>integral</a>, <a>dif_pos</a>, <a>dif_pos</a>]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [788, 17], "def_end_pos": [788, 25]}, {"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}, {"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal - (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 Integrable (fun a => f a) \u03bc\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d"}, {"tactic": "exact L1.integral_eq_norm_posPart_sub _", "annotated_tactic": ["exact <a>L1.integral_eq_norm_posPart_sub</a> _", [{"full_name": "MeasureTheory.L1.integral_eq_norm_posPart_sub", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [750, 9], "def_end_pos": [750, 37]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 Integrable (fun a => f a) \u03bc\n\ncase hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\neq\u2082 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016\n\u22a2 CompleteSpace \u211d", "state_after": "no goals"}, {"tactic": "rw [L1.norm_def]", "annotated_tactic": ["rw [<a>L1.norm_def</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1444, 9], "def_end_pos": [1444, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc).toReal"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc).toReal", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [328, 9], "def_end_pos": [328, 27]}]], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc", "state_after": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 (fun a => ENNReal.ofReal (f a)) =\u1da0[ae \u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a"}, {"tactic": "filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [<a>Lp.coeFn_posPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", [{"full_name": "MeasureTheory.Lp.coeFn_posPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1316, 9], "def_end_pos": [1316, 22]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\n\u22a2 (fun a => ENNReal.ofReal (f a)) =\u1da0[ae \u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a"}, {"tactic": "rw [h\u2081, h\u2082, ENNReal.ofReal]", "annotated_tactic": ["rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 (f a\u271d).toNNReal = \u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "apply NNReal.eq", "annotated_tactic": ["apply <a>NNReal.eq</a>", [{"full_name": "NNReal.eq", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [97, 26], "def_end_pos": [97, 28]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 (f a\u271d).toNNReal = \u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u2016max (f a\u271d) 0\u2016\u208a"}, {"tactic": "rw [Real.nnnorm_of_nonneg (le_max_right _ _)]", "annotated_tactic": ["rw [<a>Real.nnnorm_of_nonneg</a> (<a>le_max_right</a> _ _)]", [{"full_name": "Real.nnnorm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1476, 9], "def_end_pos": [1476, 25]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u2016max (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u27e8max (f a\u271d) 0, \u22ef\u27e9"}, {"tactic": "rw [Real.coe_toNNReal', NNReal.coe_mk]", "annotated_tactic": ["rw [<a>Real.coe_toNNReal'</a>, <a>NNReal.coe_mk</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 34]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(f a\u271d).toNNReal = \u2191\u27e8max (f a\u271d) 0, \u22ef\u27e9", "state_after": "no goals"}, {"tactic": "rw [L1.norm_def]", "annotated_tactic": ["rw [<a>L1.norm_def</a>]", [{"full_name": "MeasureTheory.L1.norm_def", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1444, 9], "def_end_pos": [1444, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = \u2016Lp.negPart f\u2081\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc).toReal"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc).toReal = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc).toReal", "state_after": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [328, 9], "def_end_pos": [328, 27]}]], "state_before": "case e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc", "state_after": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1da0[ae \u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a"}, {"tactic": "filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", "annotated_tactic": ["filter_upwards [<a>Lp.coeFn_negPart</a> f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082", [{"full_name": "MeasureTheory.Lp.coeFn_negPart", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 22]}]], "state_before": "case e_a.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\n\u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1da0[ae \u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a"}, {"tactic": "rw [h\u2081, h\u2082, ENNReal.ofReal]", "annotated_tactic": ["rw [h\u2081, h\u2082, <a>ENNReal.ofReal</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(-f a\u271d).toNNReal = \u2191\u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(-f a\u271d).toNNReal = \u2191\u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 (-f a\u271d).toNNReal = \u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "apply NNReal.eq", "annotated_tactic": ["apply <a>NNReal.eq</a>", [{"full_name": "NNReal.eq", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [97, 26], "def_end_pos": [97, 28]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 (-f a\u271d).toNNReal = \u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(-f a\u271d).toNNReal = \u2191\u2016-min (f a\u271d) 0\u2016\u208a"}, {"tactic": "simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]", "annotated_tactic": ["simp only [<a>Real.coe_toNNReal'</a>, <a>coe_nnnorm</a>, <a>nnnorm_neg</a>]", [{"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [764, 41], "def_end_pos": [764, 51]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [811, 30], "def_end_pos": [811, 40]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 \u2191(-f a\u271d).toNNReal = \u2191\u2016-min (f a\u271d) 0\u2016\u208a", "state_after": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016"}, {"tactic": "rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero]", "annotated_tactic": ["rw [<a>Real.norm_of_nonpos</a> (<a>min_le_right</a> _ _), \u2190 <a>max_neg_neg</a>, <a>neg_zero</a>]", [{"full_name": "Real.norm_of_nonpos", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 23]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "max_neg_neg", "def_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "def_pos": [48, 15], "def_end_pos": [48, 26]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}]], "state_before": "case h.e_a.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nf\u2081 : \u21a5(Lp \u211d 1 \u03bc) := Integrable.toL1 f hf\neq\u2081 : (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc).toReal = \u2016Lp.posPart f\u2081\u2016\na\u271d : \u03b1\nh\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0\nh\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d\n\u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MonoidAlgebra/Degree.lean", "full_name": "AddMonoidAlgebra.supDegree_add_le", "start": [235, 1], "end": [237, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean", "full_name": "HomogeneousLocalization.val_sub", "start": [449, 1], "end": [450, 49], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, \u2190 val_neg, \u2190 val_add]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, \u2190 <a>val_neg</a>, \u2190 <a>val_add</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "HomogeneousLocalization.val_neg", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean", "def_pos": [444, 9], "def_end_pos": [444, 16]}, {"full_name": "HomogeneousLocalization.val_add", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean", "def_pos": [434, 9], "def_end_pos": [434, 16]}]], "state_before": "\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 (y1 - y2).val = y1.val - y2.val", "state_after": "\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 (y1 - y2).val = (y1 + -y2).val"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 (y1 - y2).val = (y1 + -y2).val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_add_Icc", "start": [147, 1], "end": [148, 25], "traced_tactics": [{"tactic": "simp [\u2190 Ici_inter_Iic]", "annotated_tactic": ["simp [\u2190 <a>Ici_inter_Iic</a>]", [{"full_name": "Set.Ici_inter_Iic", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [636, 9], "def_end_pos": [636, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedAddCommGroup \u03b1\na b c : \u03b1\n\u22a2 (fun x => a + x) \u207b\u00b9' Icc b c = Icc (b - a) (c - a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.floor_eq_iff", "start": [248, 1], "end": [250, 31], "traced_tactics": [{"tactic": "rw [\u2190 le_floor_iff ha, \u2190 Nat.cast_one, \u2190 Nat.cast_add, \u2190 floor_lt ha, Nat.lt_add_one_iff,\n  le_antisymm_iff, and_comm]", "annotated_tactic": ["rw [\u2190 <a>le_floor_iff</a> ha, \u2190 <a>Nat.cast_one</a>, \u2190 <a>Nat.cast_add</a>, \u2190 <a>floor_lt</a> ha, <a>Nat.lt_add_one_iff</a>,\n    <a>le_antisymm_iff</a>, <a>and_comm</a>]", [{"full_name": "Nat.le_floor_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [131, 9], "def_end_pos": [131, 21]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.floor_lt", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}, {"full_name": "Nat.lt_add_one_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [575, 19], "def_end_pos": [575, 33]}, {"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [194, 9], "def_end_pos": [194, 24]}, {"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a = n \u2194 \u2191n \u2264 a \u2227 a < \u2191n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Infix.lean", "full_name": "List.IsPrefix.ne_nil", "start": [554, 1], "end": [555, 47], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nx y : List \u03b1\nh : x <+: y\nhx : x \u2260 []\n\u22a2 y \u2260 []", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nx : List \u03b1\nhx : x \u2260 []\nh : x <+: []\n\u22a2 False"}, {"tactic": "exact hx <| List.prefix_nil.mp h", "annotated_tactic": ["exact hx <| List.prefix_nil.mp h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nx : List \u03b1\nhx : x \u2260 []\nh : x <+: []\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_insert", "start": [1416, 1], "end": [1418, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "full_name": "OreLocalization.add_assoc", "start": [692, 11], "end": [702, 94], "traced_tactics": [{"tactic": "induction' x with r\u2081 s\u2081", "annotated_tactic": ["induction' x with r\u2081 s\u2081", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nx y z : OreLocalization S X\n\u22a2 x + y + z = x + (y + z)", "state_after": "case c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\ny z : OreLocalization S X\nr\u2081 : X\ns\u2081 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + y + z = r\u2081 /\u2092 s\u2081 + (y + z)"}, {"tactic": "induction' y with r\u2082 s\u2082", "annotated_tactic": ["induction' y with r\u2082 s\u2082", []], "state_before": "case c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\ny z : OreLocalization S X\nr\u2081 : X\ns\u2081 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + y + z = r\u2081 /\u2092 s\u2081 + (y + z)", "state_after": "case c.c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nz : OreLocalization S X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + z = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + z)"}, {"tactic": "induction' z with r\u2083 s\u2083", "annotated_tactic": ["induction' z with r\u2083 s\u2083", []], "state_before": "case c.c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nz : OreLocalization S X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + z = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + z)", "state_after": "case c.c.c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "rcases oreDivAddChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9", "annotated_tactic": ["rcases <a>oreDivAddChar'</a> r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9", [{"full_name": "OreLocalization.oreDivAddChar'", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [681, 5], "def_end_pos": [681, 19]}]], "state_before": "case c.c.c\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081)\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [ha']", "annotated_tactic": ["rw [ha']", []], "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081)\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081)\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "clear ha'", "annotated_tactic": ["clear ha'", []], "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081)\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "rcases oreDivAddChar' (sa \u2022 r\u2081 + ra \u2022 r\u2082) r\u2083 (sa * s\u2081) s\u2083 with \u27e8rc, sc, hc, q\u27e9", "annotated_tactic": ["rcases <a>oreDivAddChar'</a> (sa \u2022 r\u2081 + ra \u2022 r\u2082) r\u2083 (sa * s\u2081) s\u2083 with \u27e8rc, sc, hc, q\u27e9", [{"full_name": "OreLocalization.oreDivAddChar'", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [681, 5], "def_end_pos": [681, 19]}]], "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\nq : (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081))\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [q]", "annotated_tactic": ["rw [q]", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\nq : (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081))\n\u22a2 (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\nq : (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081))\n\u22a2 (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "clear q", "annotated_tactic": ["clear q", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\nq : (sa \u2022 r\u2081 + ra \u2022 r\u2082) /\u2092 (sa * s\u2081) + r\u2083 /\u2092 s\u2083 = (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081))\n\u22a2 (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "simp only [smul_add, mul_assoc, add_assoc]", "annotated_tactic": ["simp only [<a>smul_add</a>, <a>mul_assoc</a>, <a>add_assoc</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}]], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 (sc \u2022 (sa \u2022 r\u2081 + ra \u2022 r\u2082) + rc \u2022 r\u2083) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 (sc \u2022 sa \u2022 r\u2081 + (sc \u2022 ra \u2022 r\u2082 + rc \u2022 r\u2083)) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "simp_rw [\u2190 add_oreDiv, \u2190 OreLocalization.expand']", "annotated_tactic": ["simp_rw [\u2190 <a>add_oreDiv</a>, \u2190 <a>OreLocalization.expand'</a>]", [{"full_name": "OreLocalization.add_oreDiv", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [688, 9], "def_end_pos": [688, 19]}, {"full_name": "OreLocalization.expand'", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [132, 19], "def_end_pos": [132, 26]}]], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 (sc \u2022 sa \u2022 r\u2081 + (sc \u2022 ra \u2022 r\u2082 + rc \u2022 r\u2083)) /\u2092 (sc * (sa * s\u2081)) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 r\u2081 /\u2092 s\u2081 + (ra \u2022 r\u2082 /\u2092 (sa * s\u2081) + rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081))) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 r\u2081 /\u2092 s\u2081 + (ra \u2022 r\u2082 /\u2092 (sa * s\u2081) + rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081))) = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 ra \u2022 r\u2082 /\u2092 (sa * s\u2081) = r\u2082 /\u2092 s\u2082\n\ncase c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081)) = r\u2083 /\u2092 s\u2083"}, {"tactic": "rw [OreLocalization.expand r\u2082 s\u2082 ra (ha.symm \u25b8 (sa * s\u2081).2)]", "annotated_tactic": ["rw [<a>OreLocalization.expand</a> r\u2082 s\u2082 ra (ha.symm \u25b8 (sa * s\u2081).2)]", [{"full_name": "OreLocalization.expand", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [124, 19], "def_end_pos": [124, 25]}]], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 ra \u2022 r\u2082 /\u2092 (sa * s\u2081) = r\u2082 /\u2092 s\u2082", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 ra \u2022 r\u2082 /\u2092 (sa * s\u2081) = ra \u2022 r\u2082 /\u2092 \u27e8ra * \u2191s\u2082, \u22ef\u27e9"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 ra \u2022 r\u2082 /\u2092 (sa * s\u2081) = ra \u2022 r\u2082 /\u2092 \u27e8ra * \u2191s\u2082, \u22ef\u27e9", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 sa * s\u2081 = \u27e8ra * \u2191s\u2082, \u22ef\u27e9"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 sa * s\u2081 = \u27e8ra * \u2191s\u2082, \u22ef\u27e9", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s.a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 \u2191(sa * s\u2081) = \u2191\u27e8ra * \u2191s\u2082, \u22ef\u27e9"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s.a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 \u2191(sa * s\u2081) = \u2191\u27e8ra * \u2191s\u2082, \u22ef\u27e9", "state_after": "no goals"}, {"tactic": "rw [OreLocalization.expand r\u2083 s\u2083 rc (hc.symm \u25b8 (sc * (sa * s\u2081)).2)]", "annotated_tactic": ["rw [<a>OreLocalization.expand</a> r\u2083 s\u2083 rc (hc.symm \u25b8 (sc * (sa * s\u2081)).2)]", [{"full_name": "OreLocalization.expand", "def_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "def_pos": [124, 19], "def_end_pos": [124, 25]}]], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081)) = r\u2083 /\u2092 s\u2083", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081)) = rc \u2022 r\u2083 /\u2092 \u27e8rc * \u2191s\u2083, \u22ef\u27e9"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 rc \u2022 r\u2083 /\u2092 (sc * (sa * s\u2081)) = rc \u2022 r\u2083 /\u2092 \u27e8rc * \u2191s\u2083, \u22ef\u27e9", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 sc * (sa * s\u2081) = \u27e8rc * \u2191s\u2083, \u22ef\u27e9"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 sc * (sa * s\u2081) = \u27e8rc * \u2191s\u2083, \u22ef\u27e9", "state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s.a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 \u2191(sc * (sa * s\u2081)) = \u2191\u27e8rc * \u2191s\u2083, \u22ef\u27e9"}, {"tactic": "exact hc", "annotated_tactic": ["exact hc", []], "state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.e_a.e_a.e_s.a\nR : Type u_1\ninst\u271d\u00b3 : Monoid R\nS : Submonoid R\ninst\u271d\u00b2 : OreSet S\nX : Type u_2\ninst\u271d\u00b9 : AddMonoid X\ninst\u271d : DistribMulAction R X\nr\u2081 : X\ns\u2081 : \u21a5S\nr\u2082 : X\ns\u2082 : \u21a5S\nr\u2083 : X\ns\u2083 : \u21a5S\nra : R\nsa : \u21a5S\nha : \u2191sa * \u2191s\u2081 = ra * \u2191s\u2082\nrc : R\nsc : \u21a5S\nhc : \u2191sc * \u2191(sa * s\u2081) = rc * \u2191s\u2083\n\u22a2 \u2191(sc * (sa * s\u2081)) = \u2191\u27e8rc * \u2191s\u2083, \u22ef\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Category/Preorder.lean", "full_name": "CategoryTheory.Equivalence.toOrderIso_apply", "start": [215, 1], "end": [216, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Real.lean", "full_name": "QuadraticForm.equivalent_signType_weighted_sum_squared", "start": [75, 1], "end": [80, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "inv_mem_iff", "start": [129, 1], "end": [131, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "bddBelow_iff_exists_le", "start": [500, 1], "end": [502, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.toReal_le_toReal", "start": [76, 1], "end": [79, 12], "traced_tactics": [{"tactic": "lift a to \u211d\u22650 using ha", "annotated_tactic": ["lift a to \u211d\u22650 using ha", []], "state_before": "a b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 a.toReal \u2264 b.toReal \u2194 a \u2264 b", "state_after": "case intro\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 (\u2191a).toReal \u2264 b.toReal \u2194 \u2191a \u2264 b"}, {"tactic": "lift b to \u211d\u22650 using hb", "annotated_tactic": ["lift b to \u211d\u22650 using hb", []], "state_before": "case intro\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 (\u2191a).toReal \u2264 b.toReal \u2194 \u2191a \u2264 b", "state_after": "case intro.intro\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 (\u2191a).toReal \u2264 (\u2191b).toReal \u2194 \u2191a \u2264 \u2191b"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case intro.intro\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 (\u2191a).toReal \u2264 (\u2191b).toReal \u2194 \u2191a \u2264 \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Units.lean", "full_name": "Subgroup.mem_ofUnits_iff_toUnits_mem", "start": [314, 1], "end": [316, 91], "traced_tactics": [{"tactic": "simp_rw [mem_ofUnits_iff, toUnits.surjective.exists, val_toUnits_apply, exists_eq_right]", "annotated_tactic": ["simp_rw [<a>mem_ofUnits_iff</a>, toUnits.surjective.exists, <a>val_toUnits_apply</a>, <a>exists_eq_right</a>]", [{"full_name": "Subgroup.mem_ofUnits_iff", "def_path": "Mathlib/Algebra/Group/Submonoid/Units.lean", "def_pos": [188, 7], "def_end_pos": [188, 22]}, {"full_name": "val_toUnits_apply", "def_path": "Mathlib/Algebra/Group/Units/Equiv.lean", "def_pos": [21, 30], "def_end_pos": [21, 39]}, {"full_name": "exists_eq_right", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 32]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : Monoid M\nG : Type u_2\ninst\u271d : Group G\nH : Subgroup G\u02e3\nx : G\n\u22a2 x \u2208 H.ofUnits \u2194 toUnits x \u2208 H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.toHomeomorph_injective", "start": [676, 1], "end": [677, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.card_cycleType_eq_zero", "start": [87, 1], "end": [88, 39], "traced_tactics": [{"tactic": "rw [card_eq_zero, cycleType_eq_zero]", "annotated_tactic": ["rw [<a>card_eq_zero</a>, <a>cycleType_eq_zero</a>]", [{"full_name": "Multiset.card_eq_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [831, 9], "def_end_pos": [831, 21]}, {"full_name": "Equiv.Perm.cycleType_eq_zero", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [79, 9], "def_end_pos": [79, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 card \u03c3.cycleType = 0 \u2194 \u03c3 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.map_preimage_singleton", "start": [333, 1], "end": [335, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.top_coe_submodule", "start": [450, 1], "end": [451, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/LittleWedderburn.lean", "full_name": "Finite.isDomain_to_isField", "start": [167, 1], "end": [172, 26], "traced_tactics": [{"tactic": "classical\ncases nonempty_fintype D\nlet _ := Fintype.divisionRingOfIsDomain D\nlet _ := littleWedderburn D\nexact Field.toIsField D", "annotated_tactic": ["classical\n  cases <a>nonempty_fintype</a> D\n  let _ := <a>Fintype.divisionRingOfIsDomain</a> D\n  let _ := <a>littleWedderburn</a> D\n  exact <a>Field.toIsField</a> D", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}, {"full_name": "Fintype.divisionRingOfIsDomain", "def_path": "Mathlib/RingTheory/IntegralDomain.lean", "def_pos": [97, 5], "def_end_pos": [97, 35]}, {"full_name": "littleWedderburn", "def_path": "Mathlib/RingTheory/LittleWedderburn.lean", "def_pos": [160, 28], "def_end_pos": [160, 44]}, {"full_name": "Field.toIsField", "def_path": "Mathlib/Algebra/Field/IsField.lean", "def_pos": [47, 9], "def_end_pos": [47, 24]}]], "state_before": "D : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\n\u22a2 IsField D", "state_after": "no goals"}, {"tactic": "cases nonempty_fintype D", "annotated_tactic": ["cases <a>nonempty_fintype</a> D", [{"full_name": "nonempty_fintype", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "D : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\n\u22a2 IsField D", "state_after": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\n\u22a2 IsField D"}, {"tactic": "let _ := Fintype.divisionRingOfIsDomain D", "annotated_tactic": ["let _ := <a>Fintype.divisionRingOfIsDomain</a> D", [{"full_name": "Fintype.divisionRingOfIsDomain", "def_path": "Mathlib/RingTheory/IntegralDomain.lean", "def_pos": [97, 5], "def_end_pos": [97, 35]}]], "state_before": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\n\u22a2 IsField D", "state_after": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\nx\u271d : DivisionRing D := divisionRingOfIsDomain D\n\u22a2 IsField D"}, {"tactic": "let _ := littleWedderburn D", "annotated_tactic": ["let _ := <a>littleWedderburn</a> D", [{"full_name": "littleWedderburn", "def_path": "Mathlib/RingTheory/LittleWedderburn.lean", "def_pos": [160, 28], "def_end_pos": [160, 44]}]], "state_before": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\nx\u271d : DivisionRing D := divisionRingOfIsDomain D\n\u22a2 IsField D", "state_after": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\nx\u271d\u00b9 : DivisionRing D := divisionRingOfIsDomain D\nx\u271d : Field D := littleWedderburn D\n\u22a2 IsField D"}, {"tactic": "exact Field.toIsField D", "annotated_tactic": ["exact <a>Field.toIsField</a> D", [{"full_name": "Field.toIsField", "def_path": "Mathlib/Algebra/Field/IsField.lean", "def_pos": [47, 9], "def_end_pos": [47, 24]}]], "state_before": "case intro\nD : Type u_1\ninst\u271d\u00b2 : Finite D\ninst\u271d\u00b9 : Ring D\ninst\u271d : IsDomain D\nval\u271d : Fintype D\nx\u271d\u00b9 : DivisionRing D := divisionRingOfIsDomain D\nx\u271d : Field D := littleWedderburn D\n\u22a2 IsField D", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/TorusIntegral.lean", "full_name": "TorusIntegrable.sub", "start": [128, 18], "end": [130, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/CompactlySupported.lean", "full_name": "CompactlySupportedContinuousMap.coe_mk", "start": [97, 1], "end": [98, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.iInf_principal_finite", "start": [1074, 1], "end": [1077, 43], "traced_tactics": [{"tactic": "lift s to Finset \u03b9 using hs", "annotated_tactic": ["lift s to <a>Finset</a> \u03b9 using hs", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9\u271d : Sort x\nf\u271d g : Filter \u03b1\ns\u271d t : Set \u03b1\n\u03b9 : Type w\ns : Set \u03b9\nhs : s.Finite\nf : \u03b9 \u2192 Set \u03b1\n\u22a2 \u2a05 i \u2208 s, \ud835\udcdf (f i) = \ud835\udcdf (\u22c2 i \u2208 s, f i)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9\u271d : Sort x\nf\u271d g : Filter \u03b1\ns\u271d t : Set \u03b1\n\u03b9 : Type w\nf : \u03b9 \u2192 Set \u03b1\ns : Finset \u03b9\n\u22a2 \u2a05 i \u2208 \u2191s, \ud835\udcdf (f i) = \ud835\udcdf (\u22c2 i \u2208 \u2191s, f i)"}, {"tactic": "exact mod_cast iInf_principal_finset s f", "annotated_tactic": ["exact mod_cast <a>iInf_principal_finset</a> s f", [{"full_name": "Filter.iInf_principal_finset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 30]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9\u271d : Sort x\nf\u271d g : Filter \u03b1\ns\u271d t : Set \u03b1\n\u03b9 : Type w\nf : \u03b9 \u2192 Set \u03b1\ns : Finset \u03b9\n\u22a2 \u2a05 i \u2208 \u2191s, \ud835\udcdf (f i) = \ud835\udcdf (\u22c2 i \u2208 \u2191s, f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm'", "start": [1609, 1], "end": [1646, 44], "traced_tactics": [{"tactic": "have h_summable : \u2200\u1d50 x \u2202\u03bc, Summable fun i : \u2115 => f (i + 1) x - f i x := by\n  have h1 :\n    \u2200 n, snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n    snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau\n  have h2 :\n    \u2200 n,\n      (\u222b\u207b a, (\u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n        (\u2211' i, B i) ^ p :=\n    fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)\n  have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n    lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2\n  have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n    tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3\n  exact h4.mono fun x hx => .of_nnnorm <| ENNReal.tsum_coe_ne_top_iff_summable.mp hx.ne", "annotated_tactic": ["have h_summable : \u2200\u1d50 x \u2202\u03bc, <a>Summable</a> fun i : \u2115 => f (i + 1) x - f i x := by\n    have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i \u2208 <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau\n    have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i \u2208 <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)\n    have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2\n    have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3\n    exact h4.mono fun x hx => .of_nnnorm <| ENNReal.tsum_coe_ne_top_iff_summable.mp hx.ne", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Defs.lean", "def_pos": [91, 3], "def_end_pos": [91, 14]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [71, 5], "def_end_pos": [71, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1519, 17], "def_end_pos": [1519, 61]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1535, 17], "def_end_pos": [1535, 63]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1559, 17], "def_end_pos": [1559, 59]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1593, 17], "def_end_pos": [1593, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h :\n  \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n    atTop.Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n  refine h_summable.mono fun x hx => ?_\n  let hx_sum := hx.hasSum.tendsto_sum_nat\n  exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["have h :\n    \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n      atTop.Tendsto (fun n => \u2211 i \u2208 <a>Finset.range</a> n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n    refine h_summable.mono fun x hx => ?_\n    let hx_sum := hx.hasSum.tendsto_sum_nat\n    exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "refine h.mono fun x hx => ?_", "annotated_tactic": ["refine h.mono fun x hx => ?_", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "cases' hx with l hx", "annotated_tactic": ["cases' hx with l hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_rw_sum :\n    (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n  ext1 n\n  change\n    (\u2211 i \u2208 Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n  rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["have h_rw_sum :\n      (fun n => \u2211 i \u2208 <a>Finset.range</a> n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n    ext1 n\n    change\n      (\u2211 i \u2208 <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n    rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1693, 3], "def_end_pos": [1693, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [h_rw_sum] at hx", "annotated_tactic": ["rw [h_rw_sum] at hx", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n  ext1 n\n  abel", "annotated_tactic": ["have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n    ext1 n\n    abel", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [hf_rw]", "annotated_tactic": ["rw [hf_rw]", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8l + f 0 x, Tendsto.add_const _ hx\u27e9", "annotated_tactic": ["exact \u27e8l + f 0 x, <a>Tendsto.add_const</a> _ hx\u27e9", [{"full_name": "Filter.Tendsto.add_const", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [142, 3], "def_end_pos": [142, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "have h1 :\n  \u2200 n, snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n  snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau", "annotated_tactic": ["have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i \u2208 <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [71, 5], "def_end_pos": [71, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1519, 17], "def_end_pos": [1519, 61]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h2 :\n  \u2200 n,\n    (\u222b\u207b a, (\u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n      (\u2211' i, B i) ^ p :=\n  fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)", "annotated_tactic": ["have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i \u2208 <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1535, 17], "def_end_pos": [1535, 63]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n  lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2", "annotated_tactic": ["have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1559, 17], "def_end_pos": [1559, 59]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n  tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3", "annotated_tactic": ["have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1593, 17], "def_end_pos": [1593, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "exact h4.mono fun x hx => .of_nnnorm <| ENNReal.tsum_coe_ne_top_iff_summable.mp hx.ne", "annotated_tactic": ["exact h4.mono fun x hx => .of_nnnorm <| ENNReal.tsum_coe_ne_top_iff_summable.mp hx.ne", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i \u2208 Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i \u2208 Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "no goals"}, {"tactic": "refine h_summable.mono fun x hx => ?_", "annotated_tactic": ["refine h_summable.mono fun x hx => ?_", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "let hx_sum := hx.hasSum.tendsto_sum_nat", "annotated_tactic": ["let hx_sum := hx.hasSum.tendsto_sum_nat", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x"}, {"tactic": "change\n  (\u2211 i \u2208 Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", "annotated_tactic": ["change\n      (\u2211 i \u2208 <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i \u2208 Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x"}, {"tactic": "rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1693, 3], "def_end_pos": [1693, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i \u2208 Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 (fun n => f n x) = fun n => f n x - f 0 x + f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i \u2208 Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.isUnit_iff", "start": [1318, 1], "end": [1321, 87], "traced_tactics": [{"tactic": "rw [mul_top, h]", "annotated_tactic": ["rw [<a>mul_top</a>, h]", [{"full_name": "Ideal.mul_top", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [743, 9], "def_end_pos": [743, 16]}]], "state_before": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nh : I = \u22a4\n\u22a2 \u22a4 = I * \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.not_prime_zero", "start": [53, 24], "end": [54, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isPredLimit_of_pred_lt", "start": [387, 1], "end": [388, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.nodup_enum_map_fst", "start": [152, 1], "end": [152, 100], "traced_tactics": [{"tactic": "simp [List.nodup_range]", "annotated_tactic": ["simp [<a>List.nodup_range</a>]", [{"full_name": "List.nodup_range", "def_path": "Mathlib/Data/List/Range.lean", "def_pos": [92, 9], "def_end_pos": [92, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\nl : List \u03b1\n\u22a2 (map Prod.fst l.enum).Nodup", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "full_name": "Projectivization.Subspace.sup_span", "start": [192, 1], "end": [193, 28], "traced_tactics": [{"tactic": "rw [span_union, span_coe]", "annotated_tactic": ["rw [<a>span_union</a>, <a>span_coe</a>]", [{"full_name": "Projectivization.Subspace.span_union", "def_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "Projectivization.Subspace.span_coe", "def_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "def_pos": [113, 9], "def_end_pos": [113, 17]}]], "state_before": "K : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nW : Subspace K V\n\u22a2 W \u2294 span S = span (\u2191W \u222a S)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean", "full_name": "CategoryTheory.LocalizerMorphism.inverts", "start": [75, 1], "end": [76, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean", "full_name": "FiniteDimensional.finrank_self", "start": [449, 1], "end": [450, 34], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type w\n\u03b9' : Type w'\ninst\u271d : StrongRankCondition R\n\u22a2 Module.rank R R = \u21911", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basis.lean", "full_name": "Matrix.mem_range_scalar_iff_commute_stdBasisMatrix", "start": [265, 1], "end": [269, 7], "traced_tactics": [{"tactic": "refine \u27e8fun \u27e8r, hr\u27e9 i j _ => hr \u25b8 Commute.symm ?_, mem_range_scalar_of_commute_stdBasisMatrix\u27e9", "annotated_tactic": ["refine \u27e8fun \u27e8r, hr\u27e9 i j _ => hr \u25b8 <a>Commute.symm</a> ?_, <a>mem_range_scalar_of_commute_stdBasisMatrix</a>\u27e9", [{"full_name": "Commute.symm", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [70, 19], "def_end_pos": [70, 23]}, {"full_name": "Matrix.mem_range_scalar_of_commute_stdBasisMatrix", "def_path": "Mathlib/Data/Matrix/Basis.lean", "def_pos": [247, 9], "def_end_pos": [247, 51]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n\nM : Matrix n n \u03b1\n\u22a2 M \u2208 Set.range \u21d1(scalar n) \u2194 \u2200 (i j : n), i \u2260 j \u2192 Commute (stdBasisMatrix i j 1) M", "state_after": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n\nM : Matrix n n \u03b1\nx\u271d\u00b9 : M \u2208 Set.range \u21d1(scalar n)\ni j : n\nx\u271d : i \u2260 j\nr : \u03b1\nhr : (scalar n) r = M\n\u22a2 Commute ((scalar n) r) (stdBasisMatrix i j 1)"}, {"tactic": "rw [scalar_commute_iff]", "annotated_tactic": ["rw [<a>scalar_commute_iff</a>]", [{"full_name": "Matrix.scalar_commute_iff", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1303, 9], "def_end_pos": [1303, 27]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n\nM : Matrix n n \u03b1\nx\u271d\u00b9 : M \u2208 Set.range \u21d1(scalar n)\ni j : n\nx\u271d : i \u2260 j\nr : \u03b1\nhr : (scalar n) r = M\n\u22a2 Commute ((scalar n) r) (stdBasisMatrix i j 1)", "state_after": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n\nM : Matrix n n \u03b1\nx\u271d\u00b9 : M \u2208 Set.range \u21d1(scalar n)\ni j : n\nx\u271d : i \u2260 j\nr : \u03b1\nhr : (scalar n) r = M\n\u22a2 r \u2022 stdBasisMatrix i j 1 = MulOpposite.op r \u2022 stdBasisMatrix i j 1"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n\nM : Matrix n n \u03b1\nx\u271d\u00b9 : M \u2208 Set.range \u21d1(scalar n)\ni j : n\nx\u271d : i \u2260 j\nr : \u03b1\nhr : (scalar n) r = M\n\u22a2 r \u2022 stdBasisMatrix i j 1 = MulOpposite.op r \u2022 stdBasisMatrix i j 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "le_csInf_iff'", "start": [1153, 1], "end": [1154, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Distributions/Gaussian.lean", "full_name": "ProbabilityTheory.gaussianReal_map_add_const", "start": [254, 1], "end": [267, 75], "traced_tactics": [{"tactic": "by_cases hv : v = 0", "annotated_tactic": ["by_cases hv : v = 0", []], "state_before": "\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case pos\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : v = 0\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v\n\ncase neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v"}, {"tactic": "let e : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv", "annotated_tactic": ["let e : \u211d \u2243\u1d50 \u211d := (<a>Homeomorph.addRight</a> y).symm.toMeasurableEquiv", [{"full_name": "Homeomorph.addRight", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 14]}]], "state_before": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v"}, {"tactic": "have he' : \u2200 x, HasDerivAt e ((fun _ \u21a6 1) x) x := fun _ \u21a6 (hasDerivAt_id _).sub_const y", "annotated_tactic": ["have he' : \u2200 x, <a>HasDerivAt</a> e ((fun _ \u21a6 1) x) x := fun _ \u21a6 (<a>hasDerivAt_id</a> _).<a>sub_const</a> y", [{"full_name": "HasDerivAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "hasDerivAt_id", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 22]}, {"full_name": "HasDerivAt.sub_const", "def_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "def_pos": [339, 16], "def_end_pos": [339, 36]}]], "state_before": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v"}, {"tactic": "change (gaussianReal \u03bc v).map e.symm = gaussianReal (\u03bc + y) v", "annotated_tactic": ["change (<a>gaussianReal</a> \u03bc v).<a>map</a> e.symm = <a>gaussianReal</a> (\u03bc + y) v", [{"full_name": "ProbabilityTheory.gaussianReal", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [188, 5], "def_end_pos": [188, 17]}, {"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1212, 17], "def_end_pos": [1212, 20]}, {"full_name": "ProbabilityTheory.gaussianReal", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [188, 5], "def_end_pos": [188, 17]}]], "state_before": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\n\u22a2 Measure.map (\u21d1e.symm) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v"}, {"tactic": "ext s' hs'", "annotated_tactic": ["ext s' hs'", []], "state_before": "case neg\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\n\u22a2 Measure.map (\u21d1e.symm) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 (Measure.map (\u21d1e.symm) (gaussianReal \u03bc v)) s' = (gaussianReal (\u03bc + y) v) s'"}, {"tactic": "rw [MeasurableEquiv.gaussianReal_map_symm_apply hv e he' hs']", "annotated_tactic": ["rw [<a>MeasurableEquiv.gaussianReal_map_symm_apply</a> hv e he' hs']", [{"full_name": "MeasurableEquiv.gaussianReal_map_symm_apply", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [246, 7], "def_end_pos": [246, 57]}]], "state_before": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 (Measure.map (\u21d1e.symm) (gaussianReal \u03bc v)) s' = (gaussianReal (\u03bc + y) v) s'", "state_after": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', |1| * gaussianPDFReal \u03bc v (e x)) = (gaussianReal (\u03bc + y) v) s'"}, {"tactic": "simp only [abs_neg, abs_one, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, one_mul, ne_eq]", "annotated_tactic": ["simp only [<a>abs_neg</a>, <a>abs_one</a>, <a>MeasurableEquiv.coe_mk</a>, <a>Equiv.coe_fn_mk</a>, <a>one_mul</a>, <a>ne_eq</a>]", [{"full_name": "abs_neg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [87, 3], "def_end_pos": [87, 14]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [42, 15], "def_end_pos": [42, 22]}, {"full_name": "MeasurableEquiv.coe_mk", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean", "def_pos": [179, 9], "def_end_pos": [179, 15]}, {"full_name": "Equiv.coe_fn_mk", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [113, 17], "def_end_pos": [113, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}]], "state_before": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', |1| * gaussianPDFReal \u03bc v (e x)) = (gaussianReal (\u03bc + y) v) s'", "state_after": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal \u03bc v (e x)) = (gaussianReal (\u03bc + y) v) s'"}, {"tactic": "rw [gaussianReal_apply_eq_integral _ hv s']", "annotated_tactic": ["rw [<a>gaussianReal_apply_eq_integral</a> _ hv s']", [{"full_name": "ProbabilityTheory.gaussianReal_apply_eq_integral", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [205, 7], "def_end_pos": [205, 37]}]], "state_before": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal \u03bc v (e x)) = (gaussianReal (\u03bc + y) v) s'", "state_after": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal \u03bc v (e x)) =\n    ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal (\u03bc + y) v x)"}, {"tactic": "simp [e, gaussianPDFReal_sub _ y, Homeomorph.addRight, \u2190 sub_eq_add_neg]", "annotated_tactic": ["simp [e, <a>gaussianPDFReal_sub</a> _ y, <a>Homeomorph.addRight</a>, \u2190 <a>sub_eq_add_neg</a>]", [{"full_name": "ProbabilityTheory.gaussianPDFReal_sub", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [123, 7], "def_end_pos": [123, 26]}, {"full_name": "Homeomorph.addRight", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 14]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}]], "state_before": "case neg.h\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : \u00acv = 0\ne : \u211d \u2243\u1d50 \u211d := (Homeomorph.addRight y).symm.toMeasurableEquiv\nhe' : \u2200 (x : \u211d), HasDerivAt (\u21d1e) ((fun x => 1) x) x\ns' : Set \u211d\nhs' : MeasurableSet s'\n\u22a2 ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal \u03bc v (e x)) =\n    ENNReal.ofReal (\u222b (x : \u211d) in s', gaussianPDFReal (\u03bc + y) v x)", "state_after": "no goals"}, {"tactic": "simp only [hv, ne_eq, not_true, gaussianReal_zero_var]", "annotated_tactic": ["simp only [hv, <a>ne_eq</a>, <a>not_true</a>, <a>gaussianReal_zero_var</a>]", [{"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "not_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 17]}, {"full_name": "ProbabilityTheory.gaussianReal_zero_var", "def_path": "Mathlib/Probability/Distributions/Gaussian.lean", "def_pos": [195, 7], "def_end_pos": [195, 28]}]], "state_before": "case pos\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : v = 0\n\u22a2 Measure.map (fun x => x + y) (gaussianReal \u03bc v) = gaussianReal (\u03bc + y) v", "state_after": "case pos\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : v = 0\n\u22a2 Measure.map (fun x => x + y) (Measure.dirac \u03bc) = Measure.dirac (\u03bc + y)"}, {"tactic": "exact Measure.map_dirac (measurable_id'.add_const _) _", "annotated_tactic": ["exact <a>Measure.map_dirac</a> (measurable_id'.add_const _) _", [{"full_name": "MeasureTheory.Measure.map_dirac", "def_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "def_pos": [62, 9], "def_end_pos": [62, 18]}]], "state_before": "case pos\n\u03bc : \u211d\nv : \u211d\u22650\ny : \u211d\nhv : v = 0\n\u22a2 Measure.map (fun x => x + y) (Measure.dirac \u03bc) = Measure.dirac (\u03bc + y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "OrthonormalBasis.coe_of_orthogonal_eq_bot_mk", "start": [628, 11], "end": [630, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "full_name": "MultilinearMap.mkContinuousLinear_norm_le", "start": [1079, 1], "end": [1081, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.pred_wcovBy", "start": [648, 1], "end": [649, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean", "full_name": "StrictMonoOn.continuousWithinAt_right_of_surjOn", "start": [124, 1], "end": [129, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.pow_end", "start": [289, 11], "end": [294, 48], "traced_tactics": [{"tactic": "simpa only [pow_zero] using LipschitzWith.id", "annotated_tactic": ["simpa only [<a>pow_zero</a>] using <a>LipschitzWith.id</a>", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "LipschitzWith.id", "def_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "def_pos": [189, 19], "def_end_pos": [189, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\n\u22a2 LipschitzWith (K ^ 0) (f ^ 0)", "state_after": "no goals"}, {"tactic": "rw [pow_succ, pow_succ]", "annotated_tactic": ["rw [<a>pow_succ</a>, <a>pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ (n + 1)) (f ^ (n + 1))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) (f ^ n * f)"}, {"tactic": "exact (LipschitzWith.pow_end h n).mul_end h", "annotated_tactic": ["exact (LipschitzWith.pow_end h n).<a>mul_end</a> h", [{"full_name": "LipschitzWith.mul_end", "def_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "def_pos": [274, 19], "def_end_pos": [274, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) (f ^ n * f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Basic.lean", "full_name": "Polynomial.toFinsupp_sum", "start": [396, 1], "end": [398, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "full_name": "YoungDiagram.mk_mem_col_iff", "start": [351, 1], "end": [351, 101], "traced_tactics": [{"tactic": "simp [col]", "annotated_tactic": ["simp [<a>col</a>]", [{"full_name": "YoungDiagram.col", "def_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "def_pos": [343, 5], "def_end_pos": [343, 8]}]], "state_before": "\u03bc : YoungDiagram\ni j : \u2115\n\u22a2 (i, j) \u2208 \u03bc.col j \u2194 (i, j) \u2208 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Colex.lean", "full_name": "Finset.geomSum_ofColex_strictMono", "start": [401, 1], "end": [407, 27], "traced_tactics": [{"tactic": "rintro \u27e8s\u27e9 \u27e8t\u27e9 hst", "annotated_tactic": ["rintro \u27e8s\u27e9 \u27e8t\u27e9 hst", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictMono fun s => \u2211 k \u2208 s.ofColex, n ^ k", "state_after": "case toColex.toColex\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\nhst : { ofColex := s } < { ofColex := t }\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }"}, {"tactic": "rw [toColex_lt_toColex_iff_exists_forall_lt] at hst", "annotated_tactic": ["rw [<a>toColex_lt_toColex_iff_exists_forall_lt</a>] at hst", [{"full_name": "Finset.Colex.toColex_lt_toColex_iff_exists_forall_lt", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [273, 7], "def_end_pos": [273, 46]}]], "state_before": "case toColex.toColex\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\nhst : { ofColex := s } < { ofColex := t }\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }", "state_after": "case toColex.toColex\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\nhst : \u2203 a \u2208 t, a \u2209 s \u2227 \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }"}, {"tactic": "obtain \u27e8a, hat, has, ha\u27e9 := hst", "annotated_tactic": ["obtain \u27e8a, hat, has, ha\u27e9 := hst", []], "state_before": "case toColex.toColex\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\nhst : \u2203 a \u2208 t, a \u2209 s \u2227 \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }", "state_after": "case toColex.toColex.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }"}, {"tactic": "rw [\u2190 sum_sdiff_lt_sum_sdiff]", "annotated_tactic": ["rw [\u2190 <a>sum_sdiff_lt_sum_sdiff</a>]", [{"full_name": "Finset.sum_sdiff_lt_sum_sdiff", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}]], "state_before": "case toColex.toColex.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := s } < (fun s => \u2211 k \u2208 s.ofColex, n ^ k) { ofColex := t }", "state_after": "case toColex.toColex.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 \u2211 i \u2208 { ofColex := s }.ofColex \\ { ofColex := t }.ofColex, n ^ i <\n    \u2211 i \u2208 { ofColex := t }.ofColex \\ { ofColex := s }.ofColex, n ^ i"}, {"tactic": "exact (Nat.geomSum_lt hn <| by simpa).trans_le <| single_le_sum (fun _ _ \u21a6 by positivity) <|\n  mem_sdiff.2 \u27e8hat, has\u27e9", "annotated_tactic": ["exact (<a>Nat.geomSum_lt</a> hn <| by simpa).<a>trans_le</a> <| <a>single_le_sum</a> (fun _ _ \u21a6 by positivity) <|\n    <a>mem_sdiff</a>.2 \u27e8hat, has\u27e9", [{"full_name": "Nat.geomSum_lt", "def_path": "Mathlib/Algebra/GeomSum.lean", "def_pos": [600, 7], "def_end_pos": [600, 21]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "Finset.single_le_sum", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [198, 15], "def_end_pos": [198, 28]}, {"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2102, 9], "def_end_pos": [2102, 18]}]], "state_before": "case toColex.toColex.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 \u2211 i \u2208 { ofColex := s }.ofColex \\ { ofColex := t }.ofColex, n ^ i <\n    \u2211 i \u2208 { ofColex := t }.ofColex \\ { ofColex := s }.ofColex, n ^ i", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\n\u22a2 \u2200 k \u2208 { ofColex := s }.ofColex \\ { ofColex := t }.ofColex, k < a", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u2115\nn : \u2115\nhn : 2 \u2264 n\ns t : Finset \u2115\na : \u2115\nhat : a \u2208 t\nhas : a \u2209 s\nha : \u2200 b \u2208 s, b \u2209 t \u2192 b < a\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 { ofColex := t }.ofColex \\ { ofColex := s }.ofColex\n\u22a2 0 \u2264 n ^ x\u271d\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sym/Basic.lean", "full_name": "Sym.replicate_right_inj", "start": [359, 1], "end": [360, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Hom.lean", "full_name": "AlgHom.map_mul", "start": [246, 11], "end": [247, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/InducedShiftSequence.lean", "full_name": "CategoryTheory.Functor.ShiftSequence.induced_shiftIso_hom_app_obj", "start": [114, 1], "end": [120, 37], "traced_tactics": [{"tactic": "apply induced.shiftIso_hom_app_obj", "annotated_tactic": ["apply <a>induced.shiftIso_hom_app_obj</a>", [{"full_name": "CategoryTheory.Functor.ShiftSequence.induced.shiftIso_hom_app_obj", "def_path": "Mathlib/CategoryTheory/Shift/InducedShiftSequence.lean", "def_pos": [64, 7], "def_end_pos": [64, 27]}]], "state_before": "C : Type u_1\nD : Type u_2\nA : Type u_3\ninst\u271d\u2079 : Category.{u_7, u_1} C\ninst\u271d\u2078 : Category.{u_6, u_2} D\ninst\u271d\u2077 : Category.{u_5, u_3} A\nL : C \u2964 D\nF : D \u2964 A\nG : C \u2964 A\ne : L \u22d9 F \u2245 G\nM : Type u_4\ninst\u271d\u2076 : AddMonoid M\ninst\u271d\u2075 : HasShift C M\ninst\u271d\u2074 : HasShift D M\ninst\u271d\u00b3 : L.CommShift M\ninst\u271d\u00b2 : G.ShiftSequence M\nF' : M \u2192 D \u2964 A\ne' : (m : M) \u2192 L \u22d9 F' m \u2245 G.shift m\ninst\u271d\u00b9 : ((whiskeringLeft C D A).obj L).Full\ninst\u271d : ((whiskeringLeft C D A).obj L).Faithful\nn a a' : M\nha' : n + a = a'\nX : C\n\u22a2 (F.shiftIso n a a' ha').hom.app (L.obj X) =\n    (F.shift a).map ((L.commShiftIso n).inv.app X) \u226b\n      (e' a).hom.app ((shiftFunctor C n).obj X) \u226b (G.shiftIso n a a' ha').hom.app X \u226b (e' a').inv.app X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Spec.lean", "full_name": "AlgebraicGeometry.Spec.topMap_id", "start": [72, 1], "end": [73, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "lowerBounds_singleton", "start": [655, 1], "end": [656, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Adjoin/Basic.lean", "full_name": "AlgHom.ext_of_adjoin_eq_top", "start": [476, 1], "end": [478, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.add", "start": [466, 1], "end": [467, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Kleene.lean", "full_name": "Prod.snd_kstar", "start": [317, 1], "end": [318, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Splits.lean", "full_name": "Polynomial.degree_eq_card_roots", "start": [319, 1], "end": [321, 69], "traced_tactics": [{"tactic": "rw [degree_eq_natDegree p_ne_zero, natDegree_eq_card_roots hsplit]", "annotated_tactic": ["rw [<a>degree_eq_natDegree</a> p_ne_zero, <a>natDegree_eq_card_roots</a> hsplit]", [{"full_name": "Polynomial.degree_eq_natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [132, 9], "def_end_pos": [132, 28]}, {"full_name": "Polynomial.natDegree_eq_card_roots", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [314, 9], "def_end_pos": [314, 32]}]], "state_before": "R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\np_ne_zero : p \u2260 0\nhsplit : Splits i p\n\u22a2 p.degree = \u2191(Multiset.card (map i p).roots)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.restrictTotalDegree_le_restrictDegree", "start": [128, 1], "end": [131, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Multiset.lean", "full_name": "Sym.coe_equivNatSum_symm_apply", "start": [275, 1], "end": [277, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/EditDistance/Defs.lean", "full_name": "levenshtein_cons_cons", "start": [293, 1], "end": [299, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "full_name": "UniformOnFun.uniformContinuous_toFun", "start": [994, 11], "end": [997, 33], "traced_tactics": [{"tactic": "rw [uniformContinuous_pi]", "annotated_tactic": ["rw [<a>uniformContinuous_pi</a>]", [{"full_name": "uniformContinuous_pi", "def_path": "Mathlib/Topology/UniformSpace/Pi.lean", "def_pos": [46, 9], "def_end_pos": [46, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d : UniformSpace \u03b2\n\ud835\udd16 : Set (Set \u03b1)\nh : \u22c3\u2080 \ud835\udd16 = univ\n\u22a2 UniformContinuous \u21d1(toFun \ud835\udd16)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d : UniformSpace \u03b2\n\ud835\udd16 : Set (Set \u03b1)\nh : \u22c3\u2080 \ud835\udd16 = univ\n\u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => (toFun \ud835\udd16) x i"}, {"tactic": "exact uniformContinuous_eval h", "annotated_tactic": ["exact <a>uniformContinuous_eval</a> h", [{"full_name": "UniformOnFun.uniformContinuous_eval", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [864, 9], "def_end_pos": [864, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d : UniformSpace \u03b2\n\ud835\udd16 : Set (Set \u03b1)\nh : \u22c3\u2080 \ud835\udd16 = univ\n\u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => (toFun \ud835\udd16) x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.derivBFamily_isNormal", "start": [361, 1], "end": [363, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean", "full_name": "AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec", "start": [798, 1], "end": [805, 73], "traced_tactics": [{"tactic": "rw [awayTo\u0393_\u0393ToStalk, \u2190 toStalk_specStalkEquiv, Category.assoc]", "annotated_tactic": ["rw [<a>awayTo\u0393_\u0393ToStalk</a>, \u2190 <a>toStalk_specStalkEquiv</a>, <a>Category.assoc</a>]", [{"full_name": "AlgebraicGeometry.ProjectiveSpectrum.Proj.awayTo\u0393_\u0393ToStalk", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean", "def_pos": [632, 7], "def_end_pos": [632, 23]}, {"full_name": "AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean", "def_pos": [786, 7], "def_end_pos": [786, 29]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 awayTo\u0393 \ud835\udc9c f \u226b (Proj.restrict \u22ef).\u0393ToStalk x =\n    RingHom.comp\n      ((specStalkEquiv \ud835\udc9c f x f_deg hm).hom \u226b\n        (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv \u226b (Proj.restrictStalkIso \u22ef x).inv)\n      (algebraMap (A\u2070_ f) \u2191((Spec.structureSheaf (A\u2070_ f)).presheaf.stalk ((toSpec \ud835\udc9c f).val.base x)))", "state_after": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 StructureSheaf.toStalk (A\u2070_ f) ((toSpec \ud835\udc9c f).val.base x) \u226b\n      (specStalkEquiv \ud835\udc9c f x ?f_deg ?hm).hom \u226b\n        (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv \u226b (Proj.restrictStalkIso \u22ef x).inv =\n    RingHom.comp\n      ((specStalkEquiv \ud835\udc9c f x f_deg hm).hom \u226b\n        (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv \u226b (Proj.restrictStalkIso \u22ef x).inv)\n      (algebraMap (A\u2070_ f) \u2191((Spec.structureSheaf (A\u2070_ f)).presheaf.stalk ((toSpec \ud835\udc9c f).val.base x)))\n\ncase m\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 \u2115\n\ncase f_deg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 f \u2208 \ud835\udc9c ?m\n\ncase hm\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 0 < ?m"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 StructureSheaf.toStalk (A\u2070_ f) ((toSpec \ud835\udc9c f).val.base x) \u226b\n      (specStalkEquiv \ud835\udc9c f x ?f_deg ?hm).hom \u226b\n        (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv \u226b (Proj.restrictStalkIso \u22ef x).inv =\n    RingHom.comp\n      ((specStalkEquiv \ud835\udc9c f x f_deg hm).hom \u226b\n        (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv \u226b (Proj.restrictStalkIso \u22ef x).inv)\n      (algebraMap (A\u2070_ f) \u2191((Spec.structureSheaf (A\u2070_ f)).presheaf.stalk ((toSpec \ud835\udc9c f).val.base x)))\n\ncase m\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 \u2115\n\ncase f_deg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 f \u2208 \ud835\udc9c ?m\n\ncase hm\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\n\u22a2 0 < ?m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "Dense.Iio_eq_biUnion", "start": [929, 1], "end": [931, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsOpen.rightCoset", "start": [128, 1], "end": [129, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/DirectSum/Finsupp.lean", "full_name": "finsuppTensorFinsupp'_apply_apply", "start": [338, 1], "end": [340, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Defs.lean", "full_name": "Finset.mem_Ioo", "start": [328, 1], "end": [329, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "AnalyticAt.congr", "start": [549, 1], "end": [551, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "ConvexOn.convex_lt", "start": [550, 1], "end": [558, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "full_name": "MeasureTheory.lintegral_tilted", "start": [185, 1], "end": [188, 91], "traced_tactics": [{"tactic": "rw [\u2190 setLIntegral_univ, setLIntegral_tilted' f g MeasurableSet.univ, setLIntegral_univ]", "annotated_tactic": ["rw [\u2190 <a>setLIntegral_univ</a>, <a>setLIntegral_tilted'</a> f g <a>MeasurableSet.univ</a>, <a>setLIntegral_univ</a>]", [{"full_name": "MeasureTheory.setLIntegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [684, 9], "def_end_pos": [684, 26]}, {"full_name": "MeasureTheory.setLIntegral_tilted'", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [144, 7], "def_end_pos": [144, 27]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [103, 19], "def_end_pos": [103, 37]}, {"full_name": "MeasureTheory.setLIntegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [684, 9], "def_end_pos": [684, 26]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1), ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Choose/Basic.lean", "full_name": "Nat.choose_two_right", "start": [99, 1], "end": [103, 24], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "n : \u2115\n\u22a2 n.choose 2 = n * (n - 1) / 2", "state_after": "case zero\n\n\u22a2 choose 0 2 = 0 * (0 - 1) / 2\n\ncase succ\nn : \u2115\nih : n.choose 2 = n * (n - 1) / 2\n\u22a2 (n + 1).choose 2 = (n + 1) * (n + 1 - 1) / 2"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\n\n\u22a2 choose 0 2 = 0 * (0 - 1) / 2", "state_after": "no goals"}, {"tactic": "rw [triangle_succ n, choose, ih]", "annotated_tactic": ["rw [<a>triangle_succ</a> n, <a>choose</a>, ih]", [{"full_name": "Nat.triangle_succ", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [93, 9], "def_end_pos": [93, 22]}, {"full_name": "Nat.choose", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 11]}]], "state_before": "case succ\nn : \u2115\nih : n.choose 2 = n * (n - 1) / 2\n\u22a2 (n + 1).choose 2 = (n + 1) * (n + 1 - 1) / 2", "state_after": "case succ\nn : \u2115\nih : n.choose 2 = n * (n - 1) / 2\n\u22a2 n.choose 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n"}, {"tactic": "simp [Nat.add_comm]", "annotated_tactic": ["simp [<a>Nat.add_comm</a>]", [{"full_name": "Nat.add_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [160, 19], "def_end_pos": [160, 27]}]], "state_before": "case succ\nn : \u2115\nih : n.choose 2 = n * (n - 1) / 2\n\u22a2 n.choose 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.core_def", "start": [459, 1], "end": [460, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Vector/Snoc.lean", "full_name": "Vector.replicate_succ_to_snoc", "start": [54, 1], "end": [62, 23], "traced_tactics": [{"tactic": "clear xs", "annotated_tactic": ["clear xs", []], "state_before": "\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nval : \u03b1\n\u22a2 replicate (n + 1) val = (replicate n val).snoc val", "state_after": "\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (n + 1) val = (replicate n val).snoc val"}, {"tactic": "induction n with\n| zero => rfl\n| succ n ih =>\n  rw [replicate_succ]\n  conv => rhs; rw [replicate_succ]\n  rw [snoc_cons, ih]", "annotated_tactic": ["induction n with\n  | zero => rfl\n  | succ n ih =>\n    rw [<a>replicate_succ</a>]\n    conv => rhs; rw [<a>replicate_succ</a>]\n    rw [<a>snoc_cons</a>, ih]", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 23]}, {"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 23]}, {"full_name": "Vector.snoc_cons", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [34, 9], "def_end_pos": [34, 18]}]], "state_before": "\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (n + 1) val = (replicate n val).snoc val", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (0 + 1) val = (replicate 0 val).snoc val", "state_after": "no goals"}, {"tactic": "rw [replicate_succ]", "annotated_tactic": ["rw [<a>replicate_succ</a>]", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 23]}]], "state_before": "case succ\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = (replicate n val).snoc val\n\u22a2 replicate (n + 1 + 1) val = (replicate (n + 1) val).snoc val", "state_after": "case succ\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = (replicate n val).snoc val\n\u22a2 val ::\u1d65 replicate (n + 1) val = (replicate (n + 1) val).snoc val"}, {"tactic": "conv => rhs; rw [replicate_succ]", "annotated_tactic": ["conv => rhs; rw [<a>replicate_succ</a>]", [{"full_name": "Vector.replicate_succ", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 23]}]], "state_before": "case succ\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = (replicate n val).snoc val\n\u22a2 val ::\u1d65 replicate (n + 1) val = (replicate (n + 1) val).snoc val", "state_after": "case succ\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = (replicate n val).snoc val\n\u22a2 val ::\u1d65 replicate (n + 1) val = (val ::\u1d65 replicate n val).snoc val"}, {"tactic": "rw [snoc_cons, ih]", "annotated_tactic": ["rw [<a>snoc_cons</a>, ih]", [{"full_name": "Vector.snoc_cons", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [34, 9], "def_end_pos": [34, 18]}]], "state_before": "case succ\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = (replicate n val).snoc val\n\u22a2 val ::\u1d65 replicate (n + 1) val = (val ::\u1d65 replicate n val).snoc val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.list_ofFn", "start": [741, 1], "end": [749, 63], "traced_tactics": [{"tactic": "simp only [List.ofFn_zero]", "annotated_tactic": ["simp only [<a>List.ofFn_zero</a>]", [{"full_name": "List.ofFn_zero", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [124, 9], "def_end_pos": [124, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nx\u271d\u00b9 : Fin 0 \u2192 \u03b1 \u2192 \u03c3\nx\u271d : \u2200 (i : Fin 0), Computable (x\u271d\u00b9 i)\n\u22a2 Computable fun a => List.ofFn fun i => x\u271d\u00b9 i a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nx\u271d\u00b9 : Fin 0 \u2192 \u03b1 \u2192 \u03c3\nx\u271d : \u2200 (i : Fin 0), Computable (x\u271d\u00b9 i)\n\u22a2 Computable fun a => []"}, {"tactic": "exact const []", "annotated_tactic": ["exact <a>const</a> []", [{"full_name": "Computable.const", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [287, 9], "def_end_pos": [287, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nx\u271d\u00b9 : Fin 0 \u2192 \u03b1 \u2192 \u03c3\nx\u271d : \u2200 (i : Fin 0), Computable (x\u271d\u00b9 i)\n\u22a2 Computable fun a => []", "state_after": "no goals"}, {"tactic": "simp only [List.ofFn_succ]", "annotated_tactic": ["simp only [<a>List.ofFn_succ</a>]", [{"full_name": "List.ofFn_succ", "def_path": "Mathlib/Data/List/OfFn.lean", "def_pos": [129, 9], "def_end_pos": [129, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin (n + 1)), Computable (f i)\n\u22a2 Computable fun a => List.ofFn fun i => f i a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin (n + 1)), Computable (f i)\n\u22a2 Computable fun a => f 0 a :: List.ofFn fun i => f i.succ a"}, {"tactic": "exact list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ)", "annotated_tactic": ["exact list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin (n + 1)), Computable (f i)\n\u22a2 Computable fun a => f 0 a :: List.ofFn fun i => f i.succ a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Congruence/Basic.lean", "full_name": "Con.eq", "start": [351, 11], "end": [352, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Even.lean", "full_name": "IsSquare.pow", "start": [125, 1], "end": [126, 61], "traced_tactics": [{"tactic": "rintro \u27e8a, rfl\u27e9", "annotated_tactic": ["rintro \u27e8a, rfl\u27e9", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn\u271d : \u2115\na : \u03b1\nn : \u2115\n\u22a2 IsSquare a \u2192 IsSquare (a ^ n)", "state_after": "case intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn\u271d n : \u2115\na : \u03b1\n\u22a2 IsSquare ((a * a) ^ n)"}, {"tactic": "exact \u27e8a ^ n, (Commute.refl _).mul_pow _\u27e9", "annotated_tactic": ["exact \u27e8a ^ n, (<a>Commute.refl</a> _).<a>mul_pow</a> _\u27e9", [{"full_name": "Commute.refl", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [63, 19], "def_end_pos": [63, 23]}, {"full_name": "Commute.mul_pow", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [221, 22], "def_end_pos": [221, 29]}]], "state_before": "case intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn\u271d n : \u2115\na : \u03b1\n\u22a2 IsSquare ((a * a) ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.coequalizerComparison_map_desc", "start": [1168, 1], "end": [1173, 22], "traced_tactics": [{"tactic": "simp only [\u2190 G.map_comp, w]", "annotated_tactic": ["simp only [\u2190 G.map_comp, w]", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y : C\nf g : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 G.map f \u226b G.map h = G.map g \u226b G.map h", "state_after": "no goals"}, {"tactic": "apply coequalizer.hom_ext", "annotated_tactic": ["apply <a>coequalizer.hom_ext</a>", [{"full_name": "CategoryTheory.Limits.coequalizer.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 28]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y : C\nf g : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 coequalizerComparison f g G \u226b G.map (coequalizer.desc h w) = coequalizer.desc (G.map h) \u22ef", "state_after": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y : C\nf g : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G \u226b G.map (coequalizer.desc h w) =\n    coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizer.desc (G.map h) \u22ef"}, {"tactic": "simp [\u2190 G.map_comp]", "annotated_tactic": ["simp [\u2190 G.map_comp]", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y : C\nf g : X \u27f6 Y\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\nZ : C\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G \u226b G.map (coequalizer.desc h w) =\n    coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizer.desc (G.map h) \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/DiscreteQuotient.lean", "full_name": "DiscreteQuotient.fiber_eq", "start": 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"Mathlib/Algebra/Group/Hom/Defs.lean", "full_name": "MonoidHom.copy_eq", "start": [744, 1], "end": [746, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "sbtw_one_zero_iff", "start": [458, 1], "end": [459, 36], "traced_tactics": [{"tactic": "rw [sbtw_comm, sbtw_zero_one_iff]", "annotated_tactic": ["rw [<a>sbtw_comm</a>, <a>sbtw_zero_one_iff</a>]", [{"full_name": "sbtw_comm", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [275, 9], "def_end_pos": [275, 18]}, {"full_name": "sbtw_zero_one_iff", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [450, 9], "def_end_pos": [450, 26]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx : R\n\u22a2 Sbtw R 1 x 0 \u2194 x \u2208 Set.Ioo 0 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/JordanHolder.lean", "full_name": "CompositionSeries.isMaximal_eraseLast_last", "start": [287, 1], "end": [292, 60], "traced_tactics": [{"tactic": "have : s.length - 1 + 1 = s.length := by\n  conv_rhs => rw [\u2190 Nat.add_one_sub_one s.length]; rw [Nat.succ_sub h]", "annotated_tactic": ["have : s.length - 1 + 1 = s.length := by\n    conv_rhs => rw [\u2190 <a>Nat.add_one_sub_one</a> s.length]; rw [<a>Nat.succ_sub</a> h]", [{"full_name": "Nat.add_one_sub_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [118, 19], "def_end_pos": [118, 34]}, {"full_name": "Nat.succ_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [1013, 9], "def_end_pos": [1013, 17]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\n\u22a2 IsMaximal (eraseLast s).last (last s)", "state_after": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 IsMaximal (eraseLast s).last (last s)"}, {"tactic": "rw [last_eraseLast, last]", "annotated_tactic": ["rw [<a>last_eraseLast</a>, <a>last</a>]", [{"full_name": "RelSeries.last_eraseLast", "def_path": "Mathlib/Order/RelSeries.lean", "def_pos": [478, 15], "def_end_pos": [478, 29]}, {"full_name": "RelSeries.last", "def_path": "Mathlib/Order/RelSeries.lean", "def_pos": [213, 5], "def_end_pos": [213, 9]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 IsMaximal (eraseLast s).last (last s)", "state_after": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 IsMaximal (s.toFun \u27e8s.length.pred, \u22ef\u27e9) (s.toFun (Fin.last s.length))"}, {"tactic": "convert s.step \u27e8s.length - 1, by omega\u27e9", "annotated_tactic": ["convert s.step \u27e8s.length - 1, by omega\u27e9", []], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 IsMaximal (s.toFun \u27e8s.length.pred, \u22ef\u27e9) (s.toFun (Fin.last s.length))", "state_after": "case h.e'_5.h.e'_4\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 Fin.last s.length = \u27e8s.length - 1, \u22ef\u27e9.succ"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.e'_5.h.e'_4\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 Fin.last s.length = \u27e8s.length - 1, \u22ef\u27e9.succ", "state_after": "case h.e'_5.h.e'_4.h\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 \u2191(Fin.last s.length) = \u2191\u27e8s.length - 1, \u22ef\u27e9.succ"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case h.e'_5.h.e'_4.h\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 \u2191(Fin.last s.length) = \u2191\u27e8s.length - 1, \u22ef\u27e9.succ", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [\u2190 Nat.add_one_sub_one s.length]; rw [Nat.succ_sub h]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Nat.add_one_sub_one</a> s.length]; rw [<a>Nat.succ_sub</a> h]", [{"full_name": "Nat.add_one_sub_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [118, 19], "def_end_pos": [118, 34]}, {"full_name": "Nat.succ_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [1013, 9], "def_end_pos": [1013, 17]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\n\u22a2 s.length - 1 + 1 = s.length", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\nh : 0 < s.length\nthis : s.length - 1 + 1 = s.length\n\u22a2 s.length - 1 < s.length", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Operations.lean", "full_name": "ENNReal.eq_sub_of_add_eq", "start": [390, 11], "end": [391, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "full_name": "Complex.cpow_ofNat_inv_pow", "start": [169, 1], "end": [173, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "full_name": "MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top", "start": [263, 1], "end": [281, 54], "traced_tactics": [{"tactic": "refine measure_iUnion_of_tendsto_zero m atTop ?_", "annotated_tactic": ["refine <a>measure_iUnion_of_tendsto_zero</a> m <a>atTop</a> ?_", [{"full_name": "MeasureTheory.measure_iUnion_of_tendsto_zero", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 39]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\n\u22a2 m (\u22c3 n, s n) = \u2a06 n, m (s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\n\u22a2 Tendsto (fun k => m ((\u22c3 n, s n) \\ s k)) atTop (\ud835\udcdd 0)"}, {"tactic": "refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_", "annotated_tactic": ["refine <a>tendsto_nhds_bot_mono'</a> (<a>ENNReal.tendsto_sum_nat_add</a> _ h0) fun n => ?_", [{"full_name": "tendsto_nhds_bot_mono'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 31]}, {"full_name": "ENNReal.tendsto_sum_nat_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1293, 9], "def_end_pos": [1293, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\n\u22a2 Tendsto (fun k => m ((\u22c3 n, s n) \\ s k)) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\n\u22a2 m ((\u22c3 n, s n) \\ s n) \u2264 \u2211' (k : \u2115), m (s (k + n + 1) \\ s (k + n))"}, {"tactic": "refine (m.mono ?_).trans (measure_iUnion_le _)", "annotated_tactic": ["refine (m.mono ?_).<a>trans</a> (<a>measure_iUnion_le</a> _)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\n\u22a2 m ((\u22c3 n, s n) \\ s n) \u2264 \u2211' (k : \u2115), m (s (k + n + 1) \\ s (k + n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have h' : Monotone s := @monotone_nat_of_le_succ (Set \u03b1) _ _ h_mono", "annotated_tactic": ["have h' : <a>Monotone</a> s := @<a>monotone_nat_of_le_succ</a> (<a>Set</a> \u03b1) _ _ h_mono", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "monotone_nat_of_le_succ", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 32]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "simp only [diff_subset_iff, iUnion_subset_iff]", "annotated_tactic": ["simp only [<a>diff_subset_iff</a>, <a>iUnion_subset_iff</a>]", [{"full_name": "Set.diff_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1890, 9], "def_end_pos": [1890, 24]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [258, 9], "def_end_pos": [258, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\n\u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\n\u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "intro i x hx", "annotated_tactic": ["intro i x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\n\u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have : \u2203i, x \u2208 s i := by exists i", "annotated_tactic": ["have : \u2203i, x \u2208 s i := by exists i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "rcases Nat.findX this with \u27e8j, hj, hlt\u27e9", "annotated_tactic": ["rcases <a>Nat.findX</a> this with \u27e8j, hj, hlt\u27e9", [{"full_name": "Nat.findX", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [681, 15], "def_end_pos": [681, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "clear hx i", "annotated_tactic": ["clear hx i", []], "state_before": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "rcases le_or_lt j n with hjn | hnj", "annotated_tactic": ["rcases <a>le_or_lt</a> j n with hjn | hnj", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}]], "state_before": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhjn : j \u2264 n\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)\n\ncase mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "have : j - (n + 1) + n + 1 = j := by omega", "annotated_tactic": ["have : j - (n + 1) + n + 1 = j := by omega", []], "state_before": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)"}, {"tactic": "refine Or.inr (mem_iUnion.2 \u27e8j - (n + 1), ?_, hlt _ ?_\u27e9)", "annotated_tactic": ["refine <a>Or.inr</a> (<a>mem_iUnion</a>.2 \u27e8j - (n + 1), ?_, hlt _ ?_\u27e9)", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}]], "state_before": "case mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "case mk.intro.inr.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s (j - (n + 1) + n + 1)\n\ncase mk.intro.inr.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 j - (n + 1) + n < j"}, {"tactic": "exists i", "annotated_tactic": ["exists i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 \u2203 i, x \u2208 s i", "state_after": "no goals"}, {"tactic": "exact Or.inl (h' hjn hj)", "annotated_tactic": ["exact <a>Or.inl</a> (h' hjn hj)", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhjn : j \u2264 n\n\u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n)", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\n\u22a2 j - (n + 1) + n + 1 = j", "state_after": "no goals"}, {"tactic": "rwa [this]", "annotated_tactic": ["rwa [this]", []], "state_before": "case mk.intro.inr.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s (j - (n + 1) + n + 1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, this]", "annotated_tactic": ["rw [\u2190 <a>Nat.succ_le_iff</a>, <a>Nat.succ_eq_add_one</a>, this]", [{"full_name": "Nat.succ_le_iff", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [128, 7], "def_end_pos": [128, 18]}, {"full_name": "Nat.succ_eq_add_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [154, 17], "def_end_pos": [154, 32]}]], "state_before": "case mk.intro.inr.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), m (s (k + 1) \\ s k) \u2260 \u22a4\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 m < j, x \u2209 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 j - (n + 1) + n < j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Units/Regulator.lean", "full_name": "NumberField.Units.regulator_pos", "start": [45, 1], "end": [45, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.id", "start": [1758, 11], "end": [1759, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "full_name": "RingHom.mul_def", "start": [691, 1], "end": [691, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Perfect.lean", "full_name": "bijective_iterateFrobenius", "start": [65, 1], "end": [66, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_cons_of_ne", "start": [2460, 1], "end": [2461, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "le_iSup'", "start": [710, 1], "end": [711, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.diffs_subset_left", "start": [643, 1], "end": [643, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.subsingleton", "start": [158, 11], "end": [159, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.contLinear_map_vsub", "start": [93, 1], "end": [94, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "full_name": "Batteries.RBNode.lowerBound?_le'", "start": [420, 1], "end": [423, 87], "traced_tactics": [{"tactic": "rw [\u2190 reverse_reverse t, lowerBound?_reverse, Ne, \u2190 Ordering.swap_inj]", "annotated_tactic": ["rw [\u2190 <a>reverse_reverse</a> t, <a>lowerBound?_reverse</a>, <a>Ne</a>, \u2190 <a>Ordering.swap_inj</a>]", [{"full_name": "Batteries.RBNode.reverse_reverse", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/WF.lean", "def_pos": [48, 17], "def_end_pos": [48, 32]}, {"full_name": "Batteries.RBNode.lowerBound?_reverse", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "def_pos": [386, 17], "def_end_pos": [386, 36]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Ordering.swap_inj", "def_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "def_pos": [14, 17], "def_end_pos": [14, 25]}]], "state_before": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nH : \u2200 {x : \u03b1}, x \u2208 lb \u2192 cut x \u2260 Ordering.lt\n\u22a2 lowerBound? cut t lb = some x \u2192 cut x \u2260 Ordering.lt", "state_after": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nH : \u2200 {x : \u03b1}, x \u2208 lb \u2192 cut x \u2260 Ordering.lt\n\u22a2 upperBound? (fun x => (cut x).swap) t.reverse lb = some x \u2192 \u00ac(cut x).swap = Ordering.lt.swap"}, {"tactic": "exact upperBound?_ge' fun h => by specialize H h; rwa [Ne, \u2190 Ordering.swap_inj] at H", "annotated_tactic": ["exact <a>upperBound?_ge'</a> fun h => by specialize H h; rwa [<a>Ne</a>, \u2190 <a>Ordering.swap_inj</a>] at H", [{"full_name": "Batteries.RBNode.upperBound?_ge'", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "def_pos": [405, 9], "def_end_pos": [405, 24]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Ordering.swap_inj", "def_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "def_pos": [14, 17], "def_end_pos": [14, 25]}]], "state_before": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nH : \u2200 {x : \u03b1}, x \u2208 lb \u2192 cut x \u2260 Ordering.lt\n\u22a2 upperBound? (fun x => (cut x).swap) t.reverse lb = some x \u2192 \u00ac(cut x).swap = Ordering.lt.swap", "state_after": "no goals"}, {"tactic": "specialize H h", "annotated_tactic": ["specialize H h", []], "state_before": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nH : \u2200 {x : \u03b1}, x \u2208 lb \u2192 cut x \u2260 Ordering.lt\nx\u271d : \u03b1\nh : x\u271d \u2208 lb\n\u22a2 (cut x\u271d).swap \u2260 Ordering.gt", "state_after": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nx\u271d : \u03b1\nh : x\u271d \u2208 lb\nH : cut x\u271d \u2260 Ordering.lt\n\u22a2 (cut x\u271d).swap \u2260 Ordering.gt"}, {"tactic": "rwa [Ne, \u2190 Ordering.swap_inj] at H", "annotated_tactic": ["rwa [<a>Ne</a>, \u2190 <a>Ordering.swap_inj</a>] at H", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Ordering.swap_inj", "def_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "def_pos": [14, 17], "def_end_pos": [14, 25]}]], "state_before": "\u03b1 : Type u_1\nlb : Option \u03b1\ncut : \u03b1 \u2192 Ordering\nx : \u03b1\nt : RBNode \u03b1\nx\u271d : \u03b1\nh : x\u271d \u2208 lb\nH : cut x\u271d \u2260 Ordering.lt\n\u22a2 (cut x\u271d).swap \u2260 Ordering.gt", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.zero_num", "start": [19, 9], "end": [19, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "full_name": "LieAlgebra.IsKilling.traceForm_eq_zero_of_mem_ker_of_mem_span_coroot", "start": [419, 1], "end": [432, 31], "traced_tactics": [{"tactic": "rw [\u2190 coe_corootSpace_eq_span_singleton, LieSubmodule.mem_coeSubmodule, mem_corootSpace'] at hy", "annotated_tactic": ["rw [\u2190 <a>coe_corootSpace_eq_span_singleton</a>, <a>LieSubmodule.mem_coeSubmodule</a>, <a>mem_corootSpace'</a>] at hy", [{"full_name": "LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton", "def_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "def_pos": [391, 7], "def_end_pos": [391, 40]}, {"full_name": "LieSubmodule.mem_coeSubmodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [108, 9], "def_end_pos": [108, 25]}, {"full_name": "LieAlgebra.mem_corootSpace'", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [298, 7], "def_end_pos": [298, 23]}]], "state_before": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nhy : y \u2208 span K {coroot \u03b1}\n\u22a2 ((traceForm K (\u21a5H) L) x) y = 0", "state_after": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nhy : y \u2208 span K {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\n\u22a2 ((traceForm K (\u21a5H) L) x) y = 0"}, {"tactic": "induction hy using Submodule.span_induction' with\n| mem z hz =>\n  obtain \u27e8u, hu, v, -, huv\u27e9 := hz\n  change killingForm K L (x : L) (z : L) = 0\n  replace hx : \u03b1 x = 0 := by simpa using hx\n  rw [\u2190 huv, \u2190 traceForm_apply_lie_apply, \u2190 LieSubalgebra.coe_bracket_of_module,\n    lie_eq_smul_of_mem_rootSpace hu, hx, zero_smul, map_zero, LinearMap.zero_apply]\n| zero => simp\n| add _ _ _ _ hx hy => simp [hx, hy]\n| smul _ _ _ hz => simp [hz]", "annotated_tactic": ["induction hy using <a>Submodule.span_induction'</a> with\n  | mem z hz =>\n    obtain \u27e8u, hu, v, -, huv\u27e9 := hz\n    change <a>killingForm</a> K L (x : L) (z : L) = 0\n    replace hx : \u03b1 x = 0 := by simpa using hx\n    rw [\u2190 huv, \u2190 <a>traceForm_apply_lie_apply</a>, \u2190 <a>LieSubalgebra.coe_bracket_of_module</a>,\n      <a>lie_eq_smul_of_mem_rootSpace</a> hu, hx, <a>zero_smul</a>, <a>map_zero</a>, <a>LinearMap.zero_apply</a>]\n  | <a>zero</a> => simp\n  | <a>add</a> _ _ _ _ hx hy => simp [hx, hy]\n  | <a>smul</a> _ _ _ hz => simp [hz]", [{"full_name": "Submodule.span_induction'", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [203, 9], "def_end_pos": [203, 24]}, {"full_name": "killingForm", "def_path": "Mathlib/Algebra/Lie/TraceForm.lean", "def_pos": [330, 22], "def_end_pos": [330, 33]}, {"full_name": "LieModule.traceForm_apply_lie_apply", "def_path": "Mathlib/Algebra/Lie/TraceForm.lean", "def_pos": [66, 7], "def_end_pos": [66, 32]}, {"full_name": "LieSubalgebra.coe_bracket_of_module", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [239, 9], "def_end_pos": [239, 30]}, {"full_name": "LieAlgebra.IsKilling.lie_eq_smul_of_mem_rootSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "def_pos": [295, 7], "def_end_pos": [295, 35]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "LinearMap.zero_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [871, 9], "def_end_pos": [871, 19]}, {"full_name": "Submodule.zero", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [252, 10], "def_end_pos": [252, 14]}, {"full_name": "Submodule.add", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [248, 10], "def_end_pos": [248, 13]}, {"full_name": "Submodule.smul", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [260, 10], "def_end_pos": [260, 14]}]], "state_before": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nhy : y \u2208 span K {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\n\u22a2 ((traceForm K (\u21a5H) L) x) y = 0", "state_after": "no goals"}, {"tactic": "obtain \u27e8u, hu, v, -, huv\u27e9 := hz", "annotated_tactic": ["obtain \u27e8u, hu, v, -, huv\u27e9 := hz", []], "state_before": "case mem\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nhz : z \u2208 {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\n\u22a2 ((traceForm K (\u21a5H) L) x) z = 0", "state_after": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\n\u22a2 ((traceForm K (\u21a5H) L) x) z = 0"}, {"tactic": "change killingForm K L (x : L) (z : L) = 0", "annotated_tactic": ["change <a>killingForm</a> K L (x : L) (z : L) = 0", [{"full_name": "killingForm", "def_path": "Mathlib/Algebra/Lie/TraceForm.lean", "def_pos": [330, 22], "def_end_pos": [330, 33]}]], "state_before": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\n\u22a2 ((traceForm K (\u21a5H) L) x) z = 0", "state_after": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\n\u22a2 ((killingForm K L) \u2191x) \u2191z = 0"}, {"tactic": "replace hx : \u03b1 x = 0 := by simpa using hx", "annotated_tactic": ["replace hx : \u03b1 x = 0 := by simpa using hx", []], "state_before": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\n\u22a2 ((killingForm K L) \u2191x) \u2191z = 0", "state_after": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y z : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\nhx : \u03b1 x = 0\n\u22a2 ((killingForm K L) \u2191x) \u2191z = 0"}, {"tactic": "rw [\u2190 huv, \u2190 traceForm_apply_lie_apply, \u2190 LieSubalgebra.coe_bracket_of_module,\n  lie_eq_smul_of_mem_rootSpace hu, hx, zero_smul, map_zero, LinearMap.zero_apply]", "annotated_tactic": ["rw [\u2190 huv, \u2190 <a>traceForm_apply_lie_apply</a>, \u2190 <a>LieSubalgebra.coe_bracket_of_module</a>,\n      <a>lie_eq_smul_of_mem_rootSpace</a> hu, hx, <a>zero_smul</a>, <a>map_zero</a>, <a>LinearMap.zero_apply</a>]", [{"full_name": "LieModule.traceForm_apply_lie_apply", "def_path": "Mathlib/Algebra/Lie/TraceForm.lean", "def_pos": [66, 7], "def_end_pos": [66, 32]}, {"full_name": "LieSubalgebra.coe_bracket_of_module", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [239, 9], "def_end_pos": [239, 30]}, {"full_name": "LieAlgebra.IsKilling.lie_eq_smul_of_mem_rootSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "def_pos": [295, 7], "def_end_pos": [295, 35]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "LinearMap.zero_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [871, 9], "def_end_pos": [871, 19]}]], "state_before": "case mem.intro.intro.intro.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y z : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\nhx : \u03b1 x = 0\n\u22a2 ((killingForm K L) \u2191x) \u2191z = 0", "state_after": "no goals"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "state_before": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\nz : \u21a5H\nu : L\nhu : u \u2208 rootSpace H \u21d1\u03b1\nv : L\nhuv : \u2045u, v\u2046 = \u2191z\n\u22a2 \u03b1 x = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\n\u22a2 ((traceForm K (\u21a5H) L) x) 0 = 0", "state_after": "no goals"}, {"tactic": "simp [hx, hy]", "annotated_tactic": ["simp [hx, hy]", []], "state_before": "case add\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx\u271d\u00b9 : x \u2208 Weight.ker\nx\u271d : \u21a5H\nhx\u271d : x\u271d \u2208 span K {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\ny\u271d : \u21a5H\nhy\u271d : y\u271d \u2208 span K {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\nhx : ((traceForm K (\u21a5H) L) x) x\u271d = 0\nhy : ((traceForm K (\u21a5H) L) x) y\u271d = 0\n\u22a2 ((traceForm K (\u21a5H) L) x) (x\u271d + y\u271d) = 0", "state_after": "no goals"}, {"tactic": "simp [hz]", "annotated_tactic": ["simp [hz]", []], "state_before": "case smul\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : LieAlgebra K L\ninst\u271d\u2074 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b3 : H.IsCartanSubalgebra\ninst\u271d\u00b2 : IsTriangularizable K (\u21a5H) L\ninst\u271d\u00b9 : IsKilling K L\ninst\u271d : CharZero K\n\u03b1 : Weight K (\u21a5H) L\nx y : \u21a5H\nhx : x \u2208 Weight.ker\na\u271d : K\nx\u271d : \u21a5H\nhx\u271d : x\u271d \u2208 span K {x | \u2203 y \u2208 rootSpace H \u21d1\u03b1, \u2203 z \u2208 rootSpace H (-\u21d1\u03b1), \u2045y, z\u2046 = \u2191x}\nhz : ((traceForm K (\u21a5H) L) x) x\u271d = 0\n\u22a2 ((traceForm K (\u21a5H) L) x) (a\u271d \u2022 x\u271d) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/DenomsClearable.lean", "full_name": "one_le_pow_mul_abs_eval_div", "start": [90, 1], "end": [107, 25], "traced_tactics": [{"tactic": "obtain \u27e8ev, bi, bu, hF\u27e9 :=\n  denomsClearable_natDegree (b := b) (algebraMap \u2124 K) f a\n    (by\n      rw [eq_intCast, one_div_mul_cancel]\n      rw [Int.cast_ne_zero]\n      exact b0.ne.symm)", "annotated_tactic": ["obtain \u27e8ev, bi, bu, hF\u27e9 :=\n    <a>denomsClearable_natDegree</a> (b := b) (<a>algebraMap</a> \u2124 K) f a\n      (by\n        rw [<a>eq_intCast</a>, <a>one_div_mul_cancel</a>]\n        rw [<a>Int.cast_ne_zero</a>]\n        exact b0.ne.symm)", [{"full_name": "denomsClearable_natDegree", "def_path": "Mathlib/Algebra/Polynomial/DenomsClearable.lean", "def_pos": [76, 9], "def_end_pos": [76, 34]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "one_div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [348, 7], "def_end_pos": [348, 25]}, {"full_name": "Int.cast_ne_zero", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [82, 7], "def_end_pos": [82, 19]}]], "state_before": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|"}, {"tactic": "obtain Fa := _root_.congr_arg abs hF", "annotated_tactic": ["obtain Fa := <a>_root_.congr_arg</a> <a>abs</a> hF", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "abs", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [38, 3], "def_end_pos": [38, 14]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa :\n  |(algebraMap \u2124 K) ev| =\n    |(algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|"}, {"tactic": "rw [eq_one_div_of_mul_eq_one_left bu, eq_intCast, eq_intCast, abs_mul] at Fa", "annotated_tactic": ["rw [<a>eq_one_div_of_mul_eq_one_left</a> bu, <a>eq_intCast</a>, <a>eq_intCast</a>, <a>abs_mul</a>] at Fa", [{"full_name": "eq_one_div_of_mul_eq_one_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 38]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [48, 7], "def_end_pos": [48, 14]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa :\n  |(algebraMap \u2124 K) ev| =\n    |(algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = |\u2191b ^ f.natDegree| * |eval ((algebraMap \u2124 K) a * (1 / \u2191b)) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|"}, {"tactic": "rw [abs_of_pos (pow_pos (Int.cast_pos.mpr b0) _ : 0 < (b : K) ^ _), one_div, eq_intCast] at Fa", "annotated_tactic": ["rw [<a>abs_of_pos</a> (<a>pow_pos</a> (Int.cast_pos.mpr b0) _ : 0 < (b : K) ^ _), <a>one_div</a>, <a>eq_intCast</a>] at Fa", [{"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = |\u2191b ^ f.natDegree| * |eval ((algebraMap \u2124 K) a * (1 / \u2191b)) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|"}, {"tactic": "rw [div_eq_mul_inv, \u2190 Fa, \u2190 Int.cast_abs, \u2190 Int.cast_one, Int.cast_le]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, \u2190 Fa, \u2190 <a>Int.cast_abs</a>, \u2190 <a>Int.cast_one</a>, <a>Int.cast_le</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "Int.cast_abs", "def_path": "Mathlib/Algebra/Order/Ring/Cast.lean", "def_pos": [83, 7], "def_end_pos": [83, 15]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 17]}, {"full_name": "Int.cast_le", "def_path": "Mathlib/Algebra/Order/Ring/Cast.lean", "def_pos": [49, 26], "def_end_pos": [49, 33]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 \u2191b ^ f.natDegree * |eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f)|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 |ev|"}, {"tactic": "refine Int.le_of_lt_add_one ((lt_add_iff_pos_left 1).mpr (abs_pos.mpr fun F0 => fab ?_))", "annotated_tactic": ["refine <a>Int.le_of_lt_add_one</a> ((<a>lt_add_iff_pos_left</a> 1).<a>mpr</a> (abs_pos.mpr fun F0 => fab ?_))", [{"full_name": "Int.le_of_lt_add_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [861, 9], "def_end_pos": [861, 25]}, {"full_name": "lt_add_iff_pos_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [571, 30], "def_end_pos": [571, 49]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\n\u22a2 1 \u2264 |ev|", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0"}, {"tactic": "rw [eq_one_div_of_mul_eq_one_left bu, F0, one_div, eq_intCast, Int.cast_zero, zero_eq_mul] at hF", "annotated_tactic": ["rw [<a>eq_one_div_of_mul_eq_one_left</a> bu, F0, <a>one_div</a>, <a>eq_intCast</a>, <a>Int.cast_zero</a>, <a>zero_eq_mul</a>] at hF", [{"full_name": "eq_one_div_of_mul_eq_one_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [515, 9], "def_end_pos": [515, 38]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "Int.cast_zero", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [59, 9], "def_end_pos": [59, 18]}, {"full_name": "zero_eq_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [290, 9], "def_end_pos": [290, 20]}]], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) ev =\n    (algebraMap \u2124 K) b ^ f.natDegree * eval ((algebraMap \u2124 K) a * bi) (Polynomial.map (algebraMap \u2124 K) f)\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) b ^ f.natDegree = 0 \u2228\n    eval ((algebraMap \u2124 K) a * ((algebraMap \u2124 K) b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f) = 0\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0"}, {"tactic": "cases' hF with hF hF", "annotated_tactic": ["cases' hF with hF hF", []], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nhF :\n  (algebraMap \u2124 K) b ^ f.natDegree = 0 \u2228\n    eval ((algebraMap \u2124 K) a * ((algebraMap \u2124 K) b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f) = 0\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0", "state_after": "case intro.intro.intro.inl\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\nhF : (algebraMap \u2124 K) b ^ f.natDegree = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0\n\ncase intro.intro.intro.inr\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\nhF : eval ((algebraMap \u2124 K) a * ((algebraMap \u2124 K) b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f) = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0"}, {"tactic": "rw [eq_intCast, one_div_mul_cancel]", "annotated_tactic": ["rw [<a>eq_intCast</a>, <a>one_div_mul_cancel</a>]", [{"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [412, 9], "def_end_pos": [412, 19]}, {"full_name": "one_div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [348, 7], "def_end_pos": [348, 25]}]], "state_before": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 ?m.22808 * (algebraMap \u2124 K) b = 1", "state_after": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 \u2191b \u2260 0"}, {"tactic": "rw [Int.cast_ne_zero]", "annotated_tactic": ["rw [<a>Int.cast_ne_zero</a>]", [{"full_name": "Int.cast_ne_zero", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [82, 7], "def_end_pos": [82, 19]}]], "state_before": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 \u2191b \u2260 0", "state_after": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 b \u2260 0"}, {"tactic": "exact b0.ne.symm", "annotated_tactic": ["exact b0.ne.symm", []], "state_before": "K : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\n\u22a2 b \u2260 0", "state_after": "no goals"}, {"tactic": "exact (not_le.mpr b0 (le_of_eq (Int.cast_eq_zero.mp (pow_eq_zero hF)))).elim", "annotated_tactic": ["exact (not_le.mpr b0 (<a>le_of_eq</a> (Int.cast_eq_zero.mp (<a>pow_eq_zero</a> hF)))).<a>elim</a>", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case intro.intro.intro.inl\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\nhF : (algebraMap \u2124 K) b ^ f.natDegree = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0", "state_after": "no goals"}, {"tactic": "rwa [div_eq_mul_inv]", "annotated_tactic": ["rwa [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}]], "state_before": "case intro.intro.intro.inr\nK : Type u_1\ninst\u271d : LinearOrderedField K\nf : \u2124[X]\na b : \u2124\nb0 : 0 < b\nfab : eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) \u2260 0\nev : \u2124\nbi : K\nbu : bi * (algebraMap \u2124 K) b = 1\nFa : |\u2191ev| = \u2191b ^ f.natDegree * |eval (\u2191a * (\u2191b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f)|\nF0 : ev = 0\nhF : eval ((algebraMap \u2124 K) a * ((algebraMap \u2124 K) b)\u207b\u00b9) (Polynomial.map (algebraMap \u2124 K) f) = 0\n\u22a2 eval (\u2191a / \u2191b) (Polynomial.map (algebraMap \u2124 K) f) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean", "full_name": "CategoryTheory.NatTrans.app_sub", "start": [103, 1], "end": [104, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.iSup_add_iSup", "start": [626, 1], "end": [633, 29], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b9", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b9", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [221, 9], "def_end_pos": [221, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\n\u22a2 iSup f + iSup g = \u2a06 a, f a + g a", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : IsEmpty \u03b9\n\u22a2 iSup f + iSup g = \u2a06 a, f a + g a\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f + iSup g = \u2a06 a, f a + g a"}, {"tactic": "simp only [iSup_of_empty, bot_eq_zero, zero_add]", "annotated_tactic": ["simp only [<a>iSup_of_empty</a>, <a>bot_eq_zero</a>, <a>zero_add</a>]", [{"full_name": "iSup_of_empty", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1473, 9], "def_end_pos": [1473, 22]}, {"full_name": "ENNReal.bot_eq_zero", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 20]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : IsEmpty \u03b9\n\u22a2 iSup f + iSup g = \u2a06 a, f a + g a", "state_after": "no goals"}, {"tactic": "refine le_antisymm ?_ (iSup_le fun a => add_le_add (le_iSup _ _) (le_iSup _ _))", "annotated_tactic": ["refine <a>le_antisymm</a> ?_ (<a>iSup_le</a> fun a => <a>add_le_add</a> (<a>le_iSup</a> _ _) (<a>le_iSup</a> _ _))", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [760, 9], "def_end_pos": [760, 16]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [702, 9], "def_end_pos": [702, 16]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [702, 9], "def_end_pos": [702, 16]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f + iSup g = \u2a06 a, f a + g a", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f + iSup g \u2264 \u2a06 a, f a + g a"}, {"tactic": "refine iSup_add_iSup_le fun i j => ?_", "annotated_tactic": ["refine <a>iSup_add_iSup_le</a> fun i j => ?_", [{"full_name": "ENNReal.iSup_add_iSup_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [608, 9], "def_end_pos": [608, 25]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f + iSup g \u2264 \u2a06 a, f a + g a", "state_after": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\ni j : \u03b9\n\u22a2 f i + g j \u2264 \u2a06 a, f a + g a"}, {"tactic": "rcases h i j with \u27e8k, hk\u27e9", "annotated_tactic": ["rcases h i j with \u27e8k, hk\u27e9", []], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\ni j : \u03b9\n\u22a2 f i + g j \u2264 \u2a06 a, f a + g a", "state_after": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\ni j k : \u03b9\nhk : f i + g j \u2264 f k + g k\n\u22a2 f i + g j \u2264 \u2a06 a, f a + g a"}, {"tactic": "exact le_iSup_of_le k hk", "annotated_tactic": ["exact <a>le_iSup_of_le</a> k hk", [{"full_name": "le_iSup_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [734, 9], "def_end_pos": [734, 22]}]], "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f i + g j \u2264 f k + g k\nh\u271d : Nonempty \u03b9\ni j k : \u03b9\nhk : f i + g j \u2264 f k + g k\n\u22a2 f i + g j \u2264 \u2a06 a, f a + g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/EGauge.lean", "full_name": "le_egauge_smul_right", "start": [137, 1], "end": [146, 49], "traced_tactics": [{"tactic": "rw [le_egauge_iff]", "annotated_tactic": ["rw [<a>le_egauge_iff</a>]", [{"full_name": "le_egauge_iff", "def_path": "Mathlib/Analysis/Convex/EGauge.lean", "def_pos": [57, 7], "def_end_pos": [57, 20]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\n\u22a2 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 egauge \ud835\udd5c s (c \u2022 x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\n\u22a2 \u2200 (c_1 : \ud835\udd5c), c \u2022 x \u2208 c_1 \u2022 s \u2192 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016c_1\u2016\u208a"}, {"tactic": "rintro a \u27e8y, hy, hxy\u27e9", "annotated_tactic": ["rintro a \u27e8y, hy, hxy\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\n\u22a2 \u2200 (c_1 : \ud835\udd5c), c \u2022 x \u2208 c_1 \u2022 s \u2192 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016c_1\u2016\u208a", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\n\u22a2 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a"}, {"tactic": "rcases eq_or_ne c 0 with rfl | hc", "annotated_tactic": ["rcases <a>eq_or_ne</a> c 0 with rfl | hc", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\n\u22a2 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a", "state_after": "case intro.intro.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = 0 \u2022 x\n\u22a2 \u2191\u20160\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a\n\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case intro.intro.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = 0 \u2022 x\n\u22a2 \u2191\u20160\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a", "state_after": "no goals"}, {"tactic": "refine ENNReal.mul_le_of_le_div' <| le_trans ?_ ENNReal.coe_div_le", "annotated_tactic": ["refine <a>ENNReal.mul_le_of_le_div'</a> <| <a>le_trans</a> ?_ <a>ENNReal.coe_div_le</a>", [{"full_name": "ENNReal.mul_le_of_le_div'", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [351, 9], "def_end_pos": [351, 26]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "ENNReal.coe_div_le", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [76, 7], "def_end_pos": [76, 17]}]], "state_before": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 \u2191\u2016c\u2016\u208a * egauge \ud835\udd5c s x \u2264 \u2191\u2016a\u2016\u208a", "state_after": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 egauge \ud835\udd5c s x \u2264 \u2191(\u2016a\u2016\u208a / \u2016c\u2016\u208a)"}, {"tactic": "rw [div_eq_inv_mul, \u2190 nnnorm_inv, \u2190 nnnorm_mul]", "annotated_tactic": ["rw [<a>div_eq_inv_mul</a>, \u2190 <a>nnnorm_inv</a>, \u2190 <a>nnnorm_mul</a>]", [{"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}, {"full_name": "nnnorm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [759, 9], "def_end_pos": [759, 19]}, {"full_name": "nnnorm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [703, 9], "def_end_pos": [703, 19]}]], "state_before": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 egauge \ud835\udd5c s x \u2264 \u2191(\u2016a\u2016\u208a / \u2016c\u2016\u208a)", "state_after": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 egauge \ud835\udd5c s x \u2264 \u2191\u2016c\u207b\u00b9 * a\u2016\u208a"}, {"tactic": "refine egauge_le_of_mem_smul \u27e8y, hy, ?_\u27e9", "annotated_tactic": ["refine <a>egauge_le_of_mem_smul</a> \u27e8y, hy, ?_\u27e9", [{"full_name": "egauge_le_of_mem_smul", "def_path": "Mathlib/Analysis/Convex/EGauge.lean", "def_pos": [55, 7], "def_end_pos": [55, 28]}]], "state_before": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 egauge \ud835\udd5c s x \u2264 \u2191\u2016c\u207b\u00b9 * a\u2016\u208a", "state_after": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 (fun x => (c\u207b\u00b9 * a) \u2022 x) y = x"}, {"tactic": "simp only [mul_smul, hxy, inv_smul_smul\u2080 hc]", "annotated_tactic": ["simp only [<a>mul_smul</a>, hxy, <a>inv_smul_smul\u2080</a> hc]", [{"full_name": "MulAction.mul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [114, 3], "def_end_pos": [114, 11]}, {"full_name": "inv_smul_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [207, 9], "def_end_pos": [207, 23]}]], "state_before": "case intro.intro.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\n\u03b1 : Type u_2\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nc\u271d : \ud835\udd5c\ns\u271d t : Set E\nx\u271d y\u271d : E\nr : \u211d\u22650\u221e\nc : \ud835\udd5c\ns : Set E\nx : E\na : \ud835\udd5c\ny : E\nhy : y \u2208 s\nhxy : (fun x => a \u2022 x) y = c \u2022 x\nhc : c \u2260 0\n\u22a2 (fun x => (c\u207b\u00b9 * a) \u2022 x) y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.mul_apply", "start": [717, 1], "end": [718, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Nontrivial/Defs.lean", "full_name": "subsingleton_or_nontrivial", "start": [99, 1], "end": [101, 23], "traced_tactics": [{"tactic": "rw [\u2190 not_nontrivial_iff_subsingleton, or_comm]", "annotated_tactic": ["rw [\u2190 <a>not_nontrivial_iff_subsingleton</a>, <a>or_comm</a>]", [{"full_name": "not_nontrivial_iff_subsingleton", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 40]}, {"full_name": "or_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [823, 9], "def_end_pos": [823, 16]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\n\u22a2 Subsingleton \u03b1 \u2228 Nontrivial \u03b1", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\n\u22a2 Nontrivial \u03b1 \u2228 \u00acNontrivial \u03b1"}, {"tactic": "exact Classical.em _", "annotated_tactic": ["exact <a>Classical.em</a> _", [{"full_name": "Classical.em", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [32, 9], "def_end_pos": [32, 11]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\n\u22a2 Nontrivial \u03b1 \u2228 \u00acNontrivial \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "LinearMap.finiteDimensional_of_det_ne_one", "start": [335, 1], "end": [340, 48], "traced_tactics": [{"tactic": "by_cases H : \u2203 s : Finset M, Nonempty (Basis s \ud835\udd5c M)", "annotated_tactic": ["by_cases H : \u2203 s : <a>Finset</a> M, <a>Nonempty</a> (<a>Basis</a> s \ud835\udd5c M)", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}, {"full_name": "Basis", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [88, 11], "def_end_pos": [88, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : LinearMap.det f \u2260 1\n\u22a2 FiniteDimensional \ud835\udd5c M", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : LinearMap.det f \u2260 1\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 FiniteDimensional \ud835\udd5c M\n\ncase neg\nR : Type 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"annotated_tactic": ["rw [<a>lt_iff_not_ge</a>] at h", [{"full_name": "lt_iff_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [365, 9], "def_end_pos": [365, 22]}]], "state_before": "case inl\nx : \u211d\u22650\u221e\ny : \u211d\nhy0 : 0 \u2264 y\nh : x = 0 \u2227 y < 0\n\u22a2 False", "state_after": "case inl\nx : \u211d\u22650\u221e\ny : \u211d\nhy0 : 0 \u2264 y\nh : x = 0 \u2227 \u00acy \u2265 0\n\u22a2 False"}, {"tactic": "exact h.right hy0", "annotated_tactic": ["exact h.right hy0", []], "state_before": "case inl\nx : \u211d\u22650\u221e\ny : \u211d\nhy0 : 0 \u2264 y\nh : x = 0 \u2227 \u00acy \u2265 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact h.left", "annotated_tactic": ["exact h.left", []], "state_before": "case inr\nx : \u211d\u22650\u221e\ny : \u211d\nhy0 : 0 \u2264 y\nh : x = \u22a4 \u2227 0 < y\n\u22a2 x = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.cylinder_eq_cylinder_of_le_firstDiff", "start": [168, 1], "end": [172, 50], "traced_tactics": [{"tactic": "rw [\u2190 mem_cylinder_iff_eq]", "annotated_tactic": ["rw [\u2190 <a>mem_cylinder_iff_eq</a>]", [{"full_name": "PiNat.mem_cylinder_iff_eq", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [134, 9], "def_end_pos": [134, 28]}]], "state_before": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\n\u22a2 cylinder x n = cylinder y n", "state_after": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\n\u22a2 x \u2208 cylinder y n"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\n\u22a2 x \u2208 cylinder y n", "state_after": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\ni : \u2115\nhi : i < n\n\u22a2 x i = y i"}, {"tactic": "exact apply_eq_of_lt_firstDiff (hi.trans_le hn)", "annotated_tactic": ["exact <a>apply_eq_of_lt_firstDiff</a> (hi.trans_le hn)", [{"full_name": "PiNat.apply_eq_of_lt_firstDiff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [80, 9], "def_end_pos": [80, 33]}]], "state_before": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\ni : \u2115\nhi : i < n\n\u22a2 x i = y i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.invFun_eq_of_injective_of_rightInverse", "start": [453, 1], "end": [459, 37], "traced_tactics": [{"tactic": "rw [hg b]", "annotated_tactic": ["rw [hg b]", []], "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb\u271d : \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : RightInverse g f\nb : \u03b2\n\u22a2 f (invFun f b) = f (g b)", "state_after": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb\u271d : \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : RightInverse g f\nb : \u03b2\n\u22a2 f (invFun f b) = b"}, {"tactic": "exact invFun_eq \u27e8g b, hg b\u27e9", "annotated_tactic": ["exact <a>invFun_eq</a> \u27e8g b, hg b\u27e9", [{"full_name": "Function.invFun_eq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [441, 9], "def_end_pos": [441, 18]}]], "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb\u271d : \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : RightInverse g f\nb : \u03b2\n\u22a2 f (invFun f b) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence", "start": [1747, 1], "end": [1768, 15], "traced_tactics": [{"tactic": "rw [tendsto_iff_seq_tendsto]", "annotated_tactic": ["rw [<a>tendsto_iff_seq_tendsto</a>]", [{"full_name": "Filter.tendsto_iff_seq_tendsto", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1971, 9], "def_end_pos": [1971, 32]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) l (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "intro x xl", "annotated_tactic": ["intro x xl", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have hxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s := by rwa [tendsto_atTop'] at xl", "annotated_tactic": ["have hxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s := by rwa [<a>tendsto_atTop'</a>] at xl", [{"full_name": "Filter.tendsto_atTop'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have h :\n  { x : \u03b9 | (fun n => AEStronglyMeasurable (fs n) \u03bc) x } \u2229\n      { x : \u03b9 | (fun n => \u2200\u1d50 a \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x } \u2208 l :=\n  inter_mem hfs_meas h_bound", "annotated_tactic": ["have h :\n    { x : \u03b9 | (fun n => <a>AEStronglyMeasurable</a> (fs n) \u03bc) x } \u2229\n        { x : \u03b9 | (fun n => \u2200\u1d50 a \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x } \u2208 l :=\n    <a>inter_mem</a> hfs_meas h_bound", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [91, 5], "def_end_pos": [91, 25]}, {"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "obtain \u27e8k, h\u27e9 := hxl _ h", "annotated_tactic": ["obtain \u27e8k, h\u27e9 := hxl _ h", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "rw [\u2190 tendsto_add_atTop_iff_nat k]", "annotated_tactic": ["rw [\u2190 <a>tendsto_add_atTop_iff_nat</a> k]", [{"full_name": "Filter.tendsto_add_atTop_iff_nat", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1795, 9], "def_end_pos": [1795, 34]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 Tendsto ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 Tendsto (fun n => ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) (n + k)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "refine tendsto_setToFun_of_dominated_convergence hT bound ?_ bound_integrable ?_ ?_", "annotated_tactic": ["refine <a>tendsto_setToFun_of_dominated_convergence</a> hT bound ?_ bound_integrable ?_ ?_", [{"full_name": "MeasureTheory.tendsto_setToFun_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1704, 9], "def_end_pos": [1704, 50]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 Tendsto (fun n => ((fun n => setToFun \u03bc T hT (fs n)) \u2218 x) (n + k)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "case intro.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (fs (x (n + k))) \u03bc\n\ncase intro.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs (x (n + k)) a\u2016 \u2264 bound a\n\ncase intro.refine_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs (x (n + k)) a) atTop (\ud835\udcdd (f a))"}, {"tactic": "rwa [tendsto_atTop'] at xl", "annotated_tactic": ["rwa [<a>tendsto_atTop'</a>] at xl", [{"full_name": "Filter.tendsto_atTop'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\n\u22a2 \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s", "state_after": "no goals"}, {"tactic": "exact fun n => (h _ (self_le_add_left _ _)).1", "annotated_tactic": ["exact fun n => (h _ (<a>self_le_add_left</a> _ _)).1", [{"full_name": "self_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [160, 3], "def_end_pos": [160, 14]}]], "state_before": "case intro.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (fs (x (n + k))) \u03bc", "state_after": "no goals"}, {"tactic": "exact fun n => (h _ (self_le_add_left _ _)).2", "annotated_tactic": ["exact fun n => (h _ (<a>self_le_add_left</a> _ _)).2", [{"full_name": "self_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [160, 3], "def_end_pos": [160, 14]}]], "state_before": "case intro.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs (x (n + k)) a\u2016 \u2264 bound a", "state_after": "no goals"}, {"tactic": "filter_upwards [h_lim]", "annotated_tactic": ["filter_upwards [h_lim]", []], "state_before": "case intro.refine_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs (x (n + k)) a) atTop (\ud835\udcdd (f a))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (a : \u03b1), Tendsto (fun n => fs n a) l (\ud835\udcdd (f a)) \u2192 Tendsto (fun n => fs (x (n + k)) a) atTop (\ud835\udcdd (f a))"}, {"tactic": "refine fun a h_lin => @Tendsto.comp _ _ _ (fun n => x (n + k)) (fun n => fs n a) _ _ _ h_lin ?_", "annotated_tactic": ["refine fun a h_lin => @<a>Tendsto.comp</a> _ _ _ (fun n => x (n + k)) (fun n => fs n a) _ _ _ h_lin ?_", [{"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3098, 9], "def_end_pos": [3098, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\n\u22a2 \u2200 (a : \u03b1), Tendsto (fun n => fs n a) l (\ud835\udcdd (f a)) \u2192 Tendsto (fun n => fs (x (n + k)) a) atTop (\ud835\udcdd (f a))", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\na : \u03b1\nh_lin : Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => x (n + k)) atTop l"}, {"tactic": "rw [tendsto_add_atTop_iff_nat]", "annotated_tactic": ["rw [<a>tendsto_add_atTop_iff_nat</a>]", [{"full_name": "Filter.tendsto_add_atTop_iff_nat", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1795, 9], "def_end_pos": [1795, 34]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\na : \u03b1\nh_lin : Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => x (n + k)) atTop l", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\na : \u03b1\nh_lin : Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto x atTop l"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\n\u03b9 : Type u_7\nl : Filter \u03b9\ninst\u271d : l.IsCountablyGenerated\nfs : \u03b9 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nhfs_meas : \u2200\u1da0 (n : \u03b9) in l, AEStronglyMeasurable (fs n) \u03bc\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nbound_integrable : Integrable bound \u03bc\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 s \u2208 l, \u2203 a, \u2200 b \u2265 a, x b \u2208 s\nh\u271d : {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x} \u2208 l\nk : \u2115\nh :\n  \u2200 b \u2265 k, x b \u2208 {x | (fun n => AEStronglyMeasurable (fs n) \u03bc) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) x}\na : \u03b1\nh_lin : Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto x atTop l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/WhitneyEmbedding.lean", "full_name": "SmoothBumpCovering.embeddingPiTangent_injective_mfderiv", "start": [110, 1], "end": [112, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "full_name": "LieAlgebra.zeroRootSubalgebra_normalizer_eq_self", "start": [218, 1], "end": [231, 63], "traced_tactics": [{"tactic": "refine le_antisymm ?_ (LieSubalgebra.le_normalizer _)", "annotated_tactic": ["refine <a>le_antisymm</a> ?_ (<a>LieSubalgebra.le_normalizer</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "LieSubalgebra.le_normalizer", "def_path": "Mathlib/Algebra/Lie/Normalizer.lean", "def_pos": [147, 9], "def_end_pos": [147, 22]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 (zeroRootSubalgebra R L H).normalizer = zeroRootSubalgebra R L H", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 (zeroRootSubalgebra R L H).normalizer \u2264 zeroRootSubalgebra R L H"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 (zeroRootSubalgebra R L H).normalizer \u2264 zeroRootSubalgebra R L H", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 (zeroRootSubalgebra R L H).normalizer\n\u22a2 x \u2208 zeroRootSubalgebra R L H"}, {"tactic": "rw [LieSubalgebra.mem_normalizer_iff] at hx", "annotated_tactic": ["rw [<a>LieSubalgebra.mem_normalizer_iff</a>] at hx", [{"full_name": "LieSubalgebra.mem_normalizer_iff", "def_path": "Mathlib/Algebra/Lie/Normalizer.lean", "def_pos": [141, 9], "def_end_pos": [141, 27]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 (zeroRootSubalgebra R L H).normalizer\n\u22a2 x \u2208 zeroRootSubalgebra R L H", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 x \u2208 zeroRootSubalgebra R L H"}, {"tactic": "rw [mem_zeroRootSubalgebra]", "annotated_tactic": ["rw [<a>mem_zeroRootSubalgebra</a>]", [{"full_name": "LieAlgebra.mem_zeroRootSubalgebra", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [175, 9], "def_end_pos": [175, 31]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 x \u2208 zeroRootSubalgebra R L H", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2200 (y : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y ^ k) x = 0"}, {"tactic": "rintro \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hy\u27e9", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2200 (y : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y ^ k) x = 0", "state_after": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\ny : L\nhy : y \u2208 H\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0"}, {"tactic": "specialize hx y (le_zeroRootSubalgebra R L H hy)", "annotated_tactic": ["specialize hx y (<a>le_zeroRootSubalgebra</a> R L H hy)", [{"full_name": "LieAlgebra.le_zeroRootSubalgebra", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [211, 9], "def_end_pos": [211, 30]}]], "state_before": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 y \u2208 zeroRootSubalgebra R L H, \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\ny : L\nhy : y \u2208 H\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0", "state_after": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0"}, {"tactic": "rw [mem_zeroRootSubalgebra] at hx", "annotated_tactic": ["rw [<a>mem_zeroRootSubalgebra</a>] at hx", [{"full_name": "LieAlgebra.mem_zeroRootSubalgebra", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [175, 9], "def_end_pos": [175, 31]}]], "state_before": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0", "state_after": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0"}, {"tactic": "obtain \u27e8k, hk\u27e9 := hx \u27e8y, hy\u27e9", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := hx \u27e8y, hy\u27e9", []], "state_before": "case mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0", "state_after": "case mk.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0"}, {"tactic": "rw [\u2190 lie_skew, LinearMap.map_neg, neg_eq_zero] at hk", "annotated_tactic": ["rw [\u2190 <a>lie_skew</a>, <a>LinearMap.map_neg</a>, <a>neg_eq_zero</a>] at hk", [{"full_name": "lie_skew", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 17]}, {"full_name": "LinearMap.map_neg", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [628, 19], "def_end_pos": [628, 26]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}]], "state_before": "case mk.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0", "state_after": "case mk.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045y, x\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0"}, {"tactic": "use k + 1", "annotated_tactic": ["use k + 1", []], "state_before": "case mk.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045y, x\u2046 = 0\n\u22a2 \u2203 k, ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) x = 0", "state_after": "case h\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045y, x\u2046 = 0\n\u22a2 ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ (k + 1)) x = 0"}, {"tactic": "rw [LinearMap.iterate_succ, LinearMap.coe_comp, Function.comp_apply, toEnd_apply_apply,\n  LieSubalgebra.coe_bracket_of_module, Submodule.coe_mk, hk]", "annotated_tactic": ["rw [<a>LinearMap.iterate_succ</a>, <a>LinearMap.coe_comp</a>, <a>Function.comp_apply</a>, <a>toEnd_apply_apply</a>,\n    <a>LieSubalgebra.coe_bracket_of_module</a>, <a>Submodule.coe_mk</a>, hk]", [{"full_name": "LinearMap.iterate_succ", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [175, 9], "def_end_pos": [175, 21]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [557, 9], "def_end_pos": [557, 17]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "LieModule.toEnd_apply_apply", "def_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "def_pos": [195, 3], "def_end_pos": [195, 8]}, {"full_name": "LieSubalgebra.coe_bracket_of_module", "def_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "def_pos": [239, 9], "def_end_pos": [239, 30]}, {"full_name": "Submodule.coe_mk", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [315, 9], "def_end_pos": [315, 15]}]], "state_before": "case h\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : \u21a5H), \u2203 k, ((toEnd R (\u21a5H) L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ k) \u2045y, x\u2046 = 0\n\u22a2 ((toEnd R (\u21a5H) L) \u27e8y, hy\u27e9 ^ (k + 1)) x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/SheafHom.lean", "full_name": "CategoryTheory.PresheafHom.IsSheafFor.exists_app", "start": [127, 1], "end": [149, 39], "traced_tactics": [{"tactic": "let c : Cone ((Presieve.diagram (Sieve.pullback g S).arrows).op \u22d9 G) :=\n  { pt := F.obj (op Y)\n    \u03c0 :=\n      { app := fun \u27e8Z, hZ\u27e9 => F.map Z.hom.op \u226b (x _ hZ).app (op (Over.mk (\ud835\udfd9 _)))\n        naturality := by\n          rintro \u27e8Z\u2081, hZ\u2081\u27e9 \u27e8Z\u2082, hZ\u2082\u27e9 \u27e8f : Z\u2082 \u27f6 Z\u2081\u27e9\n          dsimp\n          rw [id_comp, assoc]\n          have H := hx f.left (\ud835\udfd9 _) hZ\u2081 hZ\u2082 (by simp)\n          simp only [presheafHom_obj, unop_op, Functor.id_obj, op_id,\n            FunctorToTypes.map_id_apply] at H\n          let \u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left\n          have H' := (x (Z\u2081.hom \u226b g) hZ\u2081).naturality \u03c6.op\n          dsimp at H H' \u22a2\n          erw [\u2190 H, \u2190 H', presheafHom_map_app_op_mk_id, \u2190 F.map_comp_assoc,\n            \u2190 op_comp, Over.w f] } }", "annotated_tactic": ["let c : <a>Cone</a> ((<a>Presieve.diagram</a> (<a>Sieve.pullback</a> g S).<a>arrows</a>).<a>op</a> \u22d9 G) :=\n    { pt := F.obj (<a>op</a> Y)\n      \u03c0 :=\n        { app := fun \u27e8Z, hZ\u27e9 => F.map Z.hom.op \u226b (x _ hZ).<a>app</a> (<a>op</a> (<a>Over.mk</a> (\ud835\udfd9 _)))\n          naturality := by\n            rintro \u27e8Z\u2081, hZ\u2081\u27e9 \u27e8Z\u2082, hZ\u2082\u27e9 \u27e8f : Z\u2082 \u27f6 Z\u2081\u27e9\n            dsimp\n            rw [<a>id_comp</a>, <a>assoc</a>]\n            have H := hx f.left (\ud835\udfd9 _) hZ\u2081 hZ\u2082 (by simp)\n            simp only [<a>presheafHom_obj</a>, <a>unop_op</a>, <a>Functor.id_obj</a>, <a>op_id</a>,\n              <a>FunctorToTypes.map_id_apply</a>] at H\n            let \u03c6 : <a>Over.mk</a> f.left \u27f6 <a>Over.mk</a> (\ud835\udfd9 Z\u2081.left) := <a>Over.homMk</a> f.left\n            have H' := (x (Z\u2081.hom \u226b g) hZ\u2081).<a>naturality</a> \u03c6.op\n            dsimp at H H' \u22a2\n            erw [\u2190 H, \u2190 H', <a>presheafHom_map_app_op_mk_id</a>, \u2190 F.map_comp_assoc,\n              \u2190 <a>op_comp</a>, <a>Over.w</a> f] } }", [{"full_name": "CategoryTheory.Limits.Cone", "def_path": "Mathlib/CategoryTheory/Limits/Cones.lean", "def_pos": [121, 11], "def_end_pos": [121, 15]}, {"full_name": "CategoryTheory.Presieve.diagram", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [65, 8], "def_end_pos": [65, 15]}, {"full_name": "CategoryTheory.Sieve.pullback", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [516, 5], "def_end_pos": [516, 13]}, {"full_name": "CategoryTheory.Sieve.arrows", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [272, 3], "def_end_pos": [272, 9]}, {"full_name": "CategoryTheory.Functor.op", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [182, 15], "def_end_pos": [182, 17]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "CategoryTheory.NatTrans.app", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [50, 3], "def_end_pos": [50, 6]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.presheafHom_obj", "def_path": "Mathlib/CategoryTheory/Sites/SheafHom.lean", "def_pos": [42, 10], "def_end_pos": [42, 13]}, {"full_name": "Opposite.unop_op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [66, 9], "def_end_pos": [66, 16]}, {"full_name": "CategoryTheory.Functor.id_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"full_name": "CategoryTheory.op_id", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [84, 9], "def_end_pos": [84, 14]}, {"full_name": "CategoryTheory.FunctorToTypes.map_id_apply", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [156, 9], "def_end_pos": [156, 21]}, {"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}, {"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}, {"full_name": "CategoryTheory.Over.homMk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [109, 5], "def_end_pos": [109, 10]}, {"full_name": "CategoryTheory.NatTrans.naturality", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [52, 3], "def_end_pos": [52, 13]}, {"full_name": "CategoryTheory.presheafHom_map_app_op_mk_id", "def_path": "Mathlib/CategoryTheory/Sites/SheafHom.lean", "def_pos": [68, 7], "def_end_pos": [68, 35]}, {"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}, {"full_name": "CategoryTheory.Over.w", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [81, 9], "def_end_pos": [81, 10]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\n\u22a2 \u2203 \u03c6,\n    \u2200 {Z : C} (p : Z \u27f6 Y) (hp : S.arrows (p \u226b g)),\n      \u03c6 \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\n\u22a2 \u2203 \u03c6,\n    \u2200 {Z : C} (p : Z \u27f6 Y) (hp : S.arrows (p \u226b g)),\n      \u03c6 \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }"}, {"tactic": "use (hG g).lift c", "annotated_tactic": ["use (hG g).<a>lift</a> c", [{"full_name": "CategoryTheory.Limits.IsLimit.lift", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [57, 3], "def_end_pos": [57, 7]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\n\u22a2 \u2203 \u03c6,\n    \u2200 {Z : C} (p : Z \u27f6 Y) (hp : S.arrows (p \u226b g)),\n      \u03c6 \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\n\u22a2 \u2200 {Z : C} (p : Z \u27f6 Y) (hp : S.arrows (p \u226b g)),\n    (hG g).lift c \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }"}, {"tactic": "intro Z p hp", "annotated_tactic": ["intro Z p hp", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\n\u22a2 \u2200 {Z : C} (p : Z \u27f6 Y) (hp : S.arrows (p \u226b g)),\n    (hG g).lift c \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\nZ : C\np : Z \u27f6 Y\nhp : S.arrows (p \u226b g)\n\u22a2 (hG g).lift c \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }"}, {"tactic": "exact ((hG g).fac c \u27e8Over.mk p, hp\u27e9)", "annotated_tactic": ["exact ((hG g).<a>fac</a> c \u27e8<a>Over.mk</a> p, hp\u27e9)", [{"full_name": "CategoryTheory.Limits.IsLimit.fac", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nc : Cone ((Sieve.pullback g S).arrows.diagram.op \u22d9 G) :=\n  { pt := F.obj { unop := Y },\n    \u03c0 :=\n      {\n        app := fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) },\n        naturality := \u22ef } }\nZ : C\np : Z \u27f6 Y\nhp : S.arrows (p \u226b g)\n\u22a2 (hG g).lift c \u226b G.map p.op = F.map p.op \u226b (x (p \u226b g) hp).app { unop := Over.mk (\ud835\udfd9 Z) }", "state_after": "no goals"}, {"tactic": "rintro \u27e8Z\u2081, hZ\u2081\u27e9 \u27e8Z\u2082, hZ\u2082\u27e9 \u27e8f : Z\u2082 \u27f6 Z\u2081\u27e9", "annotated_tactic": ["rintro \u27e8Z\u2081, hZ\u2081\u27e9 \u27e8Z\u2082, hZ\u2082\u27e9 \u27e8f : Z\u2082 \u27f6 Z\u2081\u27e9", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\n\u22a2 \u2200 \u2983X_1 Y_1 : (Sieve.pullback g S).arrows.category\u1d52\u1d56\u2984 (f : X_1 \u27f6 Y_1),\n    ((Functor.const (Sieve.pullback g S).arrows.category\u1d52\u1d56).obj (F.obj { unop := Y })).map f \u226b\n        (fun x_1 =>\n            match x_1 with\n            | { unop := { obj := Z, property := hZ } } =>\n              F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n          Y_1 =\n      (fun x_1 =>\n            match x_1 with\n            | { unop := { obj := Z, property := hZ } } =>\n              F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n          X_1 \u226b\n        ((Sieve.pullback g S).arrows.diagram.op \u22d9 G).map f", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 ((Functor.const (Sieve.pullback g S).arrows.category\u1d52\u1d56).obj (F.obj { unop := Y })).map { unop := f } \u226b\n      (fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n        { unop := { obj := Z\u2082, property := hZ\u2082 } } =\n    (fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n        { unop := { obj := Z\u2081, property := hZ\u2081 } } \u226b\n      ((Sieve.pullback g S).arrows.diagram.op \u22d9 G).map { unop := f }"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 ((Functor.const (Sieve.pullback g S).arrows.category\u1d52\u1d56).obj (F.obj { unop := Y })).map { unop := f } \u226b\n      (fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n        { unop := { obj := Z\u2082, property := hZ\u2082 } } =\n    (fun x_1 =>\n          match x_1 with\n          | { unop := { obj := Z, property := hZ } } =>\n            F.map Z.hom.op \u226b (x (Z.hom \u226b g) hZ).app { unop := Over.mk (\ud835\udfd9 { unop := (\ud835\udfed C).obj Z.left }.unop) })\n        { unop := { obj := Z\u2081, property := hZ\u2081 } } \u226b\n      ((Sieve.pullback g S).arrows.diagram.op \u22d9 G).map { unop := f }", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 \ud835\udfd9 (F.obj { unop := Y }) \u226b F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    (F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) }) \u226b G.map f.left.op"}, {"tactic": "rw [id_comp, assoc]", "annotated_tactic": ["rw [<a>id_comp</a>, <a>assoc</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 \ud835\udfd9 (F.obj { unop := Y }) \u226b F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    (F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) }) \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "have H := hx f.left (\ud835\udfd9 _) hZ\u2081 hZ\u2082 (by simp)", "annotated_tactic": ["have H := hx f.left (\ud835\udfd9 _) hZ\u2081 hZ\u2082 (by simp)", []], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = (presheafHom F G).map (\ud835\udfd9 Z\u2082.left).op (x (Z\u2082.hom \u226b g) hZ\u2082)\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "simp only [presheafHom_obj, unop_op, Functor.id_obj, op_id,\n  FunctorToTypes.map_id_apply] at H", "annotated_tactic": ["simp only [<a>presheafHom_obj</a>, <a>unop_op</a>, <a>Functor.id_obj</a>, <a>op_id</a>,\n              <a>FunctorToTypes.map_id_apply</a>] at H", [{"full_name": "CategoryTheory.presheafHom_obj", "def_path": "Mathlib/CategoryTheory/Sites/SheafHom.lean", "def_pos": [42, 10], "def_end_pos": [42, 13]}, {"full_name": "Opposite.unop_op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [66, 9], "def_end_pos": [66, 16]}, {"full_name": "CategoryTheory.Functor.id_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 15]}, {"full_name": "CategoryTheory.op_id", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [84, 9], "def_end_pos": [84, 14]}, {"full_name": "CategoryTheory.FunctorToTypes.map_id_apply", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [156, 9], "def_end_pos": [156, 21]}]], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = (presheafHom F G).map (\ud835\udfd9 Z\u2082.left).op (x (Z\u2082.hom \u226b g) hZ\u2082)\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "let \u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left", "annotated_tactic": ["let \u03c6 : <a>Over.mk</a> f.left \u27f6 <a>Over.mk</a> (\ud835\udfd9 Z\u2081.left) := <a>Over.homMk</a> f.left", [{"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}, {"full_name": "CategoryTheory.Over.mk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [86, 5], "def_end_pos": [86, 7]}, {"full_name": "CategoryTheory.Over.homMk", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [109, 5], "def_end_pos": [109, 10]}]], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "have H' := (x (Z\u2081.hom \u226b g) hZ\u2081).naturality \u03c6.op", "annotated_tactic": ["have H' := (x (Z\u2081.hom \u226b g) hZ\u2081).<a>naturality</a> \u03c6.op", [{"full_name": "CategoryTheory.NatTrans.naturality", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [52, 3], "def_end_pos": [52, 13]}]], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\nH' :\n  ((Over.forget { unop := (\ud835\udfed C).obj Z\u2081.left }.unop).op \u22d9 F).map \u03c6.op \u226b\n      (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk f.left } =\n    (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b\n      ((Over.forget { unop := (\ud835\udfed C).obj Z\u2081.left }.unop).op \u22d9 G).map \u03c6.op\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "dsimp at H H' \u22a2", "annotated_tactic": ["dsimp at H H' \u22a2", []], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\nH' :\n  ((Over.forget { unop := (\ud835\udfed C).obj Z\u2081.left }.unop).op \u22d9 F).map \u03c6.op \u226b\n      (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk f.left } =\n    (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b\n      ((Over.forget { unop := (\ud835\udfed C).obj Z\u2081.left }.unop).op \u22d9 G).map \u03c6.op\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\nH' :\n  F.map \u03c6.left.op \u226b (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk f.left } =\n    (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map \u03c6.left.op\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op"}, {"tactic": "erw [\u2190 H, \u2190 H', presheafHom_map_app_op_mk_id, \u2190 F.map_comp_assoc,\n  \u2190 op_comp, Over.w f]", "annotated_tactic": ["erw [\u2190 H, \u2190 H', <a>presheafHom_map_app_op_mk_id</a>, \u2190 F.map_comp_assoc,\n              \u2190 <a>op_comp</a>, <a>Over.w</a> f]", [{"full_name": "CategoryTheory.presheafHom_map_app_op_mk_id", "def_path": "Mathlib/CategoryTheory/Sites/SheafHom.lean", "def_pos": [68, 7], "def_end_pos": [68, 35]}, {"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}, {"full_name": "CategoryTheory.Over.w", "def_path": "Mathlib/CategoryTheory/Comma/Over.lean", "def_pos": [81, 9], "def_end_pos": [81, 10]}]], "state_before": "case op.mk.op.mk.op\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\nH : (presheafHom F G).map f.left.op (x (Z\u2081.hom \u226b g) hZ\u2081) = x (Z\u2082.hom \u226b g) hZ\u2082\n\u03c6 : Over.mk f.left \u27f6 Over.mk (\ud835\udfd9 Z\u2081.left) := Over.homMk f.left \u22ef\nH' :\n  F.map \u03c6.left.op \u226b (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk f.left } =\n    (x (Z\u2081.hom \u226b g) hZ\u2081).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map \u03c6.left.op\n\u22a2 F.map Z\u2082.hom.op \u226b (x (Z\u2082.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2082.left) } =\n    F.map Z\u2081.hom.op \u226b (x (Z\u2081.hom \u226b g) \u22ef).app { unop := Over.mk (\ud835\udfd9 Z\u2081.left) } \u226b G.map f.left.op", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d : Category.{v', u'} A\nF G : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nhG : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y \u27f6 X\nZ\u2081 : Over Y\nhZ\u2081 : (Sieve.pullback g S).arrows Z\u2081.hom\nZ\u2082 : Over Y\nhZ\u2082 : (Sieve.pullback g S).arrows Z\u2082.hom\nf : Z\u2082 \u27f6 Z\u2081\n\u22a2 f.left \u226b Z\u2081.hom \u226b g = \ud835\udfd9 Z\u2082.left \u226b Z\u2082.hom \u226b g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/AdjoinRoot.lean", "full_name": "AdjoinRoot.isRoot_root", "start": [248, 1], "end": [249, 36], "traced_tactics": [{"tactic": "rw [IsRoot, eval_map, eval\u2082_root]", "annotated_tactic": ["rw [<a>IsRoot</a>, <a>eval_map</a>, <a>eval\u2082_root</a>]", [{"full_name": "Polynomial.IsRoot", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [488, 5], "def_end_pos": [488, 11]}, {"full_name": "Polynomial.eval_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [964, 9], "def_end_pos": [964, 17]}, {"full_name": "AdjoinRoot.eval\u2082_root", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "R : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf\u271d f : R[X]\n\u22a2 (Polynomial.map (of f) f).IsRoot (root f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "full_name": "Polynomial.Gal.restrict_surjective", "start": [142, 1], "end": [144, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "full_name": "CauSeq.abv_pos_of_not_limZero", "start": [502, 1], "end": [514, 43], "traced_tactics": [{"tactic": "haveI := Classical.propDecidable", "annotated_tactic": ["haveI := <a>Classical.propDecidable</a>", [{"full_name": "Classical.propDecidable", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [75, 49], "def_end_pos": [75, 62]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\n\u22a2 \u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 \u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)"}, {"tactic": "by_contra nk", "annotated_tactic": ["by_contra nk", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 \u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)\n\u22a2 False"}, {"tactic": "refine hf fun \u03b5 \u03b50 => ?_", "annotated_tactic": ["refine hf fun \u03b5 \u03b50 => ?_", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5"}, {"tactic": "simp? [not_forall] at nk says\n  simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, Classical.not_imp,\n    not_le] at nk", "annotated_tactic": ["simp? [not_forall] at nk says\n    simp only [<a>gt_iff_lt</a>, <a>ge_iff_le</a>, <a>not_exists</a>, <a>not_and</a>, <a>not_forall</a>, <a>Classical.not_imp</a>,\n      <a>not_le</a>] at nk", [{"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}, {"full_name": "ge_iff_le", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1947, 17], "def_end_pos": [1947, 26]}, {"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}, {"full_name": "Classical.not_forall", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [137, 21], "def_end_pos": [137, 31]}, {"full_name": "Classical.not_imp", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [161, 17], "def_end_pos": [161, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5"}, {"tactic": "cases' f.cauchy\u2083 (half_pos \u03b50) with i hi", "annotated_tactic": ["cases' f.cauchy\u2083 (<a>half_pos</a> \u03b50) with i hi", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5"}, {"tactic": "rcases nk _ (half_pos \u03b50) i with \u27e8j, ij, hj\u27e9", "annotated_tactic": ["rcases nk _ (<a>half_pos</a> \u03b50) i with \u27e8j, ij, hj\u27e9", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 17]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5"}, {"tactic": "refine \u27e8j, fun k jk => ?_\u27e9", "annotated_tactic": ["refine \u27e8j, fun k jk => ?_\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\n\u22a2 abv (\u2191f k) < \u03b5"}, {"tactic": "have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj)", "annotated_tactic": ["have := <a>lt_of_le_of_lt</a> (<a>abv_add</a> abv _ _) (<a>add_lt_add</a> (hi j ij k jk) hj)", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "IsAbsoluteValue.abv_add", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [328, 7], "def_end_pos": [328, 14]}, {"full_name": "add_lt_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [167, 7], "def_end_pos": [167, 17]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\n\u22a2 abv (\u2191f k) < \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis\u271d : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\nthis : abv (\u2191f k - \u2191f j + \u2191f j) < \u03b5 / 2 + \u03b5 / 2\n\u22a2 abv (\u2191f k) < \u03b5"}, {"tactic": "rwa [sub_add_cancel, add_halves] at this", "annotated_tactic": ["rwa [<a>sub_add_cancel</a>, <a>add_halves</a>] at this", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [997, 3], "def_end_pos": [997, 14]}, {"full_name": "add_halves", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [436, 9], "def_end_pos": [436, 19]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis\u271d : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 j \u2265 i, \u2200 k \u2265 j, abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\nthis : abv (\u2191f k - \u2191f j + \u2191f j) < \u03b5 / 2 + \u03b5 / 2\n\u22a2 abv (\u2191f k) < \u03b5", "state_after": "no goals"}, {"tactic": "simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, Classical.not_imp,\n  not_le] at nk", "annotated_tactic": ["simp only [<a>gt_iff_lt</a>, <a>ge_iff_le</a>, <a>not_exists</a>, <a>not_and</a>, <a>not_forall</a>, <a>Classical.not_imp</a>,\n      <a>not_le</a>] at nk", [{"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}, {"full_name": "ge_iff_le", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1947, 17], "def_end_pos": [1947, 26]}, {"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}, {"full_name": "Classical.not_forall", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [137, 21], "def_end_pos": [137, 31]}, {"full_name": "Classical.not_imp", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [161, 17], "def_end_pos": [161, 24]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K > 0, \u2203 i, \u2200 j \u2265 i, K \u2264 abv (\u2191f j)\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf.LimZero\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, \u2203 (_ : x_1 \u2264 x_2), abv (\u2191f x_2) < x\n\u22a2 \u2203 i, \u2200 j \u2265 i, abv (\u2191f j) < \u03b5"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.norm_condexpIndL1_le", "start": [230, 1], "end": [239, 43], "traced_tactics": [{"tactic": "by_cases hs : MeasurableSet s", "annotated_tactic": ["by_cases hs : <a>MeasurableSet</a> s", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "by_cases h\u03bcs : \u03bc s = \u221e", "annotated_tactic": ["by_cases h\u03bcs : \u03bc s = \u221e", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "simp_rw [condexpIndL1_of_not_measurableSet hs]", "annotated_tactic": ["simp_rw [<a>condexpIndL1_of_not_measurableSet</a> hs]", [{"full_name": "MeasureTheory.condexpIndL1_of_not_measurableSet", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [195, 9], "def_end_pos": [195, 42]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u20160\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "rw [Lp.norm_zero]", "annotated_tactic": ["rw [<a>Lp.norm_zero</a>]", [{"full_name": "MeasureTheory.Lp.norm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [336, 9], "def_end_pos": [336, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 \u20160\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)", "annotated_tactic": ["exact <a>mul_nonneg</a> <a>ENNReal.toReal_nonneg</a> (<a>norm_nonneg</a> _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [247, 17], "def_end_pos": [247, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : \u00acMeasurableSet s\n\u22a2 0 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [condexpIndL1_of_measure_eq_top h\u03bcs x, Lp.norm_zero]", "annotated_tactic": ["rw [<a>condexpIndL1_of_measure_eq_top</a> h\u03bcs x, <a>Lp.norm_zero</a>]", [{"full_name": "MeasureTheory.condexpIndL1_of_measure_eq_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [190, 9], "def_end_pos": [190, 39]}, {"full_name": "MeasureTheory.Lp.norm_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [336, 9], "def_end_pos": [336, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = \u22a4\n\u22a2 0 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)", "annotated_tactic": ["exact <a>mul_nonneg</a> <a>ENNReal.toReal_nonneg</a> (<a>norm_nonneg</a> _)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [247, 17], "def_end_pos": [247, 30]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = \u22a4\n\u22a2 0 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs x]", "annotated_tactic": ["rw [<a>condexpIndL1_of_measurableSet_of_measure_ne_top</a> hs h\u03bcs x]", [{"full_name": "MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [185, 9], "def_end_pos": [185, 56]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016"}, {"tactic": "exact norm_condexpIndL1Fin_le hs h\u03bcs x", "annotated_tactic": ["exact <a>norm_condexpIndL1Fin_le</a> hs h\u03bcs x", [{"full_name": "MeasureTheory.norm_condexpIndL1Fin_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [128, 9], "def_end_pos": [128, 32]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : RCLike \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2077 : NormedSpace \u211d F'\ninst\u271d\u2076 : CompleteSpace F'\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \u211d G'\ninst\u271d\u00b2 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (\u03bc.trim hm)\nx : G\nhs : MeasurableSet s\nh\u03bcs : \u00ac\u03bc s = \u22a4\n\u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 (\u03bc s).toReal * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Antiperiodic.add", "start": [587, 1], "end": [588, 81], "traced_tactics": [{"tactic": "simp_all [\u2190 add_assoc]", "annotated_tactic": ["simp_all [\u2190 <a>add_assoc</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x : \u03b1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : InvolutiveNeg \u03b2\nh1 : Antiperiodic f c\u2081\nh2 : Antiperiodic f c\u2082\n\u22a2 Periodic f (c\u2081 + c\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/LeftRightNhds.lean", "full_name": "nhdsWithin_Iic_basis_Icc", "start": [297, 1], "end": [299, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "full_name": "MulMemClass.mul_mem_add_closure", "start": [735, 1], "end": [744, 91], "traced_tactics": [{"tactic": "revert a", "annotated_tactic": ["revert a", []], "state_before": "M : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na b : R\nha : a \u2208 AddSubmonoid.closure \u2191S\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 a * b \u2208 AddSubmonoid.closure \u2191S", "state_after": "M : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * b \u2208 AddSubmonoid.closure \u2191S"}, {"tactic": "apply @AddSubmonoid.closure_induction _ _ _\n  (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S)\n    _ hb <;> clear hb b", "annotated_tactic": ["apply @<a>AddSubmonoid.closure_induction</a> _ _ _\n    (fun z => \u2200 {a : R}, a \u2208 <a>AddSubmonoid.closure</a> \u2191S \u2192 a * z \u2208 <a>AddSubmonoid.closure</a> \u2191S)\n      _ hb <;> clear hb b", [{"full_name": "AddSubmonoid.closure_induction", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [441, 3], "def_end_pos": [441, 14]}, {"full_name": "AddSubmonoid.closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [384, 3], "def_end_pos": [384, 14]}, {"full_name": "AddSubmonoid.closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [384, 3], "def_end_pos": [384, 14]}]], "state_before": "M : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * b \u2208 AddSubmonoid.closure \u2191S", "state_after": "case mem\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 x \u2208 \u2191S, \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S\n\ncase one\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * 0 \u2208 AddSubmonoid.closure \u2191S\n\ncase mul\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n    (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S) \u2192\n      (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * y \u2208 AddSubmonoid.closure \u2191S) \u2192\n        \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * (x + y) \u2208 AddSubmonoid.closure \u2191S"}, {"tactic": "exact fun r hr b hb => MulMemClass.mul_right_mem_add_closure hb hr", "annotated_tactic": ["exact fun r hr b hb => <a>MulMemClass.mul_right_mem_add_closure</a> hb hr", [{"full_name": "MulMemClass.mul_right_mem_add_closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "def_pos": [722, 9], "def_end_pos": [722, 34]}]], "state_before": "case mem\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 x \u2208 \u2191S, \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S", "state_after": "no goals"}, {"tactic": "exact fun _ => by simp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]", "annotated_tactic": ["exact fun _ => by simp only [<a>mul_zero</a>, (<a>AddSubmonoid.closure</a> (S : <a>Set</a> R)).<a>zero_mem</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "AddSubmonoid.closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [384, 3], "def_end_pos": [384, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "AddSubmonoid.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [226, 3], "def_end_pos": [226, 14]}]], "state_before": "case one\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * 0 \u2208 AddSubmonoid.closure \u2191S", "state_after": "no goals"}, {"tactic": "simp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]", "annotated_tactic": ["simp only [<a>mul_zero</a>, (<a>AddSubmonoid.closure</a> (S : <a>Set</a> R)).<a>zero_mem</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "AddSubmonoid.closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [384, 3], "def_end_pos": [384, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "AddSubmonoid.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [226, 3], "def_end_pos": [226, 14]}]], "state_before": "M : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na\u271d : R\nx\u271d : a\u271d \u2208 AddSubmonoid.closure \u2191S\n\u22a2 a\u271d * 0 \u2208 AddSubmonoid.closure \u2191S", "state_after": "no goals"}, {"tactic": "simp_rw [mul_add]", "annotated_tactic": ["simp_rw [<a>mul_add</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "case mul\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n    (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S) \u2192\n      (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * y \u2208 AddSubmonoid.closure \u2191S) \u2192\n        \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * (x + y) \u2208 AddSubmonoid.closure \u2191S", "state_after": "case mul\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n    (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S) \u2192\n      (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * y \u2208 AddSubmonoid.closure \u2191S) \u2192\n        \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x + a * y \u2208 AddSubmonoid.closure \u2191S"}, {"tactic": "exact fun r s hr hs b hb => (AddSubmonoid.closure (S : Set R)).add_mem (hr hb) (hs hb)", "annotated_tactic": ["exact fun r s hr hs b hb => (<a>AddSubmonoid.closure</a> (S : <a>Set</a> R)).<a>add_mem</a> (hr hb) (hs hb)", [{"full_name": "AddSubmonoid.closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [384, 3], "def_end_pos": [384, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "AddSubmonoid.add_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [233, 3], "def_end_pos": [233, 14]}]], "state_before": "case mul\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n    (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S) \u2192\n      (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * y \u2208 AddSubmonoid.closure \u2191S) \u2192\n        \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x + a * y \u2208 AddSubmonoid.closure \u2191S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.subgroupOf_self", "start": [1667, 1], "end": [1668, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/GaussLemma.lean", "full_name": "Polynomial.IsPrimitive.dvd_of_fraction_map_dvd_fraction_map", "start": [291, 1], "end": [311, 47], "traced_tactics": [{"tactic": "rcases h_dvd with \u27e8r, hr\u27e9", "annotated_tactic": ["rcases h_dvd with \u27e8r, hr\u27e9", []], "state_before": "R : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nh_dvd : map (algebraMap R K) p \u2223 map (algebraMap R K) q\n\u22a2 p \u2223 q", "state_after": "case intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\n\u22a2 p \u2223 q"}, {"tactic": "obtain \u27e8\u27e8s, s0\u27e9, hs\u27e9 := integerNormalization_map_to_map R\u2070 r", "annotated_tactic": ["obtain \u27e8\u27e8s, s0\u27e9, hs\u27e9 := <a>integerNormalization_map_to_map</a> R\u2070 r", [{"full_name": "IsLocalization.integerNormalization_map_to_map", "def_path": "Mathlib/RingTheory/Localization/Integral.lean", "def_pos": [94, 9], "def_end_pos": [94, 40]}]], "state_before": "case intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\n\u22a2 p \u2223 q", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = \u2191\u27e8s, s0\u27e9 \u2022 r\n\u22a2 p \u2223 q"}, {"tactic": "rw [Subtype.coe_mk, Algebra.smul_def, algebraMap_apply] at hs", "annotated_tactic": ["rw [<a>Subtype.coe_mk</a>, <a>Algebra.smul_def</a>, <a>algebraMap_apply</a>] at hs", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [326, 9], "def_end_pos": [326, 17]}, {"full_name": "Polynomial.algebraMap_apply", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = \u2191\u27e8s, s0\u27e9 \u2022 r\n\u22a2 p \u2223 q", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 p \u2223 q"}, {"tactic": "have h : p \u2223 q * C s := by\n  use integerNormalization R\u2070 r\n  apply map_injective (algebraMap R K) (IsFractionRing.injective _ _)\n  rw [Polynomial.map_mul, Polynomial.map_mul, hs, hr, mul_assoc, mul_comm r]\n  simp", "annotated_tactic": ["have h : p \u2223 q * <a>C</a> s := by\n    use <a>integerNormalization</a> R\u2070 r\n    apply <a>map_injective</a> (<a>algebraMap</a> R K) (<a>IsFractionRing.injective</a> _ _)\n    rw [<a>Polynomial.map_mul</a>, <a>Polynomial.map_mul</a>, hs, hr, <a>mul_assoc</a>, <a>mul_comm</a> r]\n    simp", [{"full_name": "Polynomial.C", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [501, 5], "def_end_pos": [501, 6]}, {"full_name": "IsLocalization.integerNormalization", "def_path": "Mathlib/RingTheory/Localization/Integral.lean", "def_pos": [69, 19], "def_end_pos": [69, 39]}, {"full_name": "Polynomial.map_injective", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [847, 9], "def_end_pos": [847, 22]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IsFractionRing.injective", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [82, 19], "def_end_pos": [82, 28]}, {"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 p \u2223 q", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 p \u2223 q"}, {"tactic": "rw [\u2190 hp.dvd_primPart_iff_dvd, primPart_mul, hq.primPart_eq, Associated.dvd_iff_dvd_right] at h", "annotated_tactic": ["rw [\u2190 hp.dvd_primPart_iff_dvd, <a>primPart_mul</a>, hq.primPart_eq, <a>Associated.dvd_iff_dvd_right</a>] at h", [{"full_name": "Polynomial.primPart_mul", "def_path": "Mathlib/RingTheory/Polynomial/Content.lean", "def_pos": [409, 9], "def_end_pos": [409, 21]}, {"full_name": "Associated.dvd_iff_dvd_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [609, 9], "def_end_pos": [609, 37]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 p \u2223 q", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh\u271d : p \u2223 q * (C s).primPart\nh : p \u2223 ?m.86067\n\u22a2 p \u2223 q\n\ncase intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\n\u22a2 Associated (q * (C s).primPart) ?m.86067\n\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\n\u22a2 R[X]\n\ncase intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 (q * C s).primPart\n\u22a2 q * C s \u2260 0\n\ncase intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 q * C s \u2260 0"}, {"tactic": "iterate 2\n  apply mul_ne_zero hq.ne_zero\n  rw [Ne, C_eq_zero]\n  contrapose! s0\n  simp [s0, mem_nonZeroDivisors_iff_ne_zero]", "annotated_tactic": ["iterate 2\n    apply <a>mul_ne_zero</a> hq.ne_zero\n    rw [<a>Ne</a>, <a>C_eq_zero</a>]\n    contrapose! s0\n    simp [s0, <a>mem_nonZeroDivisors_iff_ne_zero</a>]", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Polynomial.C_eq_zero", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 18]}, {"full_name": "mem_nonZeroDivisors_iff_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean", "def_pos": [217, 9], "def_end_pos": [217, 40]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 (q * C s).primPart\n\u22a2 q * C s \u2260 0\n\ncase intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 q * C s \u2260 0", "state_after": "no goals"}, {"tactic": "use integerNormalization R\u2070 r", "annotated_tactic": ["use <a>integerNormalization</a> R\u2070 r", [{"full_name": "IsLocalization.integerNormalization", "def_path": "Mathlib/RingTheory/Localization/Integral.lean", "def_pos": [69, 19], "def_end_pos": [69, 39]}]], "state_before": "R : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 p \u2223 q * C s", "state_after": "case h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 q * C s = p * integerNormalization R\u2070 r"}, {"tactic": "apply map_injective (algebraMap R K) (IsFractionRing.injective _ _)", "annotated_tactic": ["apply <a>map_injective</a> (<a>algebraMap</a> R K) (<a>IsFractionRing.injective</a> _ _)", [{"full_name": "Polynomial.map_injective", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [847, 9], "def_end_pos": [847, 22]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IsFractionRing.injective", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [82, 19], "def_end_pos": [82, 28]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 q * C s = p * integerNormalization R\u2070 r", "state_after": "case h.a\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 map (algebraMap R K) (q * C s) = map (algebraMap R K) (p * integerNormalization R\u2070 r)"}, {"tactic": "rw [Polynomial.map_mul, Polynomial.map_mul, hs, hr, mul_assoc, mul_comm r]", "annotated_tactic": ["rw [<a>Polynomial.map_mul</a>, <a>Polynomial.map_mul</a>, hs, hr, <a>mul_assoc</a>, <a>mul_comm</a> r]", [{"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "Polynomial.map_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [749, 19], "def_end_pos": [749, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case h.a\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 map (algebraMap R K) (q * C s) = map (algebraMap R K) (p * integerNormalization R\u2070 r)", "state_after": "case h.a\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 map (algebraMap R K) p * (map (algebraMap R K) (C s) * r) = map (algebraMap R K) p * (C ((algebraMap R K) s) * r)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.a\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\n\u22a2 map (algebraMap R K) p * (map (algebraMap R K) (C s) * r) = map (algebraMap R K) p * (C ((algebraMap R K) s) * r)", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh\u271d : p \u2223 q * (C s).primPart\nh : p \u2223 ?m.86067\n\u22a2 p \u2223 q", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\n\u22a2 Associated (q * (C s).primPart) q", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\n\u22a2 Associated q (q * (C s).primPart)"}, {"tactic": "rcases isUnit_primPart_C s with \u27e8u, hu\u27e9", "annotated_tactic": ["rcases <a>isUnit_primPart_C</a> s with \u27e8u, hu\u27e9", [{"full_name": "Polynomial.isUnit_primPart_C", "def_path": "Mathlib/RingTheory/Polynomial/Content.lean", "def_pos": [291, 9], "def_end_pos": [291, 26]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\n\u22a2 Associated q (q * (C s).primPart)", "state_after": "case intro.intro.mk.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\nu : R[X]\u02e3\nhu : \u2191u = (C s).primPart\n\u22a2 Associated q (q * (C s).primPart)"}, {"tactic": "use u", "annotated_tactic": ["use u", []], "state_before": "case intro.intro.mk.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\nu : R[X]\u02e3\nhu : \u2191u = (C s).primPart\n\u22a2 Associated q (q * (C s).primPart)", "state_after": "case h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\nu : R[X]\u02e3\nhu : \u2191u = (C s).primPart\n\u22a2 q * \u2191u = q * (C s).primPart"}, {"tactic": "rw [hu]", "annotated_tactic": ["rw [hu]", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * (C s).primPart\nu : R[X]\u02e3\nhu : \u2191u = (C s).primPart\n\u22a2 q * \u2191u = q * (C s).primPart", "state_after": "no goals"}, {"tactic": "apply mul_ne_zero hq.ne_zero", "annotated_tactic": ["apply <a>mul_ne_zero</a> hq.ne_zero", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 q * C s \u2260 0", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 C s \u2260 0"}, {"tactic": "rw [Ne, C_eq_zero]", "annotated_tactic": ["rw [<a>Ne</a>, <a>C_eq_zero</a>]", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Polynomial.C_eq_zero", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [800, 9], "def_end_pos": [800, 18]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 C s \u2260 0", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 \u00acs = 0"}, {"tactic": "contrapose! s0", "annotated_tactic": ["contrapose! s0", []], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\ns0 : s \u2208 R\u2070\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\n\u22a2 \u00acs = 0", "state_after": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\ns0 : s = 0\n\u22a2 s \u2209 R\u2070"}, {"tactic": "simp [s0, mem_nonZeroDivisors_iff_ne_zero]", "annotated_tactic": ["simp [s0, <a>mem_nonZeroDivisors_iff_ne_zero</a>]", [{"full_name": "mem_nonZeroDivisors_iff_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean", "def_pos": [217, 9], "def_end_pos": [217, 40]}]], "state_before": "case intro.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nr : K[X]\nhr : map (algebraMap R K) q = map (algebraMap R K) p * r\ns : R\nhs : map (algebraMap R K) (integerNormalization R\u2070 r) = C ((algebraMap R K) s) * r\nh : p \u2223 q * C s\ns0 : s = 0\n\u22a2 s \u2209 R\u2070", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Divisors.lean", "full_name": "Nat.swap_mem_divisorsAntidiagonal", "start": [275, 1], "end": [277, 79], "traced_tactics": [{"tactic": "rw [mem_divisorsAntidiagonal, mem_divisorsAntidiagonal, mul_comm, Prod.swap]", "annotated_tactic": ["rw [<a>mem_divisorsAntidiagonal</a>, <a>mem_divisorsAntidiagonal</a>, <a>mul_comm</a>, <a>Prod.swap</a>]", [{"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "Nat.mem_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [116, 9], "def_end_pos": [116, 33]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Prod.swap", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [160, 5], "def_end_pos": [160, 9]}]], "state_before": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.swap \u2208 n.divisorsAntidiagonal \u2194 x \u2208 n.divisorsAntidiagonal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RepresentationTheory/Basic.lean", "full_name": "Representation.asAlgebraHom_single", "start": [105, 1], "end": [106, 58], "traced_tactics": [{"tactic": "simp only [asAlgebraHom_def, MonoidAlgebra.lift_single]", "annotated_tactic": ["simp only [<a>asAlgebraHom_def</a>, <a>MonoidAlgebra.lift_single</a>]", [{"full_name": "Representation.asAlgebraHom_def", "def_path": "Mathlib/RepresentationTheory/Basic.lean", "def_pos": [100, 9], "def_end_pos": [100, 25]}, {"full_name": "MonoidAlgebra.lift_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [932, 9], "def_end_pos": [932, 20]}]], "state_before": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module k V\n\u03c1 : Representation k G V\ng : G\nr : k\n\u22a2 \u03c1.asAlgebraHom (Finsupp.single g r) = r \u2022 \u03c1 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean", "full_name": "Topology.IsLowerSet.monotone_iff_continuous", "start": [334, 1], "end": [338, 45], "traced_tactics": [{"tactic": "rw [\u2190 monotone_dual_iff]", "annotated_tactic": ["rw [\u2190 <a>monotone_dual_iff</a>]", [{"full_name": "monotone_dual_iff", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsLowerSet \u03b1\ninst\u271d : Topology.IsLowerSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2194 Continuous f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsLowerSet \u03b1\ninst\u271d : Topology.IsLowerSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone (\u21d1toDual \u2218 f \u2218 \u21d1ofDual) \u2194 Continuous f"}, {"tactic": "exact IsUpperSet.monotone_iff_continuous (\u03b1 := \u03b1\u1d52\u1d48) (\u03b2 := \u03b2\u1d52\u1d48)\n  (f := (toDual \u2218 f \u2218 ofDual : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48))", "annotated_tactic": ["exact <a>IsUpperSet.monotone_iff_continuous</a> (\u03b1 := \u03b1\u1d52\u1d48) (\u03b2 := \u03b2\u1d52\u1d48)\n    (f := (<a>toDual</a> \u2218 f \u2218 <a>ofDual</a> : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48))", [{"full_name": "Topology.IsUpperSet.monotone_iff_continuous", "def_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean", "def_pos": [260, 17], "def_end_pos": [260, 40]}, {"full_name": "OrderDual.toDual", "def_path": "Mathlib/Order/Synonym.lean", "def_pos": [50, 5], "def_end_pos": [50, 11]}, {"full_name": "OrderDual.ofDual", "def_path": "Mathlib/Order/Synonym.lean", "def_pos": [55, 5], "def_end_pos": [55, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsLowerSet \u03b1\ninst\u271d : Topology.IsLowerSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone (\u21d1toDual \u2218 f \u2218 \u21d1ofDual) \u2194 Continuous f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "DifferentiableWithinAt.mono", "start": [620, 1], "end": [623, 26], "traced_tactics": [{"tactic": "rcases h with \u27e8f', hf'\u27e9", "annotated_tactic": ["rcases h with \u27e8f', hf'\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableWithinAt \ud835\udd5c f t x\nst : s \u2286 t\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x", "state_after": "case intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nst : s \u2286 t\nf' : E \u2192L[\ud835\udd5c] F\nhf' : HasFDerivWithinAt f f' t x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x"}, {"tactic": "exact \u27e8f', hf'.mono st\u27e9", "annotated_tactic": ["exact \u27e8f', hf'.mono st\u27e9", []], "state_before": "case intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nst : s \u2286 t\nf' : E \u2192L[\ud835\udd5c] F\nhf' : HasFDerivWithinAt f f' t x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "Equiv.toFun_inducedStructureEquiv_Symm", "start": [1169, 1], "end": [1173, 6], "traced_tactics": [{"tactic": "letI : L.Structure N := inducedStructure e", "annotated_tactic": ["letI : L.Structure N := <a>inducedStructure</a> e", [{"full_name": "Equiv.inducedStructure", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [1142, 5], "def_end_pos": [1142, 21]}]], "state_before": "L : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : L.Structure M\ne : M \u2243 N\n\u22a2 N \u2192 M", "state_after": "L : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : L.Structure M\ne : M \u2243 N\nthis : L.Structure N := e.inducedStructure\n\u22a2 N \u2192 M"}, {"tactic": "exact DFunLike.coe (@inducedStructureEquiv L M N _ e).symm", "annotated_tactic": ["exact <a>DFunLike.coe</a> (@<a>inducedStructureEquiv</a> L M N _ e).<a>symm</a>", [{"full_name": "DFunLike.coe", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [147, 3], "def_end_pos": [147, 6]}, {"full_name": "Equiv.inducedStructureEquiv", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [1149, 5], "def_end_pos": [1149, 26]}, {"full_name": "FirstOrder.Language.Equiv.symm", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [799, 5], "def_end_pos": [799, 9]}]], "state_before": "L : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : L.Structure M\ne : M \u2243 N\nthis : L.Structure N := e.inducedStructure\n\u22a2 N \u2192 M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "full_name": "UpperHalfPlane.dist_coe_le", "start": [265, 1], "end": [270, 94], "traced_tactics": [{"tactic": "rw [dist_center_dist, dist_self_center, \u2190 mul_add, \u2190 add_sub_assoc, Real.sinh_add_cosh]", "annotated_tactic": ["rw [<a>dist_center_dist</a>, <a>dist_self_center</a>, \u2190 <a>mul_add</a>, \u2190 <a>add_sub_assoc</a>, <a>Real.sinh_add_cosh</a>]", [{"full_name": "UpperHalfPlane.dist_center_dist", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}, {"full_name": "UpperHalfPlane.dist_self_center", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "def_pos": [196, 9], "def_end_pos": [196, 25]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "add_sub_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [455, 3], "def_end_pos": [455, 14]}, {"full_name": "Real.sinh_add_cosh", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1104, 9], "def_end_pos": [1104, 22]}]], "state_before": "z\u271d w\u271d : \u210d\nr R : \u211d\nz w : \u210d\n\u22a2 dist \u2191z \u2191(w.center (dist z w)) + dist \u2191w \u2191(w.center (dist z w)) = w.im * (rexp (dist z w) - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.mem_iff_exists", "start": [332, 1], "end": [333, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "full_name": "Real.pow_arith_mean_le_arith_mean_pow_of_even", "start": [65, 1], "end": [68, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/MulOpposite.lean", "full_name": "Submonoid.unop_le_unop_iff", "start": [65, 1], "end": [66, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Nilpotent.lean", "full_name": "Polynomial.isNilpotent_reverse_iff", "start": [100, 1], "end": [102, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "full_name": "CategoryTheory.Functor.flip_injective", "start": [138, 1], "end": [140, 41], "traced_tactics": [{"tactic": "rw [\u2190 flip_flip F\u2081, \u2190 flip_flip F\u2082, h]", "annotated_tactic": ["rw [\u2190 <a>flip_flip</a> F\u2081, \u2190 <a>flip_flip</a> F\u2082, h]", [{"full_name": "CategoryTheory.Functor.flip_flip", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [136, 7], "def_end_pos": [136, 16]}, {"full_name": "CategoryTheory.Functor.flip_flip", "def_path": "Mathlib/CategoryTheory/Functor/Currying.lean", "def_pos": [136, 7], "def_end_pos": [136, 16]}]], "state_before": "B : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} B\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\nD : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} D\nE : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} E\nH : Type u\u2085\ninst\u271d : Category.{v\u2085, u\u2085} H\nF\u2081 F\u2082 : B \u2964 C \u2964 D\nh : F\u2081.flip = F\u2082.flip\n\u22a2 F\u2081 = F\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean", "full_name": "ProbabilityTheory.monotone_defaultRatCDF", "start": [151, 1], "end": [156, 70], "traced_tactics": [{"tactic": "unfold defaultRatCDF", "annotated_tactic": ["unfold <a>defaultRatCDF</a>", [{"full_name": "ProbabilityTheory.defaultRatCDF", "def_path": "Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean", "def_pos": [149, 5], "def_end_pos": [149, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Monotone defaultRatCDF", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Monotone fun q => if q < 0 then 0 else 1"}, {"tactic": "intro x y hxy", "annotated_tactic": ["intro x y hxy", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Monotone fun q => if q < 0 then 0 else 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (fun q => if q < 0 then 0 else 1) x \u2264 (fun q => if q < 0 then 0 else 1) y"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (fun q => if q < 0 then 0 else 1) x \u2264 (fun q => if q < 0 then 0 else 1) y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (if x < 0 then 0 else 1) \u2264 if y < 0 then 0 else 1"}, {"tactic": "split_ifs with h_1 h_2 h_2", "annotated_tactic": ["split_ifs with h_1 h_2 h_2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (if x < 0 then 0 else 1) \u2264 if y < 0 then 0 else 1", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : y < 0\n\u22a2 0 \u2264 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : \u00acy < 0\n\u22a2 0 \u2264 1\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : y < 0\n\u22a2 1 \u2264 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : \u00acy < 0\n\u22a2 1 \u2264 1"}, {"tactic": "exacts [le_rfl, zero_le_one, absurd (hxy.trans_lt h_2) h_1, le_rfl]", "annotated_tactic": ["exacts [<a>le_rfl</a>, <a>zero_le_one</a>, <a>absurd</a> (hxy.trans_lt h_2) h_1, <a>le_rfl</a>]", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "absurd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : y < 0\n\u22a2 0 \u2264 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : \u00acy < 0\n\u22a2 0 \u2264 1\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : y < 0\n\u22a2 1 \u2264 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : \u00acy < 0\n\u22a2 1 \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_mk_eq_prod", "start": [64, 1], "end": [65, 42], "traced_tactics": [{"tactic": "simp only [\u2190 map_prod_eq_map\u2082, map_id']", "annotated_tactic": ["simp only [\u2190 <a>map_prod_eq_map\u2082</a>, <a>map_id'</a>]", [{"full_name": "Filter.map_prod_eq_map\u2082", "def_path": "Mathlib/Order/Filter/NAry.lean", "def_pos": [53, 9], "def_end_pos": [53, 25]}, {"full_name": "Filter.map_id'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1966, 9], "def_end_pos": [1966, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 map\u2082 Prod.mk f g = f \u00d7\u02e2 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.IntegrableOn.congr_fun_ae", "start": [136, 1], "end": [138, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Encodable/Basic.lean", "full_name": "ULower.up_down", "start": [521, 1], "end": [522, 61], "traced_tactics": [{"tactic": "simp [up, down,Equiv.left_inv _ _, Equiv.symm_apply_apply]", "annotated_tactic": ["simp [<a>up</a>, <a>down</a>,<a>Equiv.left_inv</a> _ _, <a>Equiv.symm_apply_apply</a>]", [{"full_name": "ULower.up", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [511, 5], "def_end_pos": [511, 7]}, {"full_name": "ULower.down", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [503, 5], "def_end_pos": [503, 9]}, {"full_name": "Equiv.left_inv", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [66, 13], "def_end_pos": [66, 21]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [282, 17], "def_end_pos": [282, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\na : \u03b1\n\u22a2 (down a).up = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.maximum_mem", "start": [305, 1], "end": [306, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "full_name": "MultilinearMap.bound_of_shell_of_norm_map_coord_zero", "start": [113, 1], "end": [123, 42], "traced_tactics": [{"tactic": "rcases em (\u2203 i, \u2016m i\u2016 = 0) with (\u27e8i, hi\u27e9 | hm)", "annotated_tactic": ["rcases <a>em</a> (\u2203 i, \u2016m i\u2016 = 0) with (\u27e8i, hi\u27e9 | hm)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}]], "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "case inl.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2016m i\u2016 = 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u00ac\u2203 i, \u2016m i\u2016 = 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016"}, {"tactic": "push_neg at hm", "annotated_tactic": ["push_neg at hm", []], "state_before": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u00ac\u2203 i, \u2016m i\u2016 = 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016"}, {"tactic": "choose \u03b4 h\u03b40 h\u03b4m_lt hle_\u03b4m _ using fun i => rescale_to_shell_semi_normed (hc i) (h\u03b5 i) (hm i)", "annotated_tactic": ["choose \u03b4 h\u03b40 h\u03b4m_lt hle_\u03b4m _ using fun i => <a>rescale_to_shell_semi_normed</a> (hc i) (h\u03b5 i) (hm i)", [{"full_name": "rescale_to_shell_semi_normed", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1446, 7], "def_end_pos": [1446, 35]}]], "state_before": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40 : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\na\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016"}, {"tactic": "have h\u03b40 : 0 < \u220f i, \u2016\u03b4 i\u2016 := prod_pos fun i _ => norm_pos_iff.2 (h\u03b40 i)", "annotated_tactic": ["have h\u03b40 : 0 < \u220f <a>i</a>, \u2016\u03b4 i\u2016 := <a>prod_pos</a> fun i _ => <a>norm_pos_iff</a>.2 (h\u03b40 i)", [{"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [70, 5], "def_end_pos": [70, 9]}, {"full_name": "Finset.prod_pos", "def_path": "Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean", "def_pos": [71, 7], "def_end_pos": [71, 15]}, {"full_name": "norm_pos_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1536, 30], "def_end_pos": [1536, 42]}]], "state_before": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40 : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\na\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40\u271d : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\na\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\nh\u03b40 : 0 < \u220f i : \u03b9, \u2016\u03b4 i\u2016\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016"}, {"tactic": "simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, mul_le_mul_left h\u03b40] using\n  hf (fun i => \u03b4 i \u2022 m i) hle_\u03b4m h\u03b4m_lt", "annotated_tactic": ["simpa [<a>map_smul_univ</a>, <a>norm_smul</a>, <a>prod_mul_distrib</a>, <a>mul_left_comm</a> C, <a>mul_le_mul_left</a> h\u03b40] using\n    hf (fun i => \u03b4 i \u2022 m i) hle_\u03b4m h\u03b4m_lt", [{"full_name": "MultilinearMap.map_smul_univ", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [1151, 9], "def_end_pos": [1151, 22]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 22]}, {"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [265, 9], "def_end_pos": [265, 24]}]], "state_before": "case inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40\u271d : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\na\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\nh\u03b40 : 0 < \u220f i : \u03b9, \u2016\u03b4 i\u2016\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "no goals"}, {"tactic": "rw [hf\u2080 hi, prod_eq_zero (mem_univ i) hi, mul_zero]", "annotated_tactic": ["rw [hf\u2080 hi, <a>prod_eq_zero</a> (<a>mem_univ</a> i) hi, <a>mul_zero</a>]", [{"full_name": "Finset.prod_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean", "def_pos": [25, 7], "def_end_pos": [25, 19]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "case inl.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\ninst\u271d\u00b9\u00b9 : Fintype \u03b9'\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E\u2081 i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2075 : (i : \u03b9') \u2192 SeminormedAddCommGroup (E' i)\ninst\u271d\u2074 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SeminormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf\u2080 : \u2200 {m : (i : \u03b9) \u2192 E i} {i : \u03b9}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2016m i\u2016 = 0\n\u22a2 \u2016f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UnitDisc/Basic.lean", "full_name": "Complex.UnitDisc.conj_mul", "start": [244, 1], "end": [245, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Basic.lean", "full_name": "contMDiff_inclusion", "start": [381, 1], "end": [388, 76], "traced_tactics": [{"tactic": "rintro \u27e8x, hx : x \u2208 U\u27e9", "annotated_tactic": ["rintro \u27e8x, hx : x \u2208 U\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\n\u22a2 ContMDiff I I n (inclusion h)", "state_after": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\n\u22a2 ContMDiffAt I I n (inclusion h) \u27e8x, hx\u27e9"}, {"tactic": "apply (contDiffWithinAt_localInvariantProp I I n).liftProp_inclusion", "annotated_tactic": ["apply (<a>contDiffWithinAt_localInvariantProp</a> I I n).<a>liftProp_inclusion</a>", [{"full_name": "contDiffWithinAt_localInvariantProp", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [116, 9], "def_end_pos": [116, 44]}, {"full_name": "StructureGroupoid.LocalInvariantProp.liftProp_inclusion", "def_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "def_pos": [579, 9], "def_end_pos": [579, 27]}]], "state_before": "case mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\n\u22a2 ContMDiffAt I I n (inclusion h) \u27e8x, hx\u27e9", "state_after": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\n\u22a2 \u2200 (y : H), ContDiffWithinAtProp I I n id univ y"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\n\u22a2 \u2200 (y : H), ContDiffWithinAtProp I I n id univ y", "state_after": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAtProp I I n id univ y"}, {"tactic": "dsimp only [ContDiffWithinAtProp, id_comp, preimage_univ]", "annotated_tactic": ["dsimp only [<a>ContDiffWithinAtProp</a>, <a>id_comp</a>, <a>preimage_univ</a>]", [{"full_name": "ContDiffWithinAtProp", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Defs.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "Function.id_comp", "def_path": "Mathlib/Init/Function.lean", "def_pos": [93, 9], "def_end_pos": [93, 16]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [63, 9], "def_end_pos": [63, 22]}]], "state_before": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAtProp I I n id univ y", "state_after": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAt \ud835\udd5c n (\u2191I \u2218 \u2191I.symm) (univ \u2229 range \u2191I) (\u2191I y)"}, {"tactic": "rw [Set.univ_inter]", "annotated_tactic": ["rw [<a>Set.univ_inter</a>]", [{"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [985, 9], "def_end_pos": [985, 19]}]], "state_before": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAt \ud835\udd5c n (\u2191I \u2218 \u2191I.symm) (univ \u2229 range \u2191I) (\u2191I y)", "state_after": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAt \ud835\udd5c n (\u2191I \u2218 \u2191I.symm) (range \u2191I) (\u2191I y)"}, {"tactic": "exact contDiffWithinAt_id.congr I.rightInvOn (congr_arg I (I.left_inv y))", "annotated_tactic": ["exact contDiffWithinAt_id.congr I.rightInvOn (<a>congr_arg</a> I (I.left_inv y))", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}]], "state_before": "case mk.hQ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace H M\ninst\u271d\u00b9\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2077 : TopologicalSpace M'\ninst\u271d\u2076 : ChartedSpace H' M'\ninst\u271d\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n\u271d n : \u2115\u221e\nU V : Opens M\nh : U \u2264 V\nx : M\nhx : x \u2208 U\ny : H\n\u22a2 ContDiffWithinAt \ud835\udd5c n (\u2191I \u2218 \u2191I.symm) (range \u2191I) (\u2191I y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.prime_three", "start": [173, 1], "end": [173, 43], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n : \u2115\n\u22a2 Prime 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Trace.lean", "full_name": "LinearMap.trace_tensorProduct'", "start": [270, 1], "end": [275, 10], "traced_tactics": [{"tactic": "have h := ext_iff.1 (ext_iff.1 (trace_tensorProduct R M N) f) g", "annotated_tactic": ["have h := <a>ext_iff</a>.1 (<a>ext_iff</a>.1 (<a>trace_tensorProduct</a> R M N) f) g", [{"full_name": "LinearMap.ext_iff", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [348, 9], "def_end_pos": [348, 16]}, {"full_name": "LinearMap.ext_iff", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [348, 9], "def_end_pos": [348, 16]}, {"full_name": "LinearMap.trace_tensorProduct", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [238, 9], "def_end_pos": [238, 28]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\n\u22a2 (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nh : (((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))) f) g = (((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)) f) g\n\u22a2 (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g"}, {"tactic": "simp only [compr\u2082_apply, mapBilinear_apply, compl\u2081\u2082_apply, lsmul_apply,\n  Algebra.id.smul_eq_mul] at h", "annotated_tactic": ["simp only [<a>compr\u2082_apply</a>, <a>mapBilinear_apply</a>, <a>compl\u2081\u2082_apply</a>, <a>lsmul_apply</a>,\n    <a>Algebra.id.smul_eq_mul</a>] at h", [{"full_name": "LinearMap.compr\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [390, 9], "def_end_pos": [390, 21]}, {"full_name": "TensorProduct.mapBilinear_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 26]}, {"full_name": "LinearMap.compl\u2081\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [358, 9], "def_end_pos": [358, 22]}, {"full_name": "LinearMap.lsmul_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [413, 9], "def_end_pos": [413, 20]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nh : (((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))) f) g = (((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)) f) g\n\u22a2 (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nh : (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g\n\u22a2 (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nh : (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g\n\u22a2 (trace R (M \u2297[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Set/Monoid.lean", "full_name": "Set.Ioo_add_bij", "start": [51, 1], "end": [55, 34], "traced_tactics": [{"tactic": "rw [\u2190 Ioi_inter_Iio, \u2190 Ioi_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>Ioi_inter_Iio</a>, \u2190 <a>Ioi_inter_Iio</a>]", [{"full_name": "Set.Ioi_inter_Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [648, 9], "def_end_pos": [648, 22]}, {"full_name": "Set.Ioi_inter_Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [648, 9], "def_end_pos": [648, 22]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioo a b) (Ioo (a + d) (b + d))", "state_after": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iio b) (Ioi (a + d) \u2229 Iio (b + d))"}, {"tactic": "exact\n  (Ioi_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx =>\n    lt_of_add_lt_add_right hx.2", "annotated_tactic": ["exact\n    (<a>Ioi_add_bij</a> a d).<a>inter_mapsTo</a> (fun x hx => <a>add_lt_add_right</a> hx _) fun x hx =>\n      <a>lt_of_add_lt_add_right</a> hx.2", [{"full_name": "Set.Ioi_add_bij", "def_path": "Mathlib/Algebra/Order/Interval/Set/Monoid.lean", "def_pos": [35, 9], "def_end_pos": [35, 20]}, {"full_name": "Set.BijOn.inter_mapsTo", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 27]}, {"full_name": "add_lt_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [135, 32], "def_end_pos": [135, 48]}, {"full_name": "lt_of_add_lt_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [143, 15], "def_end_pos": [143, 37]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iio b) (Ioi (a + d) \u2229 Iio (b + d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Ioc_union_Ioi'", "start": [1342, 1], "end": [1348, 30], "traced_tactics": [{"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Ioc a b \u222a Ioi c \u2194 x \u2208 Ioi (min a c)"}, {"tactic": "simp_rw [mem_union, mem_Ioc, mem_Ioi, min_lt_iff]", "annotated_tactic": ["simp_rw [<a>mem_union</a>, <a>mem_Ioc</a>, <a>mem_Ioi</a>, <a>min_lt_iff</a>]", [{"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [734, 9], "def_end_pos": [734, 18]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "min_lt_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [63, 9], "def_end_pos": [63, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Ioc a b \u222a Ioi c \u2194 x \u2208 Ioi (min a c)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x"}, {"tactic": "by_cases hc : c < x", "annotated_tactic": ["by_cases hc : c < x", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x"}, {"tactic": "simp only [hc, or_true]", "annotated_tactic": ["simp only [hc, <a>or_true</a>]", [{"full_name": "or_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [119, 17], "def_end_pos": [119, 24]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x", "state_after": "no goals"}, {"tactic": "have hxb : x \u2264 b := (le_of_not_gt hc).trans h\u2081", "annotated_tactic": ["have hxb : x \u2264 b := (<a>le_of_not_gt</a> hc).<a>trans</a> h\u2081", [{"full_name": "le_of_not_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [330, 9], "def_end_pos": [330, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\nhxb : x \u2264 b\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x"}, {"tactic": "simp only [hxb, and_true]", "annotated_tactic": ["simp only [hxb, <a>and_true</a>]", [{"full_name": "and_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [104, 17], "def_end_pos": [104, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\nhxb : x \u2264 b\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "contDiff_prod", "start": [1538, 1], "end": [1540, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "sup_eq_inf", "start": [642, 1], "end": [642, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LucasLehmer.lean", "full_name": "LucasLehmer.\u03c9_pow_formula", "start": [464, 1], "end": [487, 19], "traced_tactics": [{"tactic": "dsimp [lucasLehmerResidue] at h", "annotated_tactic": ["dsimp [<a>lucasLehmerResidue</a>] at h", [{"full_name": "LucasLehmer.lucasLehmerResidue", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [178, 5], "def_end_pos": [178, 23]}]], "state_before": "p' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "p' : \u2115\nh : sZMod (p' + 2) (p' + 2 - 2) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "rw [sZMod_eq_s p'] at h", "annotated_tactic": ["rw [<a>sZMod_eq_s</a> p'] at h", [{"full_name": "LucasLehmer.sZMod_eq_s", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [154, 9], "def_end_pos": [154, 19]}]], "state_before": "p' : \u2115\nh : sZMod (p' + 2) (p' + 2 - 2) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "p' : \u2115\nh : \u2191(s (p' + 2 - 2)) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "simp? [ZMod.intCast_zmod_eq_zero_iff_dvd] at h says\n  simp only [add_tsub_cancel_right, ZMod.intCast_zmod_eq_zero_iff_dvd, ofNat_pos,\n    pow_pos, cast_pred, cast_pow, cast_ofNat] at h", "annotated_tactic": ["simp? [ZMod.intCast_zmod_eq_zero_iff_dvd] at h says\n    simp only [<a>add_tsub_cancel_right</a>, <a>ZMod.intCast_zmod_eq_zero_iff_dvd</a>, <a>ofNat_pos</a>,\n      <a>pow_pos</a>, <a>cast_pred</a>, <a>cast_pow</a>, <a>cast_ofNat</a>] at h", [{"full_name": "add_tsub_cancel_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [356, 9], "def_end_pos": [356, 30]}, {"full_name": "ZMod.intCast_zmod_eq_zero_iff_dvd", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 37]}, {"full_name": "Nat.ofNat_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [104, 9], "def_end_pos": [104, 18]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "Nat.cast_pred", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [39, 9], "def_end_pos": [39, 18]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "Nat.cast_ofNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [76, 28], "def_end_pos": [76, 42]}]], "state_before": "p' : \u2115\nh : \u2191(s (p' + 2 - 2)) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "p' : \u2115\nh : 2 ^ (p' + 2) - 1 \u2223 s p'\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "cases' h with k h", "annotated_tactic": ["cases' h with k h", []], "state_before": "p' : \u2115\nh : 2 ^ (p' + 2) - 1 \u2223 s p'\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case intro\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "use k", "annotated_tactic": ["use k", []], "state_before": "case intro\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "replace h := congr_arg (fun n : \u2124 => (n : X (q (p' + 2)))) h", "annotated_tactic": ["replace h := <a>congr_arg</a> (fun n : \u2124 => (n : <a>X</a> (<a>q</a> (p' + 2)))) h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "LucasLehmer.X", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [226, 5], "def_end_pos": [226, 6]}, {"full_name": "LucasLehmer.q", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [216, 5], "def_end_pos": [216, 6]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : (fun n => \u2191n) (s p') = (fun n => \u2191n) ((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "dsimp at h", "annotated_tactic": ["dsimp at h", []], "state_before": "case h\np' : \u2115\nk : \u2124\nh : (fun n => \u2191n) (s p') = (fun n => \u2191n) ((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u2191(s p') = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "rw [closed_form] at h", "annotated_tactic": ["rw [<a>closed_form</a>] at h", [{"full_name": "LucasLehmer.X.closed_form", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [433, 9], "def_end_pos": [433, 20]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u2191(s p') = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p' = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "replace h := congr_arg (fun x => \u03c9 ^ 2 ^ p' * x) h", "annotated_tactic": ["replace h := <a>congr_arg</a> (fun x => <a>\u03c9</a> ^ 2 ^ p' * x) h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "LucasLehmer.X.\u03c9", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [412, 5], "def_end_pos": [412, 6]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p' = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : (fun x => \u03c9 ^ 2 ^ p' * x) (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = (fun x => \u03c9 ^ 2 ^ p' * x) \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "dsimp at h", "annotated_tactic": ["dsimp at h", []], "state_before": "case h\np' : \u2115\nk : \u2124\nh : (fun x => \u03c9 ^ 2 ^ p' * x) (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = (fun x => \u03c9 ^ 2 ^ p' * x) \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "have t : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1) := by ring", "annotated_tactic": ["have t : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1) := by ring", []], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "rw [mul_add, \u2190 pow_add \u03c9, t, \u2190 mul_pow \u03c9 \u03c9b (2 ^ p'), \u03c9_mul_\u03c9b, one_pow] at h", "annotated_tactic": ["rw [<a>mul_add</a>, \u2190 <a>pow_add</a> <a>\u03c9</a>, t, \u2190 <a>mul_pow</a> <a>\u03c9</a> <a>\u03c9b</a> (2 ^ p'), <a>\u03c9_mul_\u03c9b</a>, <a>one_pow</a>] at h", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "LucasLehmer.X.\u03c9", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [412, 5], "def_end_pos": [412, 6]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [310, 32], "def_end_pos": [310, 39]}, {"full_name": "LucasLehmer.X.\u03c9", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [412, 5], "def_end_pos": [412, 6]}, {"full_name": "LucasLehmer.X.\u03c9b", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [417, 5], "def_end_pos": [417, 7]}, {"full_name": "LucasLehmer.X.\u03c9_mul_\u03c9b", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [421, 9], "def_end_pos": [421, 17]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "rw [mul_comm, coe_mul] at h", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>coe_mul</a>] at h", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "LucasLehmer.X.coe_mul", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [385, 9], "def_end_pos": [385, 16]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191(2 ^ (p' + 2) - 1) * \u2191k * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "rw [mul_comm _ (k : X (q (p' + 2)))] at h", "annotated_tactic": ["rw [<a>mul_comm</a> _ (k : <a>X</a> (<a>q</a> (p' + 2)))] at h", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "LucasLehmer.X", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [226, 5], "def_end_pos": [226, 6]}, {"full_name": "LucasLehmer.q", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [216, 5], "def_end_pos": [216, 6]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191(2 ^ (p' + 2) - 1) * \u2191k * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "replace h := eq_sub_of_add_eq h", "annotated_tactic": ["replace h := <a>eq_sub_of_add_eq</a> h", [{"full_name": "eq_sub_of_add_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1025, 15], "def_end_pos": [1025, 31]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "have : 1 \u2264 2 ^ (p' + 2) := Nat.one_le_pow _ _ (by decide)", "annotated_tactic": ["have : 1 \u2264 2 ^ (p' + 2) := <a>Nat.one_le_pow</a> _ _ (by decide)", [{"full_name": "Nat.one_le_pow", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [785, 7], "def_end_pos": [785, 17]}]], "state_before": "case h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "case h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\nthis : 1 \u2264 2 ^ (p' + 2)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "exact mod_cast h", "annotated_tactic": ["exact mod_cast h", []], "state_before": "case h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\nthis : 1 \u2264 2 ^ (p' + 2)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "no goals"}, {"tactic": "simp only [add_tsub_cancel_right, ZMod.intCast_zmod_eq_zero_iff_dvd, ofNat_pos,\n  pow_pos, cast_pred, cast_pow, cast_ofNat] at h", "annotated_tactic": ["simp only [<a>add_tsub_cancel_right</a>, <a>ZMod.intCast_zmod_eq_zero_iff_dvd</a>, <a>ofNat_pos</a>,\n      <a>pow_pos</a>, <a>cast_pred</a>, <a>cast_pow</a>, <a>cast_ofNat</a>] at h", [{"full_name": "add_tsub_cancel_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [356, 9], "def_end_pos": [356, 30]}, {"full_name": "ZMod.intCast_zmod_eq_zero_iff_dvd", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 37]}, {"full_name": "Nat.ofNat_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [104, 9], "def_end_pos": [104, 18]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "Nat.cast_pred", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [39, 9], "def_end_pos": [39, 18]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "Nat.cast_ofNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [76, 28], "def_end_pos": [76, 42]}]], "state_before": "p' : \u2115\nh : \u2191(s (p' + 2 - 2)) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1", "state_after": "p' : \u2115\nh : 2 ^ (p' + 2) - 1 \u2223 s p'\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "p' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "p' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 0 < 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.coe_algHom_ofAlgHom", "start": [539, 1], "end": [541, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean", "full_name": "CategoryTheory.ShortComplex.shortExact_iff_unop", "start": [88, 1], "end": [89, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe", "start": [805, 1], "end": [821, 35], "traced_tactics": [{"tactic": "refine\n  generateFrom_induction (p := fun t : Set s => NullMeasurableSet ((\u2191) '' t) \u03bc)\n    { t : Set s | \u2203 s' : Set \u03b1, MeasurableSet s' \u2227 (\u2191) \u207b\u00b9' s' = t } ?_ ?_ ?_ ?_ ht", "annotated_tactic": ["refine\n    <a>generateFrom_induction</a> (p := fun t : <a>Set</a> s => <a>NullMeasurableSet</a> ((\u2191) '' t) \u03bc)\n      { t : <a>Set</a> s | \u2203 s' : <a>Set</a> \u03b1, <a>MeasurableSet</a> s' \u2227 (\u2191) \u207b\u00b9' s' = t } ?_ ?_ ?_ ?_ ht", [{"full_name": "MeasurableSpace.generateFrom_induction", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [382, 9], "def_end_pos": [382, 31]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "MeasureTheory.NullMeasurableSet", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [92, 5], "def_end_pos": [92, 22]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 NullMeasurableSet (Subtype.val '' t) \u03bc", "state_after": "case refine_1\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t}, (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t\n\ncase refine_2\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) \u2205\n\ncase refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s),\n    (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t\u1d9c\n\ncase refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),\n    (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (f n)) \u2192\n      (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (\u22c3 i, f i)"}, {"tactic": "rintro t' \u27e8s', hs', rfl\u27e9", "annotated_tactic": ["rintro t' \u27e8s', hs', rfl\u27e9", []], "state_before": "case refine_1\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t}, (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t", "state_after": "case refine_1.intro.intro\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s')) \u03bc"}, {"tactic": "rw [Subtype.image_preimage_coe]", "annotated_tactic": ["rw [<a>Subtype.image_preimage_coe</a>]", [{"full_name": "Subtype.image_preimage_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1412, 9], "def_end_pos": [1412, 27]}]], "state_before": "case refine_1.intro.intro\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s')) \u03bc", "state_after": "case refine_1.intro.intro\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (s \u2229 s') \u03bc"}, {"tactic": "exact hs.inter (hs'.nullMeasurableSet)", "annotated_tactic": ["exact hs.inter (hs'.nullMeasurableSet)", []], "state_before": "case refine_1.intro.intro\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (s \u2229 s') \u03bc", "state_after": "no goals"}, {"tactic": "simp only [image_empty, nullMeasurableSet_empty]", "annotated_tactic": ["simp only [<a>image_empty</a>, <a>nullMeasurableSet_empty</a>]", [{"full_name": "Set.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [309, 9], "def_end_pos": [309, 20]}, {"full_name": "MeasureTheory.nullMeasurableSet_empty", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [103, 9], "def_end_pos": [103, 32]}]], "state_before": "case refine_2\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) \u2205", "state_after": "no goals"}, {"tactic": "intro t'", "annotated_tactic": ["intro t'", []], "state_before": "case refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s),\n    (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t\u1d9c", "state_after": "case refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t'\u1d9c"}, {"tactic": "simp only [\u2190 range_diff_image Subtype.coe_injective, Subtype.range_coe_subtype, setOf_mem_eq]", "annotated_tactic": ["simp only [\u2190 <a>range_diff_image</a> <a>Subtype.coe_injective</a>, <a>Subtype.range_coe_subtype</a>, <a>setOf_mem_eq</a>]", [{"full_name": "Set.range_diff_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 25]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "Subtype.range_coe_subtype", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1389, 9], "def_end_pos": [1389, 26]}, {"full_name": "Set.setOf_mem_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 21]}]], "state_before": "case refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) t'\u1d9c", "state_after": "case refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 NullMeasurableSet (Subtype.val '' t') \u03bc \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t') \u03bc"}, {"tactic": "exact hs.diff", "annotated_tactic": ["exact hs.diff", []], "state_before": "case refine_3\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 NullMeasurableSet (Subtype.val '' t') \u03bc \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t') \u03bc", "state_after": "no goals"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),\n    (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (f n)) \u2192\n      (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (\u22c3 i, f i)", "state_after": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (f n)) \u2192\n    (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (\u22c3 i, f i)"}, {"tactic": "dsimp only []", "annotated_tactic": ["dsimp only []", []], "state_before": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (f n)) \u2192\n    (fun t => NullMeasurableSet (Subtype.val '' t) \u03bc) (\u22c3 i, f i)", "state_after": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n) \u03bc) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 i, f i) \u03bc"}, {"tactic": "rw [image_iUnion]", "annotated_tactic": ["rw [<a>image_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}]], "state_before": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n) \u03bc) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 i, f i) \u03bc", "state_after": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n) \u03bc) \u2192 NullMeasurableSet (\u22c3 i, Subtype.val '' f i) \u03bc"}, {"tactic": "exact NullMeasurableSet.iUnion", "annotated_tactic": ["exact <a>NullMeasurableSet.iUnion</a>", [{"full_name": "MeasureTheory.NullMeasurableSet.iUnion", "def_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "def_pos": [140, 19], "def_end_pos": [140, 25]}]], "state_before": "case refine_4\nR : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s \u03bc\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n) \u03bc) \u2192 NullMeasurableSet (\u22c3 i, Subtype.val '' f i) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.valid_nil", "start": [1064, 1], "end": [1065, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "full_name": "MeasureTheory.Measure.integral_comp_mul_left", "start": [135, 1], "end": [137, 95], "traced_tactics": [{"tactic": "simp_rw [\u2190 smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_eq_mul</a>, <a>Measure.integral_comp_smul</a>, <a>FiniteDimensional.finrank_self</a>, <a>pow_one</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "MeasureTheory.Measure.integral_comp_smul", "def_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "def_pos": [64, 9], "def_end_pos": [64, 27]}, {"full_name": "FiniteDimensional.finrank_self", "def_path": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean", "def_pos": [449, 9], "def_end_pos": [449, 39]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MeasurableSpace E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b2 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ng : \u211d \u2192 F\na : \u211d\n\u22a2 \u222b (x : \u211d), g (a * x) = |a\u207b\u00b9| \u2022 \u222b (y : \u211d), g y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_gt_inter_not_gt", "start": [426, 1], "end": [428, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PNat/Basic.lean", "full_name": "PNat.pow_coe", "start": [266, 1], "end": [267, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.im_idem", "start": [881, 9], "end": [881, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "full_name": "WeierstrassCurve.Jacobian.Point.toAffine_of_equiv", "start": [1453, 1], "end": [1455, 33], "traced_tactics": [{"tactic": "rcases h with \u27e8u, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8u, rfl\u27e9", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP Q : Fin 3 \u2192 F\nh : P \u2248 Q\n\u22a2 toAffine W P = toAffine W Q", "state_after": "case intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nQ : Fin 3 \u2192 F\nu : F\u02e3\n\u22a2 toAffine W ((fun m => m \u2022 Q) u) = toAffine W Q"}, {"tactic": "exact toAffine_smul Q u.isUnit", "annotated_tactic": ["exact <a>toAffine_smul</a> Q u.isUnit", [{"full_name": "WeierstrassCurve.Jacobian.Point.toAffine_smul", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [1443, 7], "def_end_pos": [1443, 20]}]], "state_before": "case intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nQ : Fin 3 \u2192 F\nu : F\u02e3\n\u22a2 toAffine W ((fun m => m \u2022 Q) u) = toAffine W Q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.sigma_support", "start": [1871, 1], "end": [1875, 8], "traced_tactics": [{"tactic": "simp only [Finset.ext_iff, splitSupport, split, comapDomain, @mem_image _ _ (Classical.decEq _),\n  mem_preimage, Sigma.forall, mem_sigma]", "annotated_tactic": ["simp only [<a>Finset.ext_iff</a>, <a>splitSupport</a>, <a>split</a>, <a>comapDomain</a>, @<a>mem_image</a> _ _ (<a>Classical.decEq</a> _),\n    <a>mem_preimage</a>, <a>Sigma.forall</a>, <a>mem_sigma</a>]", [{"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"full_name": "Finsupp.splitSupport", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [1846, 5], "def_end_pos": [1846, 17]}, {"full_name": "Finsupp.split", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [1834, 5], "def_end_pos": [1834, 10]}, {"full_name": "Finsupp.comapDomain", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [685, 5], "def_end_pos": [685, 16]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}, {"full_name": "Finset.mem_preimage", "def_path": "Mathlib/Data/Finset/Preimage.lean", "def_pos": [32, 9], "def_end_pos": [32, 21]}, {"full_name": "Sigma.forall", "def_path": "Mathlib/Data/Sigma/Basic.lean", "def_pos": [95, 9], "def_end_pos": [95, 17]}, {"full_name": "Finset.mem_sigma", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u22a2 l.support = l.splitSupport.sigma fun i => (l.split i).support", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u22a2 \u2200 (a : \u03b9) (b : \u03b1s a), \u27e8a, b\u27e9 \u2208 l.support \u2194 (\u2203 a_1 \u2208 l.support, a_1.fst = a) \u2227 \u27e8a, b\u27e9 \u2208 l.support"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u22a2 \u2200 (a : \u03b9) (b : \u03b1s a), \u27e8a, b\u27e9 \u2208 l.support \u2194 (\u2203 a_1 \u2208 l.support, a_1.fst = a) \u2227 \u27e8a, b\u27e9 \u2208 l.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Icc_union_Ici_eq_Ici", "start": [1381, 1], "end": [1382, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Integral.lean", "full_name": "is_integral_localization_at_leadingCoeff", "start": [207, 1], "end": [216, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Nat.lean", "full_name": "Nat.card_Ioc", "start": [90, 1], "end": [91, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Hindman.lean", "full_name": "Hindman.exists_FP_of_finite_cover", "start": [229, 1], "end": [234, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "full_name": "List.Nat.mem_antidiagonalTuple", "start": [79, 1], "end": [90, 61], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "case h0\nk n : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 n \u2194 \u2211 i : Fin 0, i.elim0 = n", "state_after": "case h0.zero\nk : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 0 \u2194 \u2211 i : Fin 0, i.elim0 = 0\n\ncase h0.succ\nk n\u271d : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (n\u271d + 1) \u2194 \u2211 i : Fin 0, i.elim0 = n\u271d + 1"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "case h0.zero\nk : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 0 \u2194 \u2211 i : Fin 0, i.elim0 = 0", "state_after": "no goals"}, {"tactic": "simp [eq_comm]", "annotated_tactic": ["simp [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "case h0.succ\nk n\u271d : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (n\u271d + 1) \u2194 \u2211 i : Fin 0, i.elim0 = n\u271d + 1", "state_after": "no goals"}, {"tactic": "simp_rw [Fin.sum_cons, antidiagonalTuple, List.mem_bind, List.mem_map,\n  List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n  @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _),\n  \u2190 Prod.mk.inj_iff (a\u2081 := Prod.fst _), exists_eq_right]", "annotated_tactic": ["simp_rw [<a>Fin.sum_cons</a>, <a>antidiagonalTuple</a>, <a>List.mem_bind</a>, <a>List.mem_map</a>,\n      <a>List.Nat.mem_antidiagonal</a>, <a>Fin.cons_eq_cons</a>, <a>exists_eq_right_right</a>, ih,\n      @<a>eq_comm</a> _ _ (<a>Prod.snd</a> _), <a>and_comm</a> (a := <a>Prod.snd</a> _ = _),\n      \u2190 <a>Prod.mk.inj_iff</a> (a\u2081 := <a>Prod.fst</a> _), <a>exists_eq_right</a>]", [{"full_name": "Fin.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [105, 3], "def_end_pos": [105, 14]}, {"full_name": "List.Nat.antidiagonalTuple", "def_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "def_pos": [61, 5], "def_end_pos": [61, 22]}, {"full_name": "List.mem_bind", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1253, 17], "def_end_pos": [1253, 25]}, {"full_name": "List.mem_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [750, 17], "def_end_pos": [750, 24]}, {"full_name": "List.Nat.mem_antidiagonal", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [38, 9], "def_end_pos": [38, 25]}, {"full_name": "Fin.cons_eq_cons", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [157, 9], "def_end_pos": [157, 21]}, {"full_name": "exists_eq_right_right", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [301, 17], "def_end_pos": [301, 38]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "Prod.snd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [485, 3], "def_end_pos": [485, 6]}, {"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}, {"full_name": "Prod.snd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [485, 3], "def_end_pos": [485, 6]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Prod.fst", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [483, 3], "def_end_pos": [483, 6]}, {"full_name": "exists_eq_right", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 32]}]], "state_before": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 \u2211 i : Fin (n\u271d + 1), Fin.cons x\u2080 x i = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.tendsto_condexp_unique", "start": [402, 1], "end": [426, 91], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [614, 7], "def_end_pos": [614, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [614, 7], "def_end_pos": [614, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]"}, {"tactic": "refine (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans ?_).symm", "annotated_tactic": ["refine (<a>condexp_ae_eq_condexpL1</a> hm f).<a>trans</a> ((<a>condexp_ae_eq_condexpL1</a> hm g).<a>trans</a> ?_).<a>symm</a>", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 27]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc g) =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc f)"}, {"tactic": "rw [\u2190 Lp.ext_iff]", "annotated_tactic": ["rw [\u2190 <a>Lp.ext_iff</a>]", [{"full_name": "MeasureTheory.Lp.ext_iff", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [173, 9], "def_end_pos": [173, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc g) =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc f)", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hn_eq : \u2200 n, condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) := by\n  intro n\n  ext1\n  refine (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans ?_)\n  exact condexp_ae_eq_condexpL1 hm (fs n)", "annotated_tactic": ["have hn_eq : \u2200 n, <a>condexpL1</a> hm \u03bc (gs n) = <a>condexpL1</a> hm \u03bc (fs n) := by\n    intro n\n    ext1\n    refine (<a>condexp_ae_eq_condexpL1</a> hm (gs n)).symm.trans ((hfg n).symm.trans ?_)\n    exact <a>condexp_ae_eq_condexpL1</a> hm (fs n)", [{"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs\n    hfs_bound hfs", "annotated_tactic": ["have hcond_fs : <a>Tendsto</a> (fun n => <a>condexpL1</a> hm \u03bc (fs n)) <a>atTop</a> (\ud835\udcdd (<a>condexpL1</a> hm \u03bc f)) :=\n    <a>tendsto_condexpL1_of_dominated_convergence</a> hm _ (fun n => (hfs_int n).1) h_int_bound_fs\n      hfs_bound hfs", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "MeasureTheory.tendsto_condexpL1_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [389, 9], "def_end_pos": [389, 51]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "have hcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs\n    hgs_bound hgs", "annotated_tactic": ["have hcond_gs : <a>Tendsto</a> (fun n => <a>condexpL1</a> hm \u03bc (gs n)) <a>atTop</a> (\ud835\udcdd (<a>condexpL1</a> hm \u03bc g)) :=\n    <a>tendsto_condexpL1_of_dominated_convergence</a> hm _ (fun n => (hgs_int n).1) h_int_bound_gs\n      hgs_bound hgs", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [522, 5], "def_end_pos": [522, 14]}, {"full_name": "MeasureTheory.tendsto_condexpL1_of_dominated_convergence", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [389, 9], "def_end_pos": [389, 51]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\nhcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f"}, {"tactic": "exact tendsto_nhds_unique_of_eventuallyEq hcond_gs hcond_fs (eventually_of_forall hn_eq)", "annotated_tactic": ["exact <a>tendsto_nhds_unique_of_eventuallyEq</a> hcond_gs hcond_fs (<a>eventually_of_forall</a> hn_eq)", [{"full_name": "tendsto_nhds_unique_of_eventuallyEq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1518, 9], "def_end_pos": [1518, 44]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nhn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)\nhcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f))\nhcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g))\n\u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1da0[ae \u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : \u00acm \u2264 m0\n\u22a2 0 =\u1da0[ae \u03bc] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 \u03bc[f|m] =\u1da0[ae \u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 0 =\u1da0[ae \u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (\u03bc.trim hm)\n\u22a2 0 =\u1da0[ae \u03bc] 0", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\n\u22a2 \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)", "state_after": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n)", "state_after": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (gs n)) =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))"}, {"tactic": "refine (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans ?_)", "annotated_tactic": ["refine (<a>condexp_ae_eq_condexpL1</a> hm (gs n)).symm.trans ((hfg n).symm.trans ?_)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc (gs n)) =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))", "state_after": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 \u03bc[fs n|m] =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))"}, {"tactic": "exact condexp_ae_eq_condexpL1 hm (fs n)", "annotated_tactic": ["exact <a>condexp_ae_eq_condexpL1</a> hm (fs n)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}]], "state_before": "case h\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : RCLike \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nfs gs : \u2115 \u2192 \u03b1 \u2192 F'\nf g : \u03b1 \u2192 F'\nhfs_int : \u2200 (n : \u2115), Integrable (fs n) \u03bc\nhgs_int : \u2200 (n : \u2115), Integrable (gs n) \u03bc\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\nhgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x))\nbound_fs : \u03b1 \u2192 \u211d\nh_int_bound_fs : Integrable bound_fs \u03bc\nbound_gs : \u03b1 \u2192 \u211d\nh_int_bound_gs : Integrable bound_gs \u03bc\nhfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x\nhgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x\nhfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1da0[ae \u03bc] \u03bc[gs n|m]\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (\u03bc.trim hm)\nn : \u2115\n\u22a2 \u03bc[fs n|m] =\u1da0[ae \u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "full_name": "RingHom.codomain_trivial_iff_map_one_eq_zero", "start": [580, 1], "end": [580, 97], "traced_tactics": [{"tactic": "rw [map_one, eq_comm]", "annotated_tactic": ["rw [<a>map_one</a>, <a>eq_comm</a>]", [{"full_name": "map_one", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [204, 9], "def_end_pos": [204, 16]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx y : \u03b1\n\u22a2 0 = 1 \u2194 f 1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bornology/Basic.lean", "full_name": "Bornology.ext_iff'", "start": [238, 1], "end": [240, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "full_name": "cfc_nonpos", "start": [756, 1], "end": [760, 59], "traced_tactics": [{"tactic": "by_cases hf : ContinuousOn f (spectrum R a)", "annotated_tactic": ["by_cases hf : <a>ContinuousOn</a> f (<a>spectrum</a> R a)", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [170, 5], "def_end_pos": [170, 17]}, {"full_name": "spectrum", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [68, 5], "def_end_pos": [68, 13]}]], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2075 : OrderedCommSemiring R\ninst\u271d\u00b9\u2074 : StarRing R\ninst\u271d\u00b9\u00b3 : StarOrderedRing R\ninst\u271d\u00b9\u00b2 : MetricSpace R\ninst\u271d\u00b9\u00b9 : TopologicalSemiring R\ninst\u271d\u00b9\u2070 : ContinuousStar R\ninst\u271d\u2079 : \u2200 (\u03b1 : Type ?u.684485) [inst : TopologicalSpace \u03b1], StarOrderedRing C(\u03b1, R)\ninst\u271d\u2078 : TopologicalSpace A\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : PartialOrder A\ninst\u271d\u2074 : StarOrderedRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : ContinuousFunctionalCalculus R p\ninst\u271d : NonnegSpectrumClass R A\nf : R \u2192 R\na : A\nh : \u2200 x \u2208 spectrum R a, f x \u2264 0\n\u22a2 cfc f a \u2264 0", "state_after": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2075 : OrderedCommSemiring R\ninst\u271d\u00b9\u2074 : StarRing R\ninst\u271d\u00b9\u00b3 : StarOrderedRing R\ninst\u271d\u00b9\u00b2 : MetricSpace R\ninst\u271d\u00b9\u00b9 : TopologicalSemiring R\ninst\u271d\u00b9\u2070 : ContinuousStar R\ninst\u271d\u2079 : \u2200 (\u03b1 : Type ?u.684485) [inst : TopologicalSpace \u03b1], StarOrderedRing C(\u03b1, R)\ninst\u271d\u2078 : TopologicalSpace A\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : PartialOrder A\ninst\u271d\u2074 : StarOrderedRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : ContinuousFunctionalCalculus R p\ninst\u271d : NonnegSpectrumClass R A\nf : R \u2192 R\na : A\nh : \u2200 x \u2208 spectrum R a, f x \u2264 0\nhf : ContinuousOn f (spectrum R a)\n\u22a2 cfc f a \u2264 0\n\ncase neg\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2075 : OrderedCommSemiring R\ninst\u271d\u00b9\u2074 : StarRing R\ninst\u271d\u00b9\u00b3 : StarOrderedRing R\ninst\u271d\u00b9\u00b2 : MetricSpace R\ninst\u271d\u00b9\u00b9 : TopologicalSemiring R\ninst\u271d\u00b9\u2070 : ContinuousStar R\ninst\u271d\u2079 : \u2200 (\u03b1 : Type ?u.684485) [inst : TopologicalSpace \u03b1], StarOrderedRing C(\u03b1, R)\ninst\u271d\u2078 : TopologicalSpace A\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : PartialOrder A\ninst\u271d\u2074 : StarOrderedRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : ContinuousFunctionalCalculus R p\ninst\u271d : NonnegSpectrumClass R A\nf : R \u2192 R\na : A\nh : \u2200 x \u2208 spectrum R a, f x \u2264 0\nhf : \u00acContinuousOn f (spectrum R a)\n\u22a2 cfc f a \u2264 0"}, {"tactic": "simpa using cfc_mono h", "annotated_tactic": ["simpa using <a>cfc_mono</a> h", [{"full_name": "cfc_mono", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [736, 7], "def_end_pos": [736, 15]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2075 : OrderedCommSemiring R\ninst\u271d\u00b9\u2074 : StarRing R\ninst\u271d\u00b9\u00b3 : StarOrderedRing R\ninst\u271d\u00b9\u00b2 : MetricSpace R\ninst\u271d\u00b9\u00b9 : TopologicalSemiring R\ninst\u271d\u00b9\u2070 : ContinuousStar R\ninst\u271d\u2079 : \u2200 (\u03b1 : Type ?u.684485) [inst : TopologicalSpace \u03b1], StarOrderedRing C(\u03b1, R)\ninst\u271d\u2078 : TopologicalSpace A\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : PartialOrder A\ninst\u271d\u2074 : StarOrderedRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : ContinuousFunctionalCalculus R p\ninst\u271d : NonnegSpectrumClass R A\nf : R \u2192 R\na : A\nh : \u2200 x \u2208 spectrum R a, f x \u2264 0\nhf : ContinuousOn f (spectrum R a)\n\u22a2 cfc f a \u2264 0", "state_after": "no goals"}, {"tactic": "simp only [cfc_apply_of_not_continuousOn _ hf, le_rfl]", "annotated_tactic": ["simp only [<a>cfc_apply_of_not_continuousOn</a> _ hf, <a>le_rfl</a>]", [{"full_name": "cfc_apply_of_not_continuousOn", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [311, 7], "def_end_pos": [311, 36]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2075 : OrderedCommSemiring R\ninst\u271d\u00b9\u2074 : StarRing R\ninst\u271d\u00b9\u00b3 : StarOrderedRing R\ninst\u271d\u00b9\u00b2 : MetricSpace R\ninst\u271d\u00b9\u00b9 : TopologicalSemiring R\ninst\u271d\u00b9\u2070 : ContinuousStar R\ninst\u271d\u2079 : \u2200 (\u03b1 : Type u_1) [inst : TopologicalSpace \u03b1], StarOrderedRing C(\u03b1, R)\ninst\u271d\u2078 : TopologicalSpace A\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : PartialOrder A\ninst\u271d\u2074 : StarOrderedRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : ContinuousFunctionalCalculus R p\ninst\u271d : NonnegSpectrumClass R A\nf : R \u2192 R\na : A\nh : \u2200 x \u2208 spectrum R a, f x \u2264 0\nhf : \u00acContinuousOn f (spectrum R a)\n\u22a2 cfc f a \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/TFAE.lean", "full_name": "List.TFAE.out", "start": [74, 1], "end": [76, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Filter.lean", "full_name": "BoxIntegral.IntegrationParams.hasBasis_toFilterDistortion", "start": [469, 1], "end": [475, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/OrderIsoNat.lean", "full_name": "RelEmbedding.exists_not_acc_lt_of_not_acc", "start": [58, 1], "end": [62, 18], "traced_tactics": [{"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u00acAcc r a\n\u22a2 \u2203 b, \u00acAcc r b \u2227 r b a", "state_after": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\n\u22a2 Acc r a"}, {"tactic": "refine \u27e8_, fun b hr => ?_\u27e9", "annotated_tactic": ["refine \u27e8_, fun b hr => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\n\u22a2 Acc r a", "state_after": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\nb : \u03b1\nhr : r b a\n\u22a2 Acc r b"}, {"tactic": "by_contra hb", "annotated_tactic": ["by_contra hb", []], "state_before": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\nb : \u03b1\nhr : r b a\n\u22a2 Acc r b", "state_after": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\nb : \u03b1\nhr : r b a\nhb : \u00acAcc r b\n\u22a2 False"}, {"tactic": "exact h b hb hr", "annotated_tactic": ["exact h b hb hr", []], "state_before": "\u03b1 : Type u_1\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\u271d\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (b : \u03b1), \u00acAcc r b \u2192 \u00acr b a\nb : \u03b1\nhr : r b a\nhb : \u00acAcc r b\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.image_prod_mk_subset_prod_left", "start": [340, 1], "end": [342, 17], "traced_tactics": [{"tactic": "rintro _ \u27e8a, ha, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8a, ha, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (fun a => (a, b)) '' s \u2286 s \u00d7\u02e2 t", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\nhb : b \u2208 t\na : \u03b1\nha : a \u2208 s\n\u22a2 (fun a => (a, b)) a \u2208 s \u00d7\u02e2 t"}, {"tactic": "exact \u27e8ha, hb\u27e9", "annotated_tactic": ["exact \u27e8ha, hb\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\nhb : b \u2208 t\na : \u03b1\nha : a \u2208 s\n\u22a2 (fun a => (a, b)) a \u2208 s \u00d7\u02e2 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "PMF.map_apply", "start": [52, 1], "end": [52, 84], "traced_tactics": [{"tactic": "simp [map]", "annotated_tactic": ["simp [<a>map</a>]", [{"full_name": "PMF.map", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : PMF \u03b1\nb : \u03b2\n\u22a2 (map f p) b = \u2211' (a : \u03b1), if b = f a then p a else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.mk\u2082'\u209b\u2097_apply", "start": [74, 1], "end": [75, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Linear.lean", "full_name": "LinearMap.derivWithin", "start": [99, 11], "end": [101, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Opposite.lean", "full_name": "Set.unop_op", "start": [56, 1], "end": [56, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Extensive.lean", "full_name": "CategoryTheory.FinitaryExtensive.isPullback_initial_to", "start": [531, 1], "end": [535, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SupClosed.lean", "full_name": "infClosed_infClosure", "start": [339, 1], "end": [339, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieHom.mem_idealRange", "start": [1174, 1], "end": [1176, 50], "traced_tactics": [{"tactic": "rw [idealRange_eq_map]", "annotated_tactic": ["rw [<a>idealRange_eq_map</a>]", [{"full_name": "LieHom.idealRange_eq_map", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 26]}]], "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n\u22a2 f x \u2208 f.idealRange", "state_after": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n\u22a2 f x \u2208 LieIdeal.map f \u22a4"}, {"tactic": "exact LieIdeal.mem_map (LieSubmodule.mem_top x)", "annotated_tactic": ["exact <a>LieIdeal.mem_map</a> (<a>LieSubmodule.mem_top</a> x)", [{"full_name": "LieIdeal.mem_map", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 16]}, {"full_name": "LieSubmodule.mem_top", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [409, 9], "def_end_pos": [409, 16]}]], "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n\u22a2 f x \u2208 LieIdeal.map f \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "continuousWithinAt_update_of_ne", "start": [719, 1], "end": [725, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": "SimpleGraph.map_singletonSubgraph", "start": [855, 1], "end": [860, 19], "traced_tactics": [{"tactic": "ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply,\n  exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,\n  singletonSubgraph_verts, Set.image_singleton]", "annotated_tactic": ["ext <;> simp only [<a>Relation.Map</a>, <a>Subgraph.map_adj</a>, <a>singletonSubgraph_adj</a>, <a>Pi.bot_apply</a>,\n    <a>exists_and_left</a>, <a>and_iff_left_iff_imp</a>, <a>IsEmpty.forall_iff</a>, <a>Subgraph.map_verts</a>,\n    <a>singletonSubgraph_verts</a>, <a>Set.image_singleton</a>]", [{"full_name": "Relation.Map", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [217, 15], "def_end_pos": [217, 18]}, {"full_name": "SimpleGraph.Subgraph.map_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [633, 3], "def_end_pos": [633, 8]}, {"full_name": "SimpleGraph.singletonSubgraph_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [74, 3], "def_end_pos": [74, 8]}, {"full_name": "Pi.bot_apply", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [601, 9], "def_end_pos": [601, 18]}, {"full_name": "exists_and_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [288, 17], "def_end_pos": [288, 32]}, {"full_name": "and_iff_left_iff_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [164, 17], "def_end_pos": [164, 37]}, {"full_name": "IsEmpty.forall_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [128, 9], "def_end_pos": [128, 19]}, {"full_name": "SimpleGraph.Subgraph.map_verts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [633, 3], "def_end_pos": [633, 8]}, {"full_name": "SimpleGraph.singletonSubgraph_verts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [74, 3], "def_end_pos": [74, 8]}, {"full_name": "Set.image_singleton", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [335, 9], "def_end_pos": [335, 24]}]], "state_before": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G \u2192g G'\nv : V\n\u22a2 Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v)", "state_after": "case Adj.h.h.a\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G \u2192g G'\nv : V\nx\u271d\u00b9 x\u271d : W\n\u22a2 \u22a5 \u2192 \u2203 x, f x = x\u271d\u00b9 \u2227 \u2203 x, f x = x\u271d"}, {"tactic": "exact False.elim", "annotated_tactic": ["exact <a>False.elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case Adj.h.h.a\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G \u2192g G'\nv : V\nx\u271d\u00b9 x\u271d : W\n\u22a2 \u22a5 \u2192 \u2203 x, f x = x\u271d\u00b9 \u2227 \u2203 x, f x = x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Int.lean", "full_name": "Int.even_iff_not_odd", "start": [108, 1], "end": [108, 74], "traced_tactics": [{"tactic": "rw [not_odd_iff, even_iff]", "annotated_tactic": ["rw [<a>not_odd_iff</a>, <a>even_iff</a>]", [{"full_name": "Int.not_odd_iff", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [105, 7], "def_end_pos": [105, 18]}, {"full_name": "Int.even_iff", "def_path": "Mathlib/Algebra/Group/Int.lean", "def_pos": [198, 7], "def_end_pos": [198, 15]}]], "state_before": "m n : \u2124\n\u22a2 Even n \u2194 \u00acOdd n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iUnion", "start": [1378, 1], "end": [1381, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Infix.lean", "full_name": "List.mem_of_mem_suffix", "start": [550, 1], "end": [551, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.parts_nonempty", "start": [203, 1], "end": [204, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan", "start": [128, 1], "end": [132, 64], "traced_tactics": [{"tactic": "rcases hp with \u27e8p2, \u27e8hp2, \u27e8v2, \u27e8hv2, hv2p\u27e9\u27e9\u27e9\u27e9", "annotated_tactic": ["rcases hp with \u27e8p2, \u27e8hp2, \u27e8v2, \u27e8hv2, hv2p\u27e9\u27e9\u27e9\u27e9", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhp : p \u2208 spanPoints k s\nhv : v \u2208 vectorSpan k s\n\u22a2 v +\u1d65 p \u2208 spanPoints k s", "state_after": "case intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v +\u1d65 p \u2208 spanPoints k s"}, {"tactic": "rw [hv2p, vadd_vadd]", "annotated_tactic": ["rw [hv2p, <a>vadd_vadd</a>]", [{"full_name": "vadd_vadd", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [445, 3], "def_end_pos": [445, 14]}]], "state_before": "case intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v +\u1d65 p \u2208 spanPoints k s", "state_after": "case intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v + v2 +\u1d65 p2 \u2208 spanPoints k s"}, {"tactic": "exact \u27e8p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl\u27e9", "annotated_tactic": ["exact \u27e8p2, hp2, v + v2, (<a>vectorSpan</a> k s).<a>add_mem</a> hv hv2, <a>rfl</a>\u27e9", [{"full_name": "vectorSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [60, 5], "def_end_pos": [60, 15]}, {"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [220, 19], "def_end_pos": [220, 26]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v + v2 +\u1d65 p2 \u2208 spanPoints k s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_pair", "start": [1446, 1], "end": [1450, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "full_name": "HasFDerivWithinAt.mul", "start": [391, 1], "end": [395, 17], "traced_tactics": [{"tactic": "convert hc.mul' hd", "annotated_tactic": ["convert hc.mul' hd", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n\u22a2 HasFDerivWithinAt (fun y => c y * d y) (c x \u2022 d' + d x \u2022 c') s x", "state_after": "case h.e'_10.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n\u22a2 d x \u2022 c' = c'.smulRight (d x)"}, {"tactic": "ext z", "annotated_tactic": ["ext z", []], "state_before": "case h.e'_10.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n\u22a2 d x \u2022 c' = c'.smulRight (d x)", "state_after": "case h.e'_10.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\nz : E\n\u22a2 (d x \u2022 c') z = (c'.smulRight (d x)) z"}, {"tactic": "apply mul_comm", "annotated_tactic": ["apply <a>mul_comm</a>", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case h.e'_10.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\nz : E\n\u22a2 (d x \u2022 c') z = (c'.smulRight (d x)) z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "full_name": "Polynomial.mirror_leadingCoeff", "start": [157, 1], "end": [158, 64], "traced_tactics": [{"tactic": "rw [\u2190 p.mirror_mirror, mirror_trailingCoeff, p.mirror_mirror]", "annotated_tactic": ["rw [\u2190 p.mirror_mirror, <a>mirror_trailingCoeff</a>, p.mirror_mirror]", [{"full_name": "Polynomial.mirror_trailingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "def_pos": [151, 9], "def_end_pos": [151, 29]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 p.mirror.leadingCoeff = p.trailingCoeff", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.RightInverse.comp", "start": [351, 1], "end": [353, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Opposite.lean", "full_name": "Opposite.op_inj_iff", "start": [72, 1], "end": [73, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/Grp/Basic.lean", "full_name": "AddCommGrp.asHom_injective", "start": [391, 1], "end": [392, 84], "traced_tactics": [{"tactic": "convert congr_arg (fun k : AddCommGrp.of \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w <;> simp", "annotated_tactic": ["convert <a>congr_arg</a> (fun k : <a>AddCommGrp.of</a> \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w <;> simp", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "AddCommGrp.of", "def_path": "Mathlib/Algebra/Category/Grp/Basic.lean", "def_pos": [257, 3], "def_end_pos": [257, 14]}]], "state_before": "G : AddCommGrp\nh k : \u2191G\nw : asHom h = asHom k\n\u22a2 h = k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Bernoulli.lean", "full_name": "bernoulli'_one", "start": [110, 1], "end": [112, 11], "traced_tactics": [{"tactic": "rw [bernoulli'_def]", "annotated_tactic": ["rw [<a>bernoulli'_def</a>]", [{"full_name": "bernoulli'_def", "def_path": "Mathlib/NumberTheory/Bernoulli.lean", "def_pos": [78, 9], "def_end_pos": [78, 23]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 bernoulli' 1 = 1 / 2", "state_after": "A : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 1 - \u2211 k \u2208 range 1, \u2191(Nat.choose 1 k) / (\u21911 - \u2191k + 1) * bernoulli' k = 1 / 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "A : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 1 - \u2211 k \u2208 range 1, \u2191(Nat.choose 1 k) / (\u21911 - \u2191k + 1) * bernoulli' k = 1 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_lookup_union_middle", "start": [604, 1], "end": [606, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/IsConnected.lean", "full_name": "CategoryTheory.induct_on_objects", "start": [183, 1], "end": [187, 10], "traced_tactics": [{"tactic": "let aux (j\u2081 j\u2082 : J) (f : j\u2081 \u27f6 j\u2082) := congrArg ULift.up <| (h1 f).eq", "annotated_tactic": ["let aux (j\u2081 j\u2082 : J) (f : j\u2081 \u27f6 j\u2082) := <a>congrArg</a> <a>ULift.up</a> <| (h1 f).<a>eq</a>", [{"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "ULift.up", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [810, 41], "def_end_pos": [810, 43]}, {"full_name": "Iff.eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [958, 26], "def_end_pos": [958, 32]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\np : Set J\nj\u2080 : J\nh0 : j\u2080 \u2208 p\nh1 : \u2200 {j\u2081 j\u2082 : J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : J\n\u22a2 j \u2208 p", "state_after": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\np : Set J\nj\u2080 : J\nh0 : j\u2080 \u2208 p\nh1 : \u2200 {j\u2081 j\u2082 : J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : J\naux : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 { down := j\u2081 \u2208 p } = { down := j\u2082 \u2208 p } :=\n  fun j\u2081 j\u2082 f => congrArg ULift.up (Iff.eq (h1 f))\n\u22a2 j \u2208 p"}, {"tactic": "injection constant_of_preserves_morphisms (fun k => ULift.up.{u\u2081} (k \u2208 p)) aux j j\u2080 with i", "annotated_tactic": ["injection <a>constant_of_preserves_morphisms</a> (fun k => <a>ULift.up</a>.{u\u2081} (k \u2208 p)) aux j j\u2080 with i", [{"full_name": "CategoryTheory.constant_of_preserves_morphisms", "def_path": "Mathlib/CategoryTheory/IsConnected.lean", "def_pos": [142, 9], "def_end_pos": [142, 40]}, {"full_name": "ULift.up", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [810, 41], "def_end_pos": [810, 43]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\np : Set J\nj\u2080 : J\nh0 : j\u2080 \u2208 p\nh1 : \u2200 {j\u2081 j\u2082 : J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : J\naux : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 { down := j\u2081 \u2208 p } = { down := j\u2082 \u2208 p } :=\n  fun j\u2081 j\u2082 f => congrArg ULift.up (Iff.eq (h1 f))\n\u22a2 j \u2208 p", "state_after": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\np : Set J\nj\u2080 : J\nh0 : j\u2080 \u2208 p\nh1 : \u2200 {j\u2081 j\u2082 : J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : J\naux : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 { down := j\u2081 \u2208 p } = { down := j\u2082 \u2208 p } :=\n  fun j\u2081 j\u2082 f => congrArg ULift.up (Iff.eq (h1 f))\ni : (j \u2208 p) = (j\u2080 \u2208 p)\n\u22a2 j \u2208 p"}, {"tactic": "rwa [i]", "annotated_tactic": ["rwa [i]", []], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\np : Set J\nj\u2080 : J\nh0 : j\u2080 \u2208 p\nh1 : \u2200 {j\u2081 j\u2082 : J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : J\naux : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 { down := j\u2081 \u2208 p } = { down := j\u2082 \u2208 p } :=\n  fun j\u2081 j\u2082 f => congrArg ULift.up (Iff.eq (h1 f))\ni : (j \u2208 p) = (j\u2080 \u2208 p)\n\u22a2 j \u2208 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean", "full_name": "CategoryTheory.MonoidalCategory.leftUnitor_inv_tensor_id", "start": [52, 1], "end": [53, 12], "traced_tactics": [{"tactic": "coherence", "annotated_tactic": ["coherence", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 (\u03bb_ X).inv \u2297 \ud835\udfd9 Y = (\u03bb_ (X \u2297 Y)).inv \u226b (\u03b1_ (\ud835\udfd9_ C) X Y).inv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.coe_toRingHom", "start": [703, 1], "end": [704, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "hasStrictDerivAt_iff_hasStrictFDerivAt", "start": [211, 1], "end": [213, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "full_name": "DFinsupp.neLocus_zero_right", "start": [72, 1], "end": [74, 61], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 Zero (N a)\nf g : \u03a0\u2080 (a : \u03b1), N a\n\u22a2 f.neLocus 0 = f.support", "state_after": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 Zero (N a)\nf g : \u03a0\u2080 (a : \u03b1), N a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 f.neLocus 0 \u2194 a\u271d \u2208 f.support"}, {"tactic": "rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]", "annotated_tactic": ["rw [<a>mem_neLocus</a>, <a>mem_support_iff</a>, <a>coe_zero</a>, <a>Pi.zero_apply</a>]", [{"full_name": "DFinsupp.mem_neLocus", "def_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "def_pos": [41, 9], "def_end_pos": [41, 20]}, {"full_name": "DFinsupp.mem_support_iff", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1151, 9], "def_end_pos": [1151, 24]}, {"full_name": "DFinsupp.coe_zero", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [115, 26], "def_end_pos": [115, 34]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}]], "state_before": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 Zero (N a)\nf g : \u03a0\u2080 (a : \u03b1), N a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 f.neLocus 0 \u2194 a\u271d \u2208 f.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_subset_smul", "start": [165, 1], "end": [166, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.Perm.kunion_left", "start": [742, 1], "end": [745, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "full_name": "Ideal.le_toIdeal_homogeneousHull", "start": [546, 1], "end": [554, 14], "traced_tactics": [{"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 I \u2264 (homogeneousHull \ud835\udc9c I).toIdeal", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 r \u2208 (homogeneousHull \ud835\udc9c I).toIdeal"}, {"tactic": "classical\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r]\nrefine Ideal.sum_mem _ ?_\nintro j _\napply Ideal.subset_span\nuse j\nuse \u27e8r, hr\u27e9", "annotated_tactic": ["classical\n  rw [\u2190 <a>DirectSum.sum_support_decompose</a> \ud835\udc9c r]\n  refine <a>Ideal.sum_mem</a> _ ?_\n  intro j _\n  apply <a>Ideal.subset_span</a>\n  use j\n  use \u27e8r, hr\u27e9", [{"full_name": "DirectSum.sum_support_decompose", "def_path": "Mathlib/Algebra/DirectSum/Decomposition.lean", "def_pos": [197, 9], "def_end_pos": [197, 30]}, {"full_name": "Ideal.sum_mem", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [80, 9], "def_end_pos": [80, 16]}, {"full_name": "Ideal.subset_span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 r \u2208 (homogeneousHull \ud835\udc9c I).toIdeal", "state_after": "no goals"}, {"tactic": "rw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r]", "annotated_tactic": ["rw [\u2190 <a>DirectSum.sum_support_decompose</a> \ud835\udc9c r]", [{"full_name": "DirectSum.sum_support_decompose", "def_path": "Mathlib/Algebra/DirectSum/Decomposition.lean", "def_pos": [197, 9], "def_end_pos": [197, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 r \u2208 (homogeneousHull \ud835\udc9c I).toIdeal", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 \u2211 i \u2208 DFinsupp.support ((decompose \ud835\udc9c) r), \u2191(((decompose \ud835\udc9c) r) i) \u2208 (homogeneousHull \ud835\udc9c I).toIdeal"}, {"tactic": "refine Ideal.sum_mem _ ?_", "annotated_tactic": ["refine <a>Ideal.sum_mem</a> _ ?_", [{"full_name": "Ideal.sum_mem", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [80, 9], "def_end_pos": [80, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 \u2211 i \u2208 DFinsupp.support ((decompose \ud835\udc9c) r), \u2191(((decompose \ud835\udc9c) r) i) \u2208 (homogeneousHull \ud835\udc9c I).toIdeal", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 \u2200 c \u2208 DFinsupp.support ((decompose \ud835\udc9c) r), \u2191(((decompose \ud835\udc9c) r) c) \u2208 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\u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support ((decompose \ud835\udc9c) r)\n\u22a2 \u2191(((decompose \ud835\udc9c) r) j) \u2208 {r | \u2203 i x, \u2191(((decompose \ud835\udc9c) \u2191x) i) = r}", "state_after": "case h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support ((decompose \ud835\udc9c) r)\n\u22a2 \u2203 x, \u2191(((decompose \ud835\udc9c) \u2191x) j) = \u2191(((decompose \ud835\udc9c) r) j)"}, {"tactic": "use \u27e8r, hr\u27e9", "annotated_tactic": ["use \u27e8r, hr\u27e9", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support ((decompose \ud835\udc9c) r)\n\u22a2 \u2203 x, \u2191(((decompose \ud835\udc9c) \u2191x) j) = \u2191(((decompose \ud835\udc9c) r) j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean", "full_name": "Fintype.prod_boole", "start": [69, 1], "end": [69, 97], "traced_tactics": [{"tactic": "simp [Finset.prod_boole]", "annotated_tactic": ["simp [<a>Finset.prod_boole</a>]", [{"full_name": "Finset.prod_boole", "def_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean", "def_pos": [29, 7], "def_end_pos": [29, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 (\u220f i : \u03b9, if p i then 1 else 0) = if \u2200 (i : \u03b9), p i then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "full_name": "LinearMap.range_rangeRestrict", "start": [434, 9], "end": [434, 99], "traced_tactics": [{"tactic": "simp [f.range_codRestrict _]", "annotated_tactic": ["simp [f.range_codRestrict _]", []], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nK : Type u_4\nK\u2082 : Type u_5\nM : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nV : Type u_9\nV\u2082 : Type u_10\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 range f.rangeRestrict = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "full_name": "Orientation.oangle_zero_right", "start": [73, 1], "end": [73, 73], "traced_tactics": [{"tactic": "simp [oangle]", "annotated_tactic": ["simp [<a>oangle</a>]", [{"full_name": "Orientation.oangle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "def_pos": [53, 5], "def_end_pos": [53, 11]}]], "state_before": "V : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 o.oangle x 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Basic.lean", "full_name": "invMonoidHom_apply", "start": [79, 1], "end": [79, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.get_eq_iff", "start": [251, 1], "end": [252, 7], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d : Type u_1\nl : List \u03b1\u271d\nn : Fin l.length\nx : \u03b1\u271d\n\u22a2 l.get n = x \u2194 l.get? \u2191n = some x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.mulVec_one", "start": [1901, 1], "end": [1902, 33], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonAssocSemiring \u03b1\ninst\u271d : Fintype n\nA : Matrix m n \u03b1\n\u22a2 A *\u1d65 1 = fun i => \u2211 j : n, A i j", "state_after": "case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonAssocSemiring \u03b1\ninst\u271d : Fintype n\nA : Matrix m n \u03b1\nx\u271d : m\n\u22a2 (A *\u1d65 1) x\u271d = \u2211 j : n, A x\u271d j"}, {"tactic": "simp [mulVec, dotProduct]", "annotated_tactic": ["simp [<a>mulVec</a>, <a>dotProduct</a>]", [{"full_name": "Matrix.mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1695, 5], "def_end_pos": [1695, 11]}, {"full_name": "Matrix.dotProduct", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [765, 5], "def_end_pos": [765, 15]}]], "state_before": "case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonAssocSemiring \u03b1\ninst\u271d : Fintype n\nA : Matrix m n \u03b1\nx\u271d : m\n\u22a2 (A *\u1d65 1) x\u271d = \u2211 j : n, A x\u271d j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Opens.lean", "full_name": "TopologicalSpace.Opens.comap_injective", "start": [392, 1], "end": [398, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Content.lean", "full_name": "MeasureTheory.Content.innerContent_mono'", "start": [244, 1], "end": [246, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.le_edist_of_le_infsep", "start": [405, 1], "end": [411, 34], "traced_tactics": [{"tactic": "by_cases hs : s.Nontrivial", "annotated_tactic": ["by_cases hs : s.Nontrivial", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\n\u22a2 d \u2264 dist x y", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : s.Nontrivial\n\u22a2 d \u2264 dist x y\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : \u00acs.Nontrivial\n\u22a2 d \u2264 dist x y"}, {"tactic": "exact hs.le_infsep_iff.1 hd x hx y hy hxy", "annotated_tactic": ["exact hs.le_infsep_iff.1 hd x hx y hy hxy", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : s.Nontrivial\n\u22a2 d \u2264 dist x y", "state_after": "no goals"}, {"tactic": "rw [not_nontrivial_iff] at hs", "annotated_tactic": ["rw [<a>not_nontrivial_iff</a>] at hs", [{"full_name": "Set.not_nontrivial_iff", "def_path": "Mathlib/Data/Set/Subsingleton.lean", "def_pos": [305, 9], "def_end_pos": [305, 27]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : \u00acs.Nontrivial\n\u22a2 d \u2264 dist x y", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : s.Subsingleton\n\u22a2 d \u2264 dist x y"}, {"tactic": "rw [hs.infsep_zero] at hd", "annotated_tactic": ["rw [hs.infsep_zero] at hd", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 s.infsep\nhs : s.Subsingleton\n\u22a2 d \u2264 dist x y", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 0\nhs : s.Subsingleton\n\u22a2 d \u2264 dist x y"}, {"tactic": "exact le_trans hd dist_nonneg", "annotated_tactic": ["exact <a>le_trans</a> hd <a>dist_nonneg</a>", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "dist_nonneg", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [259, 9], "def_end_pos": [259, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\ns t : Set \u03b1\nd : \u211d\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nhd : d \u2264 0\nhs : s.Subsingleton\n\u22a2 d \u2264 dist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean", "full_name": "WeierstrassCurve.\u03a6_one", "start": [372, 1], "end": [374, 96], "traced_tactics": [{"tactic": "erw [\u03a6_ofNat, pre\u03a8'_one, one_pow, mul_one, mul_one, pre\u03a8'_zero, mul_zero, zero_mul, sub_zero]", "annotated_tactic": ["erw [<a>\u03a6_ofNat</a>, <a>pre\u03a8'_one</a>, <a>one_pow</a>, <a>mul_one</a>, <a>mul_one</a>, <a>pre\u03a8'_zero</a>, <a>mul_zero</a>, <a>zero_mul</a>, <a>sub_zero</a>]", [{"full_name": "WeierstrassCurve.\u03a6_ofNat", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean", "def_pos": [361, 7], "def_end_pos": [361, 14]}, {"full_name": "WeierstrassCurve.pre\u03a8'_one", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean", "def_pos": [167, 7], "def_end_pos": [167, 16]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "WeierstrassCurve.pre\u03a8'_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean", "def_pos": [163, 7], "def_end_pos": [163, 17]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}]], "state_before": "R : Type r\nS : Type s\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nW : WeierstrassCurve R\n\u22a2 W.\u03a6 1 = X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.Sum.one_le_dist_inl_inr", "start": [231, 1], "end": [233, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.not_nonempty_iff_eq_empty", "start": [611, 1], "end": [612, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "FiniteField.pow_card", "start": [225, 1], "end": [228, 42], "traced_tactics": [{"tactic": "by_cases h : a = 0", "annotated_tactic": ["by_cases h : a = 0", []], "state_before": "K : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\n\u22a2 a ^ q = a", "state_after": "case pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : a = 0\n\u22a2 a ^ q = a\n\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : \u00aca = 0\n\u22a2 a ^ q = a"}, {"tactic": "rw [\u2190 Nat.succ_pred_eq_of_pos Fintype.card_pos, pow_succ, Nat.pred_eq_sub_one,\n  pow_card_sub_one_eq_one a h, one_mul]", "annotated_tactic": ["rw [\u2190 <a>Nat.succ_pred_eq_of_pos</a> <a>Fintype.card_pos</a>, <a>pow_succ</a>, <a>Nat.pred_eq_sub_one</a>,\n    <a>pow_card_sub_one_eq_one</a> a h, <a>one_mul</a>]", [{"full_name": "Nat.succ_pred_eq_of_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 28]}, {"full_name": "Fintype.card_pos", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [564, 9], "def_end_pos": [564, 17]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "Nat.pred_eq_sub_one", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [840, 17], "def_end_pos": [840, 32]}, {"full_name": "FiniteField.pow_card_sub_one_eq_one", "def_path": "Mathlib/FieldTheory/Finite/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 32]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : \u00aca = 0\n\u22a2 a ^ q = a", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : a = 0\n\u22a2 a ^ q = a", "state_after": "case pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : a = 0\n\u22a2 0 ^ q = 0"}, {"tactic": "apply zero_pow Fintype.card_ne_zero", "annotated_tactic": ["apply <a>zero_pow</a> <a>Fintype.card_ne_zero</a>", [{"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [160, 15], "def_end_pos": [160, 23]}, {"full_name": "Fintype.card_ne_zero", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [569, 9], "def_end_pos": [569, 21]}]], "state_before": "case pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nh : a = 0\n\u22a2 0 ^ q = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.nonneg_mul", "start": [813, 1], "end": [848, 85], "traced_tactics": [{"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := -\u2191x, im := \u2191y } * { re := \u2191z, im := \u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191z, im := \u2191w } * { re := -\u2191x, im := \u2191y }).Nonneg"}, {"tactic": "exact nonneg_mul_lem ha", "annotated_tactic": ["exact <a>nonneg_mul_lem</a> ha", [{"full_name": "Zsqrtd.nonneg_mul_lem", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 23]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191z, im := \u2191w } * { re := -\u2191x, im := \u2191y }).Nonneg", "state_after": "no goals"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191x, im := -\u2191y } * { re := \u2191z, im := \u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191z, im := \u2191w } * { re := \u2191x, im := -\u2191y }).Nonneg"}, {"tactic": "exact nonneg_mul_lem ha", "annotated_tactic": ["exact <a>nonneg_mul_lem</a> ha", [{"full_name": "Zsqrtd.nonneg_mul_lem", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 23]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nx\u271d : { re := \u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191z, im := \u2191w } * { re := \u2191x, im := -\u2191y }).Nonneg", "state_after": "no goals"}, {"tactic": "rw [calc\n      (\u27e8-x, y\u27e9 * \u27e8-z, w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := rfl\n      _ = \u27e8x * z + d * y * w, -(x * w + y * z)\u27e9 := by simp [add_comm]\n      ]", "annotated_tactic": ["rw [calc\n          (\u27e8-x, y\u27e9 * \u27e8-z, w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := <a>rfl</a>\n          _ = \u27e8x * z + d * y * w, -(x * w + y * z)\u27e9 := by simp [<a>add_comm</a>]\n          ]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := -\u2191x, im := \u2191y } * { re := -\u2191z, im := \u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }.Nonneg"}, {"tactic": "exact nonnegg_pos_neg.2 (sqLe_mul.left (nonnegg_neg_pos.1 ha) (nonnegg_neg_pos.1 hb))", "annotated_tactic": ["exact <a>nonnegg_pos_neg</a>.2 (sqLe_mul.left (<a>nonnegg_neg_pos</a>.1 ha) (<a>nonnegg_neg_pos</a>.1 hb))", [{"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}, {"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }.Nonneg", "state_after": "no goals"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 {\n      re :=\n        { re := -\u2191x, im := \u2191y }.re * { re := -\u2191z, im := \u2191w }.re +\n          \u2191d * { re := -\u2191x, im := \u2191y }.im * { re := -\u2191z, im := \u2191w }.im,\n      im :=\n        { re := -\u2191x, im := \u2191y }.re * { re := -\u2191z, im := \u2191w }.im +\n          { re := -\u2191x, im := \u2191y }.im * { re := -\u2191z, im := \u2191w }.re } =\n    { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }", "state_after": "no goals"}, {"tactic": "rw [calc\n      (\u27e8-x, y\u27e9 * \u27e8z, -w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := rfl\n      _ = \u27e8-(x * z + d * y * w), x * w + y * z\u27e9 := by simp [add_comm]\n      ]", "annotated_tactic": ["rw [calc\n          (\u27e8-x, y\u27e9 * \u27e8z, -w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := <a>rfl</a>\n          _ = \u27e8-(x * z + d * y * w), x * w + y * z\u27e9 := by simp [<a>add_comm</a>]\n          ]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 ({ re := -\u2191x, im := \u2191y } * { re := \u2191z, im := -\u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }.Nonneg"}, {"tactic": "exact nonnegg_neg_pos.2 (sqLe_mul.right.left (nonnegg_neg_pos.1 ha) (nonnegg_pos_neg.1 hb))", "annotated_tactic": ["exact <a>nonnegg_neg_pos</a>.2 (sqLe_mul.right.left (<a>nonnegg_neg_pos</a>.1 ha) (<a>nonnegg_pos_neg</a>.1 hb))", [{"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}, {"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}, {"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }.Nonneg", "state_after": "no goals"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := -\u2191x, im := \u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 {\n      re :=\n        { re := -\u2191x, im := \u2191y }.re * { re := \u2191z, im := -\u2191w }.re +\n          \u2191d * { re := -\u2191x, im := \u2191y }.im * { re := \u2191z, im := -\u2191w }.im,\n      im :=\n        { re := -\u2191x, im := \u2191y }.re * { re := \u2191z, im := -\u2191w }.im +\n          { re := -\u2191x, im := \u2191y }.im * { re := \u2191z, im := -\u2191w }.re } =\n    { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }", "state_after": "no goals"}, {"tactic": "rw [calc\n      (\u27e8x, -y\u27e9 * \u27e8-z, w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := rfl\n      _ = \u27e8-(x * z + d * y * w), x * w + y * z\u27e9 := by simp [add_comm]\n      ]", "annotated_tactic": ["rw [calc\n          (\u27e8x, -y\u27e9 * \u27e8-z, w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := <a>rfl</a>\n          _ = \u27e8-(x * z + d * y * w), x * w + y * z\u27e9 := by simp [<a>add_comm</a>]\n          ]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 ({ re := \u2191x, im := -\u2191y } * { re := -\u2191z, im := \u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }.Nonneg"}, {"tactic": "exact\n    nonnegg_neg_pos.2 (sqLe_mul.right.right.left (nonnegg_pos_neg.1 ha) (nonnegg_neg_pos.1 hb))", "annotated_tactic": ["exact\n        <a>nonnegg_neg_pos</a>.2 (sqLe_mul.right.right.left (<a>nonnegg_pos_neg</a>.1 ha) (<a>nonnegg_neg_pos</a>.1 hb))", [{"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}, {"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Zsqrtd.nonnegg_neg_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [476, 9], "def_end_pos": [476, 24]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }.Nonneg", "state_after": "no goals"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := -\u2191z, im := \u2191w }.Nonneg\n\u22a2 {\n      re :=\n        { re := \u2191x, im := -\u2191y }.re * { re := -\u2191z, im := \u2191w }.re +\n          \u2191d * { re := \u2191x, im := -\u2191y }.im * { re := -\u2191z, im := \u2191w }.im,\n      im :=\n        { re := \u2191x, im := -\u2191y }.re * { re := -\u2191z, im := \u2191w }.im +\n          { re := \u2191x, im := -\u2191y }.im * { re := -\u2191z, im := \u2191w }.re } =\n    { re := -(\u2191x * \u2191z + \u2191d * \u2191y * \u2191w), im := \u2191x * \u2191w + \u2191y * \u2191z }", "state_after": "no goals"}, {"tactic": "rw [calc\n      (\u27e8x, -y\u27e9 * \u27e8z, -w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := rfl\n      _ = \u27e8x * z + d * y * w, -(x * w + y * z)\u27e9 := by simp [add_comm]\n      ]", "annotated_tactic": ["rw [calc\n          (\u27e8x, -y\u27e9 * \u27e8z, -w\u27e9 : \u2124\u221ad) = \u27e8_, _\u27e9 := <a>rfl</a>\n          _ = \u27e8x * z + d * y * w, -(x * w + y * z)\u27e9 := by simp [<a>add_comm</a>]\n          ]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 ({ re := \u2191x, im := -\u2191y } * { re := \u2191z, im := -\u2191w }).Nonneg", "state_after": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }.Nonneg"}, {"tactic": "exact\n    nonnegg_pos_neg.2\n      (sqLe_mul.right.right.right (nonnegg_pos_neg.1 ha) (nonnegg_pos_neg.1 hb))", "annotated_tactic": ["exact\n        <a>nonnegg_pos_neg</a>.2\n          (sqLe_mul.right.right.right (<a>nonnegg_pos_neg</a>.1 ha) (<a>nonnegg_pos_neg</a>.1 hb))", [{"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}, {"full_name": "Zsqrtd.nonnegg_pos_neg", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 24]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }.Nonneg", "state_after": "no goals"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "d : \u2115\na b : \u2124\u221a\u2191d\nha\u271d : a.Nonneg\nhb\u271d : b.Nonneg\nx y z w : \u2115\nha : { re := \u2191x, im := -\u2191y }.Nonneg\nhb : { re := \u2191z, im := -\u2191w }.Nonneg\n\u22a2 {\n      re :=\n        { re := \u2191x, im := -\u2191y }.re * { re := \u2191z, im := -\u2191w }.re +\n          \u2191d * { re := \u2191x, im := -\u2191y }.im * { re := \u2191z, im := -\u2191w }.im,\n      im :=\n        { re := \u2191x, im := -\u2191y }.re * { re := \u2191z, im := -\u2191w }.im +\n          { re := \u2191x, im := -\u2191y }.im * { re := \u2191z, im := -\u2191w }.re } =\n    { re := \u2191x * \u2191z + \u2191d * \u2191y * \u2191w, im := -(\u2191x * \u2191w + \u2191y * \u2191z) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "equicontinuousAt_iff_range", "start": [462, 1], "end": [464, 57], "traced_tactics": [{"tactic": "simp only [EquicontinuousAt, forall_subtype_range_iff]", "annotated_tactic": ["simp only [<a>EquicontinuousAt</a>, <a>forall_subtype_range_iff</a>]", [{"full_name": "EquicontinuousAt", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [93, 5], "def_end_pos": [93, 21]}, {"full_name": "Set.forall_subtype_range_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [667, 9], "def_end_pos": [667, 33]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\n\u22a2 EquicontinuousAt F x\u2080 \u2194 EquicontinuousAt Subtype.val x\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.le_of_lt_succ", "start": [243, 1], "end": [244, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/ComputeDegree.lean", "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_intCast_le", "start": [179, 1], "end": [179, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": 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Type u_6\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring S\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u22a2 \u2a05 a, ker (lapply a) \u2264 \u22a5", "state_after": "\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring S\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u22a2 \u2200 \u2983x : \u03b1 \u2192\u2080 M\u2984, (\u2200 (i : \u03b1), x i = 0) \u2192 x = 0"}, {"tactic": "exact fun a h => Finsupp.ext h", "annotated_tactic": ["exact fun a h => <a>Finsupp.ext</a> h", [{"full_name": "Finsupp.ext", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [133, 9], "def_end_pos": 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u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x y\na b : X\n\u03b3 : Path a b\nt : \u211d\n\u22a2 \u03b3.extend (min t t) = \u03b3.extend t", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x y\na b : X\n\u03b3 : Path a b\nt : \u211d\n\u22a2 \u03b3.truncate t t = (refl (\u03b3.extend t)).cast \u22ef \u22ef", "state_after": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 (\u03b3.truncate t t) x = ((refl (\u03b3.extend t)).cast \u22ef \u22ef) x"}, {"tactic": "rw [cast_coe]", "annotated_tactic": ["rw [<a>cast_coe</a>]", [{"full_name": "Path.cast_coe", "def_path": "Mathlib/Topology/Connected/PathConnected.lean", "def_pos": [471, 9], "def_end_pos": [471, 17]}]], "state_before": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 (\u03b3.truncate t t) x = ((refl (\u03b3.extend t)).cast \u22ef \u22ef) x", "state_after": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 (\u03b3.truncate t t) x = (refl (\u03b3.extend t)) x"}, {"tactic": "simp only [truncate, DFunLike.coe, refl, min_def, max_def]", "annotated_tactic": ["simp only [<a>truncate</a>, <a>DFunLike.coe</a>, <a>refl</a>, <a>min_def</a>, <a>max_def</a>]", [{"full_name": "Path.truncate", "def_path": "Mathlib/Topology/Connected/PathConnected.lean", "def_pos": [615, 5], "def_end_pos": [615, 13]}, {"full_name": "DFunLike.coe", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [147, 3], "def_end_pos": [147, 6]}, {"full_name": "Path.refl", "def_path": "Mathlib/Topology/Connected/PathConnected.lean", "def_pos": [157, 5], "def_end_pos": [157, 9]}, {"full_name": "min_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [25, 9], "def_end_pos": [25, 16]}, {"full_name": "max_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}]], "state_before": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 (\u03b3.truncate t t) x = (refl (\u03b3.extend t)) x", "state_after": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 \u03b3.extend (if (if \u2191x \u2264 t then t else \u2191x) \u2264 t then if \u2191x \u2264 t then t else \u2191x else t) = \u03b3.extend t"}, {"tactic": "split_ifs with h\u2081 h\u2082 <;> congr", "annotated_tactic": ["split_ifs with h\u2081 h\u2082 <;> congr", []], "state_before": "case a.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y z : X\n\u03b9 : Type u_3\n\u03b3\u271d : Path x\u271d y\na b : X\n\u03b3 : Path a b\nt : \u211d\nx : \u2191I\n\u22a2 \u03b3.extend (if (if \u2191x \u2264 t then t else \u2191x) \u2264 t then if \u2191x \u2264 t then t else \u2191x else t) = \u03b3.extend t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Types.lean", "full_name": "CategoryTheory.Limits.Types.Colimit.\u03b9_desc_apply'", "start": [567, 1], "end": [569, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.empty_div", "start": [659, 1], "end": [660, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "full_name": "WeierstrassCurve.Jacobian.map_addX", "start": [1574, 1], "end": [1574, 93], "traced_tactics": [{"tactic": "simp [addX]", "annotated_tactic": ["simp [<a>addX</a>]", [{"full_name": "WeierstrassCurve.Jacobian.addX", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [838, 5], "def_end_pos": [838, 9]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d\u00b9 : Field F\nW : Jacobian F\nS : Type u_1\ninst\u271d : CommRing S\nf : R \u2192+* S\nP Q : Fin 3 \u2192 R\n\u22a2 addX (map W' f) (\u21d1f \u2218 P) (\u21d1f \u2218 Q) = f (W'.addX P Q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean", "full_name": "DedekindDomain.ProdAdicCompletions.isFiniteAdele_iff", "start": [200, 1], "end": [202, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/BinaryProducts.lean", "full_name": "hasBinaryProducts_of_hasTerminal_and_pullbacks", "start": [96, 1], "end": [98, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.ncard_le_one_iff", "start": [1036, 1], "end": [1039, 8], "traced_tactics": [{"tactic": "rw [ncard_le_one hs]", "annotated_tactic": ["rw [<a>ncard_le_one</a> hs]", [{"full_name": "Set.ncard_le_one", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : autoParam s.Finite _auto\u271d\n\u22a2 s.ncard \u2264 1 \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : autoParam s.Finite _auto\u271d\n\u22a2 (\u2200 a \u2208 s, \u2200 b \u2208 s, a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : autoParam s.Finite _auto\u271d\n\u22a2 (\u2200 a \u2208 s, \u2200 b \u2208 s, a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean", "full_name": "NNReal.tendsto_rpow_atTop", "start": [158, 8], "end": [165, 58], "traced_tactics": [{"tactic": "rw [Filter.tendsto_atTop_atTop]", "annotated_tactic": ["rw [<a>Filter.tendsto_atTop_atTop</a>]", [{"full_name": "Filter.tendsto_atTop_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1395, 9], "def_end_pos": [1395, 28]}]], "state_before": "y : \u211d\nhy : 0 < y\n\u22a2 Tendsto (fun x => x ^ y) atTop atTop", "state_after": "y : \u211d\nhy : 0 < y\n\u22a2 \u2200 (b : \u211d\u22650), \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "y : \u211d\nhy : 0 < y\n\u22a2 \u2200 (b : \u211d\u22650), \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y", "state_after": "y : \u211d\nhy : 0 < y\nb : \u211d\u22650\n\u22a2 \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y"}, {"tactic": "obtain \u27e8c, hc\u27e9 := tendsto_atTop_atTop.mp (tendsto_rpow_atTop hy) b", "annotated_tactic": ["obtain \u27e8c, hc\u27e9 := tendsto_atTop_atTop.mp (<a>tendsto_rpow_atTop</a> hy) b", [{"full_name": "tendsto_rpow_atTop", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean", "def_pos": [36, 9], "def_end_pos": [36, 27]}]], "state_before": "y : \u211d\nhy : 0 < y\nb : \u211d\u22650\n\u22a2 \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y", "state_after": "case intro\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\n\u22a2 \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y"}, {"tactic": "use c.toNNReal", "annotated_tactic": ["use c.toNNReal", []], "state_before": "case intro\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\n\u22a2 \u2203 i, \u2200 (a : \u211d\u22650), i \u2264 a \u2192 b \u2264 a ^ y", "state_after": "case h\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\n\u22a2 \u2200 (a : \u211d\u22650), c.toNNReal \u2264 a \u2192 b \u2264 a ^ y"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "case h\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\n\u22a2 \u2200 (a : \u211d\u22650), c.toNNReal \u2264 a \u2192 b \u2264 a ^ y", "state_after": "case h\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\na : \u211d\u22650\nha : c.toNNReal \u2264 a\n\u22a2 b \u2264 a ^ y"}, {"tactic": "exact mod_cast hc a (Real.toNNReal_le_iff_le_coe.mp ha)", "annotated_tactic": ["exact mod_cast hc a (Real.toNNReal_le_iff_le_coe.mp ha)", []], "state_before": "case h\ny : \u211d\nhy : 0 < y\nb : \u211d\u22650\nc : \u211d\nhc : \u2200 (a : \u211d), c \u2264 a \u2192 \u2191b \u2264 a ^ y\na : \u211d\u22650\nha : c.toNNReal \u2264 a\n\u22a2 b \u2264 a ^ y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Partial.lean", "full_name": "pcontinuous_iff'", "start": [61, 1], "end": [83, 39], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\n\u22a2 PContinuous f \u2194 \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)", "state_after": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\n\u22a2 PContinuous f \u2192 \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\n\ncase mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\n\u22a2 (\u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)) \u2192 PContinuous f"}, {"tactic": "intro hf s os", "annotated_tactic": ["intro hf s os", []], "state_before": "case mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\n\u22a2 (\u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)) \u2192 PContinuous f", "state_after": "case mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\n\u22a2 IsOpen (f.preimage s)"}, {"tactic": "rw [isOpen_iff_nhds]", "annotated_tactic": ["rw [<a>isOpen_iff_nhds</a>]", [{"full_name": "isOpen_iff_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1190, 9], "def_end_pos": [1190, 24]}]], "state_before": "case mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\n\u22a2 IsOpen (f.preimage s)", "state_after": "case mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\n\u22a2 \u2200 x \u2208 f.preimage s, \ud835\udcdd x \u2264 \ud835\udcdf (f.preimage s)"}, {"tactic": "rintro x \u27e8y, ys, fxy\u27e9 t", "annotated_tactic": ["rintro x \u27e8y, ys, fxy\u27e9 t", []], "state_before": "case mpr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\n\u22a2 \u2200 x \u2208 f.preimage s, \ud835\udcdd x \u2264 \ud835\udcdf (f.preimage s)", "state_after": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\n\u22a2 t \u2208 \ud835\udcdf (f.preimage s) \u2192 t \u2208 \ud835\udcdd x"}, {"tactic": "rw [mem_principal]", "annotated_tactic": ["rw [<a>mem_principal</a>]", [{"full_name": "Filter.mem_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [307, 17], "def_end_pos": [307, 30]}]], "state_before": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\n\u22a2 t \u2208 \ud835\udcdf (f.preimage s) \u2192 t \u2208 \ud835\udcdd x", "state_after": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\n\u22a2 f.preimage s \u2286 t \u2192 t \u2208 \ud835\udcdd x"}, {"tactic": "intro (h : f.preimage s \u2286 t)", "annotated_tactic": ["intro (h : f.preimage s \u2286 t)", []], "state_before": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\n\u22a2 f.preimage s \u2286 t \u2192 t \u2208 \ud835\udcdd x", "state_after": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\n\u22a2 t \u2208 \ud835\udcdd x"}, {"tactic": "apply mem_of_superset _ h", "annotated_tactic": ["apply <a>mem_of_superset</a> _ h", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}]], "state_before": "case mpr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\n\u22a2 t \u2208 \ud835\udcdd x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\n\u22a2 f.preimage s \u2208 \ud835\udcdd x"}, {"tactic": "have h' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x := by\n  intro s hs\n  have : PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y) := hf fxy\n  rw [ptendsto'_def] at this\n  exact this s hs", "annotated_tactic": ["have h' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x := by\n    intro s hs\n    have : <a>PTendsto'</a> f (\ud835\udcdd x) (\ud835\udcdd y) := hf fxy\n    rw [<a>ptendsto'_def</a>] at this\n    exact this s hs", [{"full_name": "Filter.PTendsto'", "def_path": "Mathlib/Order/Filter/Partial.lean", "def_pos": [263, 5], "def_end_pos": [263, 14]}, {"full_name": "Filter.ptendsto'_def", "def_path": "Mathlib/Order/Filter/Partial.lean", "def_pos": [267, 9], "def_end_pos": [267, 22]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\n\u22a2 f.preimage s \u2208 \ud835\udcdd x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 f.preimage s \u2208 \ud835\udcdd x"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 f.preimage s \u2208 \ud835\udcdd x", "state_after": "case a\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 s \u2208 \ud835\udcdd y"}, {"tactic": "rw [mem_nhds_iff]", "annotated_tactic": ["rw [<a>mem_nhds_iff</a>]", [{"full_name": "mem_nhds_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [840, 9], "def_end_pos": [840, 21]}]], "state_before": "case a\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 s \u2208 \ud835\udcdd y", "state_after": "case a\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 \u2203 t \u2286 s, IsOpen t \u2227 y \u2208 t"}, {"tactic": "exact \u27e8s, Set.Subset.refl _, os, ys\u27e9", "annotated_tactic": ["exact \u27e8s, <a>Set.Subset.refl</a> _, os, ys\u27e9", [{"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 20]}]], "state_before": "case a\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\nh' : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 \u2203 t \u2286 s, IsOpen t \u2227 y \u2208 t", "state_after": "no goals"}, {"tactic": "intro h x y h'", "annotated_tactic": ["intro h x y h'", []], "state_before": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\n\u22a2 PContinuous f \u2192 \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)", "state_after": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\n\u22a2 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)"}, {"tactic": "simp only [ptendsto'_def, mem_nhds_iff]", "annotated_tactic": ["simp only [<a>ptendsto'_def</a>, <a>mem_nhds_iff</a>]", [{"full_name": "Filter.ptendsto'_def", "def_path": "Mathlib/Order/Filter/Partial.lean", "def_pos": [267, 9], "def_end_pos": [267, 22]}, {"full_name": "mem_nhds_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [840, 9], "def_end_pos": [840, 21]}]], "state_before": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\n\u22a2 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)", "state_after": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\n\u22a2 \u2200 (s : Set Y), (\u2203 t \u2286 s, IsOpen t \u2227 y \u2208 t) \u2192 \u2203 t \u2286 f.preimage s, IsOpen t \u2227 x \u2208 t"}, {"tactic": "rintro s \u27e8t, tsubs, opent, yt\u27e9", "annotated_tactic": ["rintro s \u27e8t, tsubs, opent, yt\u27e9", []], "state_before": "case mp\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\n\u22a2 \u2200 (s : Set Y), (\u2203 t \u2286 s, IsOpen t \u2227 y \u2208 t) \u2192 \u2203 t \u2286 f.preimage s, IsOpen t \u2227 x \u2208 t", "state_after": "case mp.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\ns t : Set Y\ntsubs : t \u2286 s\nopent : IsOpen t\nyt : y \u2208 t\n\u22a2 \u2203 t \u2286 f.preimage s, IsOpen t \u2227 x \u2208 t"}, {"tactic": "exact \u27e8f.preimage t, PFun.preimage_mono _ tsubs, h _ opent, \u27e8y, yt, h'\u27e9\u27e9", "annotated_tactic": ["exact \u27e8f.preimage t, <a>PFun.preimage_mono</a> _ tsubs, h _ opent, \u27e8y, yt, h'\u27e9\u27e9", [{"full_name": "PFun.preimage_mono", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [435, 9], "def_end_pos": [435, 22]}]], "state_before": "case mp.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nh : PContinuous f\nx : X\ny : Y\nh' : y \u2208 f x\ns t : Set Y\ntsubs : t \u2286 s\nopent : IsOpen t\nyt : y \u2208 t\n\u22a2 \u2203 t \u2286 f.preimage s, IsOpen t \u2227 x \u2208 t", "state_after": "no goals"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns : Set Y\nos : IsOpen s\nx : X\ny : Y\nys : y \u2208 s\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s \u2286 t\n\u22a2 \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\n\u22a2 f.preimage s \u2208 \ud835\udcdd x"}, {"tactic": "have : PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y) := hf fxy", "annotated_tactic": ["have : <a>PTendsto'</a> f (\ud835\udcdd x) (\ud835\udcdd y) := hf fxy", [{"full_name": "Filter.PTendsto'", "def_path": "Mathlib/Order/Filter/Partial.lean", "def_pos": [263, 5], "def_end_pos": [263, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\n\u22a2 f.preimage s \u2208 \ud835\udcdd x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\nthis : PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\n\u22a2 f.preimage s \u2208 \ud835\udcdd x"}, {"tactic": "rw [ptendsto'_def] at this", "annotated_tactic": ["rw [<a>ptendsto'_def</a>] at this", [{"full_name": "Filter.ptendsto'_def", "def_path": "Mathlib/Order/Filter/Partial.lean", "def_pos": [267, 9], "def_end_pos": [267, 22]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\nthis : PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\n\u22a2 f.preimage s \u2208 \ud835\udcdd x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\nthis : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 f.preimage s \u2208 \ud835\udcdd x"}, {"tactic": "exact this s hs", "annotated_tactic": ["exact this s hs", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192. Y\nhf : \u2200 {x : X} {y : Y}, y \u2208 f x \u2192 PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)\ns\u271d : Set Y\nos : IsOpen s\u271d\nx : X\ny : Y\nys : y \u2208 s\u271d\nfxy : y \u2208 f x\nt : Set X\nh : f.preimage s\u271d \u2286 t\ns : Set Y\nhs : s \u2208 \ud835\udcdd y\nthis : \u2200 s \u2208 \ud835\udcdd y, f.preimage s \u2208 \ud835\udcdd x\n\u22a2 f.preimage s \u2208 \ud835\udcdd x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/Intersecting.lean", "full_name": "Set.Intersecting.not_mem", "start": [151, 1], "end": [152, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "full_name": "cfc_id'", "start": [340, 1], "end": [340, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.det_nonsing_inv_mul_det", "start": [476, 1], "end": [477, 47], "traced_tactics": [{"tactic": "rw [\u2190 det_mul, A.nonsing_inv_mul h, det_one]", "annotated_tactic": ["rw [\u2190 <a>det_mul</a>, A.nonsing_inv_mul h, <a>det_one</a>]", [{"full_name": "Matrix.det_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 16]}, {"full_name": "Matrix.det_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "l : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit A.det\n\u22a2 A\u207b\u00b9.det * A.det = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/InformationTheory/Hamming.lean", "full_name": "Hamming.toHamming_symm_eq", "start": [316, 1], "end": [317, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous", "start": [992, 1], "end": [996, 75], "traced_tactics": [{"tactic": "rw [\u2190 uniformContinuousOn_univ, \u2190 uniformEquicontinuousOn_univ, tendsto_pi_nhds] at *", "annotated_tactic": ["rw [\u2190 <a>uniformContinuousOn_univ</a>, \u2190 <a>uniformEquicontinuousOn_univ</a>, <a>tendsto_pi_nhds</a>] at *", [{"full_name": "uniformContinuousOn_univ", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1099, 9], "def_end_pos": [1099, 33]}, {"full_name": "uniformEquicontinuousOn_univ", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [203, 7], "def_end_pos": [203, 35]}, {"full_name": "tendsto_pi_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1319, 9], "def_end_pos": [1319, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nl : Filter \u03b9\ninst\u271d : l.NeBot\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\nh\u2081 : Tendsto F l (\ud835\udcdd f)\nh\u2082 : UniformEquicontinuous F\n\u22a2 UniformContinuous f", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nl : Filter \u03b9\ninst\u271d : l.NeBot\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\nh\u2081 : \u2200 (x : \u03b2), Tendsto (fun i => F i x) l (\ud835\udcdd (f x))\nh\u2082 : UniformEquicontinuousOn F univ\n\u22a2 UniformContinuousOn f univ"}, {"tactic": "exact uniformContinuousOn_of_uniformEquicontinuousOn (fun x _ \u21a6 h\u2081 x) h\u2082", "annotated_tactic": ["exact <a>uniformContinuousOn_of_uniformEquicontinuousOn</a> (fun x _ \u21a6 h\u2081 x) h\u2082", [{"full_name": "Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [979, 9], "def_end_pos": [979, 70]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nl : Filter \u03b9\ninst\u271d : l.NeBot\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\nh\u2081 : \u2200 (x : \u03b2), Tendsto (fun i => F i x) l (\ud835\udcdd (f x))\nh\u2082 : UniformEquicontinuousOn F univ\n\u22a2 UniformContinuousOn f univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.range_eq_range'", "start": [1396, 1], "end": [1397, 56], "traced_tactics": [{"tactic": "rw [Nat.zero_add]", "annotated_tactic": ["rw [<a>Nat.zero_add</a>]", [{"full_name": "Nat.zero_add", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [137, 27], "def_end_pos": [137, 35]}]], "state_before": "n : Nat\n\u22a2 range' 0 (0 + n) = range' 0 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Range.lean", "full_name": "List.mem_mem_ranges_iff_lt_natSum", "start": [278, 1], "end": [282, 45], "traced_tactics": [{"tactic": "rw [\u2190 mem_range, \u2190 ranges_join', mem_join]", "annotated_tactic": ["rw [\u2190 <a>mem_range</a>, \u2190 <a>ranges_join'</a>, <a>mem_join</a>]", [{"full_name": "List.mem_range", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1421, 9], "def_end_pos": [1421, 18]}, {"full_name": "List.ranges_join'", "def_path": "Mathlib/Data/List/Range.lean", "def_pos": [274, 7], "def_end_pos": [274, 19]}, {"full_name": "List.mem_join", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1234, 17], "def_end_pos": [1234, 25]}]], "state_before": "\u03b1 : Type u\nl : List \u2115\nn : \u2115\n\u22a2 (\u2203 s \u2208 l.ranges, n \u2208 s) \u2194 n < Nat.sum l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.fp_family_unbounded", "start": [145, 1], "end": [149, 66], "traced_tactics": [{"tactic": "rw [\u2190 hi, mem_fixedPoints_iff]", "annotated_tactic": ["rw [\u2190 hi, <a>mem_fixedPoints_iff</a>]", [{"full_name": "Function.mem_fixedPoints_iff", "def_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 28]}]], "state_before": "\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nH : \u2200 (i : \u03b9), IsNormal (f i)\na : Ordinal.{max u v}\ns : Set Ordinal.{max u v}\nx\u271d : s \u2208 Set.range fun i => fixedPoints (f i)\ni : \u03b9\nhi : (fun i => fixedPoints (f i)) i = s\n\u22a2 nfpFamily f a \u2208 s", "state_after": "\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nH : \u2200 (i : \u03b9), IsNormal (f i)\na : Ordinal.{max u v}\ns : Set Ordinal.{max u v}\nx\u271d : s \u2208 Set.range fun i => fixedPoints (f i)\ni : \u03b9\nhi : (fun i => fixedPoints (f i)) i = s\n\u22a2 f i (nfpFamily f a) = nfpFamily f a"}, {"tactic": "exact nfpFamily_fp.{u, v} (H i) a", "annotated_tactic": ["exact <a>nfpFamily_fp</a>.{u, v} (H i) a", [{"full_name": "Ordinal.nfpFamily_fp", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [119, 9], "def_end_pos": [119, 21]}]], "state_before": "\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nH : \u2200 (i : \u03b9), IsNormal (f i)\na : Ordinal.{max u v}\ns : Set Ordinal.{max u v}\nx\u271d : s \u2208 Set.range fun i => fixedPoints (f i)\ni : \u03b9\nhi : (fun i => fixedPoints (f i)) i = s\n\u22a2 f i (nfpFamily f a) = nfpFamily f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Subfield.lean", "full_name": "Subfield.coe_inv", "start": [391, 1], "end": [392, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PrimitiveElement.lean", "full_name": "Field.primitive_element_inf_aux_of_finite_intermediateField", "start": [179, 9], "end": [201, 63], "traced_tactics": [{"tactic": "let f : F \u2192 IntermediateField F E := fun x \u21a6 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "annotated_tactic": ["let f : F \u2192 <a>IntermediateField</a> F E := fun x \u21a6 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", [{"full_name": "IntermediateField", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [49, 11], "def_end_pos": [49, 28]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\n\u22a2 \u2203 \u03b3, F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b3\u27ef", "state_after": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u2203 \u03b3, F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b3\u27ef"}, {"tactic": "obtain \u27e8x, y, hneq, heq\u27e9 := Finite.exists_ne_map_eq_of_infinite f", "annotated_tactic": ["obtain \u27e8x, y, hneq, heq\u27e9 := <a>Finite.exists_ne_map_eq_of_infinite</a> f", [{"full_name": "Finite.exists_ne_map_eq_of_infinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [1184, 9], "def_end_pos": [1184, 44]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u2203 \u03b3, F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b3\u27ef", "state_after": "case intro.intro.intro\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 \u2203 \u03b3, F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b3\u27ef"}, {"tactic": "use \u03b1 + x \u2022 \u03b2", "annotated_tactic": ["use \u03b1 + x \u2022 \u03b2", []], "state_before": "case intro.intro.intro\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 \u2203 \u03b3, F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b3\u27ef", "state_after": "case h\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case h\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1, \u03b2\u27ef = F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1, \u03b2\u27ef \u2264 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\ncase h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef \u2264 F\u27ee\u03b1, \u03b2\u27ef"}, {"tactic": "rw [adjoin_le_iff]", "annotated_tactic": ["rw [<a>adjoin_le_iff</a>]", [{"full_name": "IntermediateField.adjoin_le_iff", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [76, 9], "def_end_pos": [76, 22]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1, \u03b2\u27ef \u2264 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "have \u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef := mem_adjoin_simple_self F _", "annotated_tactic": ["have \u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef := <a>mem_adjoin_simple_self</a> F _", [{"full_name": "IntermediateField.mem_adjoin_simple_self", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [596, 9], "def_end_pos": [596, 31]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "have \u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef := mem_adjoin_simple_self F _", "annotated_tactic": ["have \u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef := <a>mem_adjoin_simple_self</a> F _", [{"full_name": "IntermediateField.mem_adjoin_simple_self", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [596, 9], "def_end_pos": [596, 31]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "dsimp [f] at *", "annotated_tactic": ["dsimp [f] at *", []], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2264 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "simp only [\u2190 heq] at \u03b1y\u03b2_in_K", "annotated_tactic": ["simp only [\u2190 heq] at \u03b1y\u03b2_in_K", []], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "have \u03b2_in_K := sub_mem \u03b1x\u03b2_in_K \u03b1y\u03b2_in_K", "annotated_tactic": ["have \u03b2_in_K := <a>sub_mem</a> \u03b1x\u03b2_in_K \u03b1y\u03b2_in_K", [{"full_name": "sub_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [142, 3], "def_end_pos": [142, 14]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "rw [show (\u03b1 + x \u2022 \u03b2) - (\u03b1 + y \u2022 \u03b2) = (x - y) \u2022 \u03b2 by rw [sub_smul]; abel1] at \u03b2_in_K", "annotated_tactic": ["rw [show (\u03b1 + x \u2022 \u03b2) - (\u03b1 + y \u2022 \u03b2) = (x - y) \u2022 \u03b2 by rw [<a>sub_smul</a>]; abel1] at \u03b2_in_K", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [274, 9], "def_end_pos": [274, 17]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : (x - y) \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "replace \u03b2_in_K := smul_mem _ \u03b2_in_K (x := (x - y)\u207b\u00b9)", "annotated_tactic": ["replace \u03b2_in_K := <a>smul_mem</a> _ \u03b2_in_K (x := (x - y)\u207b\u00b9)", [{"full_name": "IntermediateField.smul_mem", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : (x - y) \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : (x - y)\u207b\u00b9 \u2022 (x - y) \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "rw [smul_smul, inv_mul_eq_div, div_self (sub_ne_zero.2 hneq), one_smul] at \u03b2_in_K", "annotated_tactic": ["rw [<a>smul_smul</a>, <a>inv_mul_eq_div</a>, <a>div_self</a> (<a>sub_ne_zero</a>.2 hneq), <a>one_smul</a>] at \u03b2_in_K", [{"full_name": "smul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [446, 7], "def_end_pos": [446, 16]}, {"full_name": "inv_mul_eq_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [741, 9], "def_end_pos": [741, 23]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [295, 15], "def_end_pos": [295, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : (x - y)\u207b\u00b9 \u2022 (x - y) \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "have \u03b1_in_K : \u03b1 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef := by\n  convert \u2190 sub_mem \u03b1x\u03b2_in_K (smul_mem _ \u03b2_in_K)\n  apply add_sub_cancel_right", "annotated_tactic": ["have \u03b1_in_K : \u03b1 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef := by\n      convert \u2190 <a>sub_mem</a> \u03b1x\u03b2_in_K (<a>smul_mem</a> _ \u03b2_in_K)\n      apply <a>add_sub_cancel_right</a>", [{"full_name": "sub_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [142, 3], "def_end_pos": [142, 14]}, {"full_name": "IntermediateField.smul_mem", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}, {"full_name": "add_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1008, 3], "def_end_pos": [1008, 14]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1_in_K : \u03b1 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef"}, {"tactic": "rintro x (rfl | rfl) <;> assumption", "annotated_tactic": ["rintro x (rfl | rfl) <;> assumption", []], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1_in_K : \u03b1 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 {\u03b1, \u03b2} \u2286 \u2191F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "no goals"}, {"tactic": "rw [sub_smul]", "annotated_tactic": ["rw [<a>sub_smul</a>]", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [274, 9], "def_end_pos": [274, 17]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) = (x - y) \u2022 \u03b2", "state_after": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) = x \u2022 \u03b2 - y \u2022 \u03b2"}, {"tactic": "abel1", "annotated_tactic": ["abel1", []], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 - (\u03b1 + y \u2022 \u03b2) = x \u2022 \u03b2 - y \u2022 \u03b2", "state_after": "no goals"}, {"tactic": "convert \u2190 sub_mem \u03b1x\u03b2_in_K (smul_mem _ \u03b2_in_K)", "annotated_tactic": ["convert \u2190 <a>sub_mem</a> \u03b1x\u03b2_in_K (<a>smul_mem</a> _ \u03b2_in_K)", [{"full_name": "sub_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [142, 3], "def_end_pos": [142, 14]}, {"full_name": "IntermediateField.smul_mem", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef", "state_after": "case h.e'_4\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 - ?m.44640 \u2022 \u03b2 = \u03b1\n\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 F"}, {"tactic": "apply add_sub_cancel_right", "annotated_tactic": ["apply <a>add_sub_cancel_right</a>", [{"full_name": "add_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1008, 3], "def_end_pos": [1008, 14]}]], "state_before": "case h.e'_4\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 - ?m.44640 \u2022 \u03b2 = \u03b1\n\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : \u00acx = y\nheq : F\u27ee\u03b1 + x \u2022 \u03b2\u27ef = F\u27ee\u03b1 + y \u2022 \u03b2\u27ef\n\u03b1x\u03b2_in_K : \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b1y\u03b2_in_K : \u03b1 + y \u2022 \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u03b2_in_K : \u03b2 \u2208 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\n\u22a2 F", "state_after": "no goals"}, {"tactic": "rw [adjoin_simple_le_iff]", "annotated_tactic": ["rw [<a>adjoin_simple_le_iff</a>]", [{"full_name": "IntermediateField.adjoin_simple_le_iff", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [847, 9], "def_end_pos": [847, 29]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 F\u27ee\u03b1 + x \u2022 \u03b2\u27ef \u2264 F\u27ee\u03b1, \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef"}, {"tactic": "have \u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef := subset_adjoin F {\u03b1, \u03b2} (Set.mem_insert \u03b1 {\u03b2})", "annotated_tactic": ["have \u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef := <a>subset_adjoin</a> F {\u03b1, \u03b2} (<a>Set.mem_insert</a> \u03b1 {\u03b2})", [{"full_name": "IntermediateField.subset_adjoin", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [361, 9], "def_end_pos": [361, 22]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef"}, {"tactic": "have \u03b2_in_F\u03b1\u03b2 : \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef := subset_adjoin F {\u03b1, \u03b2} (Set.mem_insert_of_mem \u03b1 rfl)", "annotated_tactic": ["have \u03b2_in_F\u03b1\u03b2 : \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef := <a>subset_adjoin</a> F {\u03b1, \u03b2} (<a>Set.mem_insert_of_mem</a> \u03b1 <a>rfl</a>)", [{"full_name": "IntermediateField.subset_adjoin", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [361, 9], "def_end_pos": [361, 22]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef", "state_after": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u03b2_in_F\u03b1\u03b2 : \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef"}, {"tactic": "exact F\u27ee\u03b1, \u03b2\u27ef.add_mem \u03b1_in_F\u03b1\u03b2 (F\u27ee\u03b1, \u03b2\u27ef.smul_mem \u03b2_in_F\u03b1\u03b2)", "annotated_tactic": ["exact F\u27ee\u03b1, \u03b2\u27ef.<a>add_mem</a> \u03b1_in_F\u03b1\u03b2 (F\u27ee\u03b1, \u03b2\u27ef.<a>smul_mem</a> \u03b2_in_F\u03b1\u03b2)", [{"full_name": "IntermediateField.add_mem", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [178, 19], "def_end_pos": [178, 26]}, {"full_name": "IntermediateField.smul_mem", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [158, 9], "def_end_pos": [158, 17]}]], "state_before": "case h.a\nF : Type u_1\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Infinite F\nE : Type u_2\ninst\u271d\u00b2 : Field E\n\u03d5 : F \u2192+* E\n\u03b1 \u03b2 : E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Finite (IntermediateField F E)\nf : F \u2192 IntermediateField F E := fun x => F\u27ee\u03b1 + x \u2022 \u03b2\u27ef\nx y : F\nhneq : x \u2260 y\nheq : f x = f y\n\u03b1_in_F\u03b1\u03b2 : \u03b1 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u03b2_in_F\u03b1\u03b2 : \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef\n\u22a2 \u03b1 + x \u2022 \u03b2 \u2208 F\u27ee\u03b1, \u03b2\u27ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "unique_one", "start": [99, 1], "end": [100, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.one_le_pow'", "start": [788, 1], "end": [788, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Hom.lean", "full_name": "MulActionHom.map_smul", "start": [155, 11], "end": [156, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "full_name": "antilipschitzWith_lineMap", "start": [237, 1], "end": [241, 50], "traced_tactics": [{"tactic": "rw [dist_lineMap_lineMap, NNReal.coe_inv, \u2190 dist_nndist, mul_left_comm,\n  inv_mul_cancel (dist_ne_zero.2 h), mul_one]", "annotated_tactic": ["rw [<a>dist_lineMap_lineMap</a>, <a>NNReal.coe_inv</a>, \u2190 <a>dist_nndist</a>, <a>mul_left_comm</a>,\n      <a>inv_mul_cancel</a> (<a>dist_ne_zero</a>.2 h), <a>mul_one</a>]", [{"full_name": "dist_lineMap_lineMap", "def_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "def_pos": [68, 9], "def_end_pos": [68, 29]}, {"full_name": "NNReal.coe_inv", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [190, 19], "def_end_pos": [190, 26]}, {"full_name": "dist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [298, 9], "def_end_pos": [298, 20]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 22]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "dist_ne_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : Q\nh : p\u2081 \u2260 p\u2082\nc\u2081 c\u2082 : \ud835\udd5c\n\u22a2 dist c\u2081 c\u2082 \u2264 \u2191(nndist p\u2081 p\u2082)\u207b\u00b9 * dist ((lineMap p\u2081 p\u2082) c\u2081) ((lineMap p\u2081 p\u2082) c\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "full_name": "exists_setIndependent_isCompl_sSup_atoms", "start": [584, 1], "end": [636, 15], "traced_tactics": [{"tactic": "have := zorn_subset\n  {s : Set \u03b1 | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\n  fun c hc1 hc2 =>\n    \u27e8\u22c3\u2080 c,\n      \u27e8CompleteLattice.independent_sUnion_of_directed hc2.directedOn fun s hs => (hc1 hs).1, ?_,\n        fun a \u27e8s, sc, as\u27e9 => (hc1 sc).2.2 a as\u27e9,\n      fun _ => Set.subset_sUnion_of_mem\u27e9", "annotated_tactic": ["have := <a>zorn_subset</a>\n    {s : <a>Set</a> \u03b1 | <a>CompleteLattice.SetIndependent</a> s \u2227 <a>Disjoint</a> b (<a>sSup</a> s) \u2227 \u2200 a \u2208 s, <a>IsAtom</a> a}\n    fun c hc1 hc2 =>\n      \u27e8\u22c3\u2080 c,\n        \u27e8<a>CompleteLattice.independent_sUnion_of_directed</a> hc2.directedOn fun s hs => (hc1 hs).1, ?_,\n          fun a \u27e8s, sc, as\u27e9 => (hc1 sc).2.2 a as\u27e9,\n        fun _ => <a>Set.subset_sUnion_of_mem</a>\u27e9", [{"full_name": "zorn_subset", "def_path": "Mathlib/Order/Zorn.lean", "def_pos": [186, 9], "def_end_pos": [186, 20]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "CompleteLattice.SetIndependent", "def_path": "Mathlib/Order/SupIndep.lean", "def_pos": [291, 5], "def_end_pos": [291, 19]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [42, 5], "def_end_pos": [42, 13]}, {"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [42, 3], "def_end_pos": [42, 7]}, {"full_name": "IsAtom", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [64, 5], "def_end_pos": [64, 11]}, {"full_name": "CompleteLattice.independent_sUnion_of_directed", "def_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 55]}, {"full_name": "Set.subset_sUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 29]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a", "state_after": "case refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nthis :\n  \u2203 m \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a},\n    \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a\n\ncase refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 Disjoint b (sSup (\u22c3\u2080 c))"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nthis :\n  \u2203 m \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a},\n    \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a\n\ncase refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 Disjoint b (sSup (\u22c3\u2080 c))", "state_after": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 Disjoint b (sSup (\u22c3\u2080 c))\n\ncase refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nthis :\n  \u2203 m \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a},\n    \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a"}, {"tactic": "obtain \u27e8s, \u27e8s_ind, b_inf_Sup_s, s_atoms\u27e9, s_max\u27e9 := this", "annotated_tactic": ["obtain \u27e8s, \u27e8s_ind, b_inf_Sup_s, s_atoms\u27e9, s_max\u27e9 := this", []], "state_before": "case refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nthis :\n  \u2203 m \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a},\n    \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a"}, {"tactic": "refine \u27e8s, s_ind, \u27e8b_inf_Sup_s, ?_\u27e9, s_atoms\u27e9", "annotated_tactic": ["refine \u27e8s, s_ind, \u27e8b_inf_Sup_s, ?_\u27e9, s_atoms\u27e9", []], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 \u2203 s, CompleteLattice.SetIndependent s \u2227 IsCompl b (sSup s) \u2227 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 IsAtom a", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 Codisjoint b (sSup s)"}, {"tactic": "rw [codisjoint_iff_le_sup, \u2190 h, sSup_le_iff]", "annotated_tactic": ["rw [<a>codisjoint_iff_le_sup</a>, \u2190 h, <a>sSup_le_iff</a>]", [{"full_name": "codisjoint_iff_le_sup", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [325, 9], "def_end_pos": [325, 30]}, {"full_name": "sSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 Codisjoint b (sSup s)", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 \u2200 b_1 \u2208 {a | IsAtom a}, b_1 \u2264 b \u2294 sSup s"}, {"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\n\u22a2 \u2200 b_1 \u2208 {a | IsAtom a}, b_1 \u2264 b \u2294 sSup s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\n\u22a2 a \u2264 b \u2294 sSup s"}, {"tactic": "rw [\u2190 inf_eq_left]", "annotated_tactic": ["rw [\u2190 <a>inf_eq_left</a>]", [{"full_name": "inf_eq_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 20]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\n\u22a2 a \u2264 b \u2294 sSup s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\n\u22a2 a \u2293 (b \u2294 sSup s) = a"}, {"tactic": "refine (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 ?_", "annotated_tactic": ["refine (ha.le_iff.mp <a>inf_le_left</a>).<a>resolve_left</a> fun con => ha.1 ?_", [{"full_name": "inf_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [358, 9], "def_end_pos": [358, 20]}, {"full_name": "Or.resolve_left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [555, 9], "def_end_pos": [555, 24]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\n\u22a2 a \u2293 (b \u2294 sSup s) = a", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a = \u22a5"}, {"tactic": "rw [\u2190 con, eq_comm, inf_eq_left]", "annotated_tactic": ["rw [\u2190 con, <a>eq_comm</a>, <a>inf_eq_left</a>]", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "inf_eq_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 20]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a = \u22a5", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a \u2264 b \u2294 sSup s"}, {"tactic": "refine (le_sSup ?_).trans le_sup_right", "annotated_tactic": ["refine (<a>le_sSup</a> ?_).<a>trans</a> <a>le_sup_right</a>", [{"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [74, 9], "def_end_pos": [74, 16]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a \u2264 b \u2294 sSup s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a \u2208 s"}, {"tactic": "rw [\u2190 disjoint_iff] at con", "annotated_tactic": ["rw [\u2190 <a>disjoint_iff</a>] at con", [{"full_name": "disjoint_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [135, 9], "def_end_pos": [135, 21]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : a \u2293 (b \u2294 sSup s) = \u22a5\n\u22a2 a \u2208 s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\n\u22a2 a \u2208 s"}, {"tactic": "have a_dis_Sup_s : Disjoint a (sSup s) := con.mono_right le_sup_right", "annotated_tactic": ["have a_dis_Sup_s : <a>Disjoint</a> a (<a>sSup</a> s) := con.mono_right <a>le_sup_right</a>", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [42, 5], "def_end_pos": [42, 13]}, {"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [42, 3], "def_end_pos": [42, 7]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\n\u22a2 a \u2208 s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 a \u2208 s"}, {"tactic": "rw [\u2190 s_max (s \u222a {a}) \u27e8fun x hx => _, _, fun x hx => _\u27e9 Set.subset_union_left]", "annotated_tactic": ["rw [\u2190 s_max (s \u222a {a}) \u27e8fun x hx => _, _, fun x hx => _\u27e9 <a>Set.subset_union_left</a>]", [{"full_name": "Set.subset_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 26]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 a \u2208 s", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 a \u2208 s \u222a {a}\n\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 \u2200 x \u2208 s \u222a {a}, Disjoint x (sSup ((s \u222a {a}) \\ {x}))\n\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 Disjoint b (sSup (s \u222a {a}))\n\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 \u2200 x \u2208 s \u222a {a}, IsAtom x"}, {"tactic": "rw [sSup_sUnion, \u2190 sSup_image, DirectedOn.disjoint_sSup_right]", "annotated_tactic": ["rw [<a>sSup_sUnion</a>, \u2190 <a>sSup_image</a>, <a>DirectedOn.disjoint_sSup_right</a>]", [{"full_name": "sSup_sUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2268, 9], "def_end_pos": [2268, 20]}, {"full_name": "sSup_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 19]}, {"full_name": "DirectedOn.disjoint_sSup_right", "def_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "def_pos": [399, 19], "def_end_pos": [399, 49]}]], "state_before": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 Disjoint b (sSup (\u22c3\u2080 c))", "state_after": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 \u2200 \u2983b_1 : \u03b1\u2984, b_1 \u2208 sSup '' c \u2192 Disjoint b b_1\n\ncase refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 DirectedOn (fun x x_1 => x \u2264 x_1) (sSup '' c)"}, {"tactic": "rintro _ \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8s, hs, rfl\u27e9", []], "state_before": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 \u2200 \u2983b_1 : \u03b1\u2984, b_1 \u2208 sSup '' c \u2192 Disjoint b b_1", "state_after": "case refine_1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\ns : Set \u03b1\nhs : s \u2208 c\n\u22a2 Disjoint b (sSup s)"}, {"tactic": "exact (hc1 hs).2.1", "annotated_tactic": ["exact (hc1 hs).2.1", []], "state_before": "case refine_1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\ns : Set \u03b1\nhs : s \u2208 c\n\u22a2 Disjoint b (sSup s)", "state_after": "no goals"}, {"tactic": "rw [directedOn_image]", "annotated_tactic": ["rw [<a>directedOn_image</a>]", [{"full_name": "directedOn_image", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [77, 9], "def_end_pos": [77, 25]}]], "state_before": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 DirectedOn (fun x x_1 => x \u2264 x_1) (sSup '' c)", "state_after": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 DirectedOn (sSup \u207b\u00b9'o fun x x_1 => x \u2264 x_1) c"}, {"tactic": "exact hc2.directedOn.mono @fun s t => sSup_le_sSup", "annotated_tactic": ["exact hc2.directedOn.mono @fun s t => <a>sSup_le_sSup</a>", [{"full_name": "sSup_le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [97, 9], "def_end_pos": [97, 21]}]], "state_before": "case refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\nc : Set (Set \u03b1)\nhc1 : c \u2286 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}\nhc2 : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 DirectedOn (sSup \u207b\u00b9'o fun x x_1 => x \u2264 x_1) c", "state_after": "no goals"}, {"tactic": "exact Set.mem_union_right _ (Set.mem_singleton _)", "annotated_tactic": ["exact <a>Set.mem_union_right</a> _ (<a>Set.mem_singleton</a> _)", [{"full_name": "Set.mem_union_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 24]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 a \u2208 s \u222a {a}", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 \u2200 x \u2208 s \u222a {a}, Disjoint x (sSup ((s \u222a {a}) \\ {x}))", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u222a {a}\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))"}, {"tactic": "rw [Set.mem_union, Set.mem_singleton_iff] at hx", "annotated_tactic": ["rw [<a>Set.mem_union</a>, <a>Set.mem_singleton_iff</a>] at hx", [{"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [734, 9], "def_end_pos": [734, 18]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u222a {a}\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))"}, {"tactic": "obtain rfl | xa := eq_or_ne x a", "annotated_tactic": ["obtain rfl | xa := <a>eq_or_ne</a> x a", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))", "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\nhx : x \u2208 s \u2228 x = x\n\u22a2 Disjoint x (sSup ((s \u222a {x}) \\ {x}))\n\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))"}, {"tactic": "simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]", "annotated_tactic": ["simp only [<a>Set.mem_singleton</a>, <a>Set.insert_diff_of_mem</a>, <a>Set.union_singleton</a>]", [{"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}, {"full_name": "Set.insert_diff_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1937, 9], "def_end_pos": [1937, 27]}, {"full_name": "Set.union_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 24]}]], "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\nhx : x \u2208 s \u2228 x = x\n\u22a2 Disjoint x (sSup ((s \u222a {x}) \\ {x}))", "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\nhx : x \u2208 s \u2228 x = x\n\u22a2 Disjoint x (sSup (s \\ {x}))"}, {"tactic": "exact con.mono_right ((sSup_le_sSup Set.diff_subset).trans le_sup_right)", "annotated_tactic": ["exact con.mono_right ((<a>sSup_le_sSup</a> <a>Set.diff_subset</a>).<a>trans</a> <a>le_sup_right</a>)", [{"full_name": "sSup_le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [97, 9], "def_end_pos": [97, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1782, 9], "def_end_pos": [1782, 20]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\nhx : x \u2208 s \u2228 x = x\n\u22a2 Disjoint x (sSup (s \\ {x}))", "state_after": "no goals"}, {"tactic": "have h : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a} := by\n  simp only [Set.union_singleton]\n  rw [Set.insert_diff_of_not_mem]\n  rw [Set.mem_singleton_iff]\n  exact Ne.symm xa", "annotated_tactic": ["have h : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a} := by\n        simp only [<a>Set.union_singleton</a>]\n        rw [<a>Set.insert_diff_of_not_mem</a>]\n        rw [<a>Set.mem_singleton_iff</a>]\n        exact <a>Ne.symm</a> xa", [{"full_name": "Set.union_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 24]}, {"full_name": "Set.insert_diff_of_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1942, 9], "def_end_pos": [1942, 31]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))", "state_after": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))"}, {"tactic": "rw [h, sSup_union, sSup_singleton]", "annotated_tactic": ["rw [h, <a>sSup_union</a>, <a>sSup_singleton</a>]", [{"full_name": "sSup_union", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [402, 9], "def_end_pos": [402, 19]}, {"full_name": "sSup_singleton", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 Disjoint x (sSup ((s \u222a {a}) \\ {x}))", "state_after": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 Disjoint x (sSup (s \\ {x}) \u2294 a)"}, {"tactic": "apply\n  (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left\n    (a_dis_Sup_s.mono_right _).symm", "annotated_tactic": ["apply\n        (s_ind (hx.resolve_right xa)).<a>disjoint_sup_right_of_disjoint_sup_left</a>\n          (a_dis_Sup_s.mono_right _).<a>symm</a>", [{"full_name": "Disjoint.disjoint_sup_right_of_disjoint_sup_left", "def_path": "Mathlib/Order/ModularLattice.lean", "def_pos": [393, 9], "def_end_pos": [393, 48]}, {"full_name": "Disjoint.symm", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [55, 9], "def_end_pos": [55, 22]}]], "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 Disjoint x (sSup (s \\ {x}) \u2294 a)", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 x \u2294 sSup (s \\ {x}) \u2264 sSup s"}, {"tactic": "rw [\u2190 sSup_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]", "annotated_tactic": ["rw [\u2190 <a>sSup_insert</a>, <a>Set.insert_diff_singleton</a>, <a>Set.insert_eq_of_mem</a> (hx.resolve_right xa)]", [{"full_name": "sSup_insert", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [440, 9], "def_end_pos": [440, 20]}, {"full_name": "Set.insert_diff_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2020, 9], "def_end_pos": [2020, 30]}, {"full_name": "Set.insert_eq_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 25]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh\u271d : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\nh : (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}\n\u22a2 x \u2294 sSup (s \\ {x}) \u2264 sSup s", "state_after": "no goals"}, {"tactic": "simp only [Set.union_singleton]", "annotated_tactic": ["simp only [<a>Set.union_singleton</a>]", [{"full_name": "Set.union_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 24]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 (s \u222a {a}) \\ {x} = s \\ {x} \u222a {a}", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 insert a s \\ {x} = insert a (s \\ {x})"}, {"tactic": "rw [Set.insert_diff_of_not_mem]", "annotated_tactic": ["rw [<a>Set.insert_diff_of_not_mem</a>]", [{"full_name": "Set.insert_diff_of_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1942, 9], "def_end_pos": [1942, 31]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 insert a s \\ {x} = insert a (s \\ {x})", "state_after": "case h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 a \u2209 {x}"}, {"tactic": "rw [Set.mem_singleton_iff]", "annotated_tactic": ["rw [<a>Set.mem_singleton_iff</a>]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 a \u2209 {x}", "state_after": "case h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 \u00aca = x"}, {"tactic": "exact Ne.symm xa", "annotated_tactic": ["exact <a>Ne.symm</a> xa", [{"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\nxa : x \u2260 a\n\u22a2 \u00aca = x", "state_after": "no goals"}, {"tactic": "rw [sSup_union, sSup_singleton]", "annotated_tactic": ["rw [<a>sSup_union</a>, <a>sSup_singleton</a>]", [{"full_name": "sSup_union", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [402, 9], "def_end_pos": [402, 19]}, {"full_name": "sSup_singleton", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 Disjoint b (sSup (s \u222a {a}))", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 Disjoint b (sSup s \u2294 a)"}, {"tactic": "exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left con.symm", "annotated_tactic": ["exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left con.symm", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 Disjoint b (sSup s \u2294 a)", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\n\u22a2 \u2200 x \u2208 s \u222a {a}, IsAtom x", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u222a {a}\n\u22a2 IsAtom x"}, {"tactic": "rw [Set.mem_union, Set.mem_singleton_iff] at hx", "annotated_tactic": ["rw [<a>Set.mem_union</a>, <a>Set.mem_singleton_iff</a>] at hx", [{"full_name": "Set.mem_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [734, 9], "def_end_pos": [734, 18]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u222a {a}\n\u22a2 IsAtom x", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\n\u22a2 IsAtom x"}, {"tactic": "obtain hx | rfl := hx", "annotated_tactic": ["obtain hx | rfl := hx", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s \u2228 x = a\n\u22a2 IsAtom x", "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 IsAtom x\n\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\n\u22a2 IsAtom x"}, {"tactic": "exact s_atoms x hx", "annotated_tactic": ["exact s_atoms x hx", []], "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\na : \u03b1\nha : a \u2208 {a | IsAtom a}\ncon : Disjoint a (b \u2294 sSup s)\na_dis_Sup_s : Disjoint a (sSup s)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 IsAtom x", "state_after": "no goals"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : IsModularLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nh : sSup {a | IsAtom a} = \u22a4\nb : \u03b1\ns : Set \u03b1\ns_max : \u2200 a \u2208 {s | CompleteLattice.SetIndependent s \u2227 Disjoint b (sSup s) \u2227 \u2200 a \u2208 s, IsAtom a}, s \u2286 a \u2192 a = s\ns_ind : CompleteLattice.SetIndependent s\nb_inf_Sup_s : Disjoint b (sSup s)\ns_atoms : \u2200 a \u2208 s, IsAtom a\nx : \u03b1\nha : x \u2208 {a | IsAtom a}\ncon : Disjoint x (b \u2294 sSup s)\na_dis_Sup_s : Disjoint x (sSup s)\n\u22a2 IsAtom x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "full_name": "sub_div_sub_smul_slope_add_sub_div_sub_smul_slope", "start": [105, 1], "end": [119, 26], "traced_tactics": [{"tactic": "by_cases hab : a = b", "annotated_tactic": ["by_cases hab : a = b", []], "state_before": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : a = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c"}, {"tactic": "by_cases hbc : b = c", "annotated_tactic": ["by_cases hbc : b = c", []], "state_before": "case neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : b = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c"}, {"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c", "state_after": "case neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((c - b) / (c - a)) \u2022 slope f b c + ((b - a) / (c - a)) \u2022 slope f a b = slope f a c"}, {"tactic": "simp_rw [slope, div_eq_inv_mul, mul_smul, \u2190 smul_add,\n  smul_inv_smul\u2080 (sub_ne_zero.2 <| Ne.symm hab), smul_inv_smul\u2080 (sub_ne_zero.2 <| Ne.symm hbc),\n  vsub_add_vsub_cancel]", "annotated_tactic": ["simp_rw [<a>slope</a>, <a>div_eq_inv_mul</a>, <a>mul_smul</a>, \u2190 <a>smul_add</a>,\n    <a>smul_inv_smul\u2080</a> (<a>sub_ne_zero</a>.2 <| <a>Ne.symm</a> hab), <a>smul_inv_smul\u2080</a> (<a>sub_ne_zero</a>.2 <| <a>Ne.symm</a> hbc),\n    <a>vsub_add_vsub_cancel</a>]", [{"full_name": "slope", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "def_pos": [30, 5], "def_end_pos": [30, 10]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [114, 3], "def_end_pos": [114, 11]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "smul_inv_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [212, 9], "def_end_pos": [212, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "smul_inv_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [212, 9], "def_end_pos": [212, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "vsub_add_vsub_cancel", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [146, 9], "def_end_pos": [146, 29]}]], "state_before": "case neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((c - b) / (c - a)) \u2022 slope f b c + ((b - a) / (c - a)) \u2022 slope f a b = slope f a c", "state_after": "no goals"}, {"tactic": "subst hab", "annotated_tactic": ["subst hab", []], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : a = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((a - a) / (c - a)) \u2022 slope f a a + ((c - a) / (c - a)) \u2022 slope f a c = slope f a c"}, {"tactic": "rw [sub_self, zero_div, zero_smul, zero_add]", "annotated_tactic": ["rw [<a>sub_self</a>, <a>zero_div</a>, <a>zero_smul</a>, <a>zero_add</a>]", [{"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}, {"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((a - a) / (c - a)) \u2022 slope f a a + ((c - a) / (c - a)) \u2022 slope f a c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c"}, {"tactic": "by_cases hac : a = c", "annotated_tactic": ["by_cases hac : a = c", []], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : a = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c\n\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : \u00aca = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c"}, {"tactic": "simp [hac]", "annotated_tactic": ["simp [hac]", []], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : a = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c", "state_after": "no goals"}, {"tactic": "rw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul]", "annotated_tactic": ["rw [<a>div_self</a> (<a>sub_ne_zero</a>.2 <| <a>Ne.symm</a> hac), <a>one_smul</a>]", [{"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [295, 15], "def_end_pos": [295, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : \u00aca = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c", "state_after": "no goals"}, {"tactic": "subst hbc", "annotated_tactic": ["subst hbc", []], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : b = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c", "state_after": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (b - a)) \u2022 slope f a b + ((b - b) / (b - a)) \u2022 slope f b b = slope f a b"}, {"tactic": "simp [sub_ne_zero.2 (Ne.symm hab)]", "annotated_tactic": ["simp [<a>sub_ne_zero</a>.2 (<a>Ne.symm</a> hab)]", [{"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (b - a)) \u2022 slope f a b + ((b - b) / (b - a)) \u2022 slope f b b = slope f a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.SatelliteConfig.exists_normalized_aux1", "start": [322, 1], "end": [349, 16], "traced_tactics": [{"tactic": "have ah :\n    Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228\n      a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j := by\n  simpa only [dist_eq_norm] using a.h", "annotated_tactic": ["have ah :\n      <a>Pairwise</a> fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228\n        a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j := by\n    simpa only [<a>dist_eq_norm</a>] using a.h", [{"full_name": "Pairwise", "def_path": "Mathlib/Logic/Pairwise.lean", "def_pos": [33, 5], "def_end_pos": [33, 13]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [401, 7], "def_end_pos": [401, 19]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have \u03b4nonneg : 0 \u2264 \u03b4 := by linarith only [h\u03c4, h\u03b41]", "annotated_tactic": ["have \u03b4nonneg : 0 \u2264 \u03b4 := by linarith only [h\u03c4, h\u03b41]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have D : 0 \u2264 1 - \u03b4 / 4 := by linarith only [h\u03b42]", "annotated_tactic": ["have D : 0 \u2264 1 - \u03b4 / 4 := by linarith only [h\u03b42]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have \u03c4pos : 0 < \u03c4 := _root_.zero_lt_one.trans_le h\u03c4", "annotated_tactic": ["have \u03c4pos : 0 < \u03c4 := _root_.zero_lt_one.trans_le h\u03c4", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have I : (1 - \u03b4 / 4) * \u03c4 \u2264 1 :=\n  calc\n    (1 - \u03b4 / 4) * \u03c4 \u2264 (1 - \u03b4 / 4) * (1 + \u03b4 / 4) := by gcongr\n    _ = (1 : \u211d) - \u03b4 ^ 2 / 16 := by ring\n    _ \u2264 1 := by linarith only [sq_nonneg \u03b4]", "annotated_tactic": ["have I : (1 - \u03b4 / 4) * \u03c4 \u2264 1 :=\n    calc\n      (1 - \u03b4 / 4) * \u03c4 \u2264 (1 - \u03b4 / 4) * (1 + \u03b4 / 4) := by gcongr\n      _ = (1 : \u211d) - \u03b4 ^ 2 / 16 := by ring\n      _ \u2264 1 := by linarith only [<a>sq_nonneg</a> \u03b4]", [{"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have J : 1 - \u03b4 \u2264 1 - \u03b4 / 4 := by linarith only [\u03b4nonneg]", "annotated_tactic": ["have J : 1 - \u03b4 \u2264 1 - \u03b4 / 4 := by linarith only [\u03b4nonneg]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have K : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9 := by rw [inv_eq_one_div, le_div_iff \u03c4pos]; exact I", "annotated_tactic": ["have K : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9 := by rw [<a>inv_eq_one_div</a>, <a>le_div_iff</a> \u03c4pos]; exact I", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "suffices L : \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016 by linarith only [J, K, L]", "annotated_tactic": ["suffices L : \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016 by linarith only [J, K, L]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "have h\u03c4' : \u2200 k, \u03c4\u207b\u00b9 \u2264 a.r k := by\n  intro k\n  rw [inv_eq_one_div, div_le_iff \u03c4pos, \u2190 lastr, mul_comm]\n  exact a.hlast' k h\u03c4", "annotated_tactic": ["have h\u03c4' : \u2200 k, \u03c4\u207b\u00b9 \u2264 a.r k := by\n    intro k\n    rw [<a>inv_eq_one_div</a>, <a>div_le_iff</a> \u03c4pos, \u2190 lastr, <a>mul_comm</a>]\n    exact a.hlast' k h\u03c4", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "rcases ah inej with (H | H)", "annotated_tactic": ["rcases ah inej with (H | H)", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "case inl\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\nH : a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016\n\ncase inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\nH : a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016"}, {"tactic": "simpa only [dist_eq_norm] using a.h", "annotated_tactic": ["simpa only [<a>dist_eq_norm</a>] using a.h", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [401, 7], "def_end_pos": [401, 19]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\n\u22a2 Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j", "state_after": "no goals"}, {"tactic": "linarith only [h\u03c4, h\u03b41]", "annotated_tactic": ["linarith only [h\u03c4, h\u03b41]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u22a2 0 \u2264 \u03b4", "state_after": "no goals"}, {"tactic": "linarith only [h\u03b42]", "annotated_tactic": ["linarith only [h\u03b42]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\n\u22a2 0 \u2264 1 - \u03b4 / 4", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\n\u22a2 (1 - \u03b4 / 4) * \u03c4 \u2264 (1 - \u03b4 / 4) * (1 + \u03b4 / 4)", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\n\u22a2 (1 - \u03b4 / 4) * (1 + \u03b4 / 4) = 1 - \u03b4 ^ 2 / 16", "state_after": "no goals"}, {"tactic": "linarith only [sq_nonneg \u03b4]", "annotated_tactic": ["linarith only [<a>sq_nonneg</a> \u03b4]", [{"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\n\u22a2 1 - \u03b4 ^ 2 / 16 \u2264 1", "state_after": "no goals"}, {"tactic": "linarith only [\u03b4nonneg]", "annotated_tactic": ["linarith only [\u03b4nonneg]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\n\u22a2 1 - \u03b4 \u2264 1 - \u03b4 / 4", "state_after": "no goals"}, {"tactic": "rw [inv_eq_one_div, le_div_iff \u03c4pos]", "annotated_tactic": ["rw [<a>inv_eq_one_div</a>, <a>le_div_iff</a> \u03c4pos]", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\n\u22a2 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\n\u22a2 (1 - \u03b4 / 4) * \u03c4 \u2264 1"}, {"tactic": "exact I", "annotated_tactic": ["exact I", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\n\u22a2 (1 - \u03b4 / 4) * \u03c4 \u2264 1", "state_after": "no goals"}, {"tactic": "linarith only [J, K, L]", "annotated_tactic": ["linarith only [J, K, L]", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nL : \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016\n\u22a2 1 - \u03b4 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "no goals"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\n\u22a2 \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nk : Fin N.succ\n\u22a2 \u03c4\u207b\u00b9 \u2264 a.r k"}, {"tactic": "rw [inv_eq_one_div, div_le_iff \u03c4pos, \u2190 lastr, mul_comm]", "annotated_tactic": ["rw [<a>inv_eq_one_div</a>, <a>div_le_iff</a> \u03c4pos, \u2190 lastr, <a>mul_comm</a>]", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nk : Fin N.succ\n\u22a2 \u03c4\u207b\u00b9 \u2264 a.r k", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nk : Fin N.succ\n\u22a2 a.r (last N) \u2264 \u03c4 * a.r k"}, {"tactic": "exact a.hlast' k h\u03c4", "annotated_tactic": ["exact a.hlast' k h\u03c4", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nk : Fin N.succ\n\u22a2 a.r (last N) \u2264 \u03c4 * a.r k", "state_after": "no goals"}, {"tactic": "apply le_trans _ H.1", "annotated_tactic": ["apply <a>le_trans</a> _ H.1", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\nH : a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\nH : a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i\n\u22a2 \u03c4\u207b\u00b9 \u2264 a.r i"}, {"tactic": "exact h\u03c4' i", "annotated_tactic": ["exact h\u03c4' i", []], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N 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14]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig E N \u03c4\nlastr : a.r (last N) = 1\nh\u03c4 : 1 \u2264 \u03c4\n\u03b4 : \u211d\nh\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4\nh\u03b42 : \u03b4 \u2264 1\ni j : Fin N.succ\ninej : i \u2260 j\nah : Pairwise fun i j => a.r i \u2264 \u2016a.c i - a.c j\u2016 \u2227 a.r j \u2264 \u03c4 * a.r i \u2228 a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u03b4nonneg : 0 \u2264 \u03b4\nD : 0 \u2264 1 - \u03b4 / 4\n\u03c4pos : 0 < \u03c4\nI : (1 - \u03b4 / 4) * \u03c4 \u2264 1\nJ : 1 - \u03b4 \u2264 1 - \u03b4 / 4\nK : 1 - \u03b4 / 4 \u2264 \u03c4\u207b\u00b9\nh\u03c4' : \u2200 (k : Fin N.succ), \u03c4\u207b\u00b9 \u2264 a.r k\nH : a.r j \u2264 \u2016a.c j - a.c i\u2016 \u2227 a.r i \u2264 \u03c4 * a.r j\n\u22a2 \u03c4\u207b\u00b9 \u2264 \u2016a.c i - a.c j\u2016", "state_after": "case inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nN : 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u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 Mem\u2112p f 1 (\u03bc + \u03bd)"}, {"tactic": "refine \u27e8h\u03bc.aestronglyMeasurable.add_measure h\u03bd.aestronglyMeasurable, ?_\u27e9", "annotated_tactic": ["refine \u27e8h\u03bc.aestronglyMeasurable.add_measure h\u03bd.aestronglyMeasurable, ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 Mem\u2112p f 1 (\u03bc + \u03bd)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 snorm f 1 (\u03bc + \u03bd) < \u22a4"}, {"tactic": "rw [snorm_one_add_measure, ENNReal.add_lt_top]", "annotated_tactic": ["rw [<a>snorm_one_add_measure</a>, <a>ENNReal.add_lt_top</a>]", [{"full_name": "MeasureTheory.snorm_one_add_measure", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [713, 9], "def_end_pos": [713, 30]}, {"full_name": "ENNReal.add_lt_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [190, 17], "def_end_pos": [190, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 snorm f 1 (\u03bc + \u03bd) < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 snorm f 1 \u03bc < \u22a4 \u2227 snorm f 1 \u03bd < \u22a4"}, {"tactic": "exact \u27e8h\u03bc.snorm_lt_top, h\u03bd.snorm_lt_top\u27e9", "annotated_tactic": ["exact \u27e8h\u03bc.snorm_lt_top, h\u03bd.snorm_lt_top\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nh\u03bc : Mem\u2112p f 1 \u03bc\nh\u03bd : Mem\u2112p f 1 \u03bd\n\u22a2 snorm f 1 \u03bc < \u22a4 \u2227 snorm f 1 \u03bd < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.update_idem", "start": [691, 1], "end": [694, 42], "traced_tactics": [{"tactic": "funext b", "annotated_tactic": ["funext b", []], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na\u271d : \u03b1\u271d\nb : \u03b2\u271d a\u271d\n\u03b1 : Sort u_2\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Sort u_1\na : \u03b1\nv w : \u03b2 a\nf : (a : \u03b1) \u2192 \u03b2 a\n\u22a2 update (update f a v) a w = update f a w", "state_after": "case h\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d a\u271d\n\u03b1 : Sort u_2\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Sort u_1\na : \u03b1\nv w : \u03b2 a\nf : (a : \u03b1) \u2192 \u03b2 a\nb : \u03b1\n\u22a2 update (update f a v) a w b = update f a w b"}, {"tactic": "by_cases h : b = a <;> simp [update, h]", "annotated_tactic": ["by_cases h : b = a <;> simp [<a>update</a>, h]", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [553, 5], "def_end_pos": [553, 11]}]], "state_before": "case h\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d a\u271d\n\u03b1 : Sort u_2\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Sort u_1\na : \u03b1\nv w : \u03b2 a\nf : (a : \u03b1) \u2192 \u03b2 a\nb : \u03b1\n\u22a2 update (update f a v) a w b = update f a w b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.one_lt_omega", "start": [2475, 1], "end": [2475, 82], "traced_tactics": [{"tactic": "simpa only [Nat.cast_one] using nat_lt_omega 1", "annotated_tactic": ["simpa only [<a>Nat.cast_one</a>] using <a>nat_lt_omega</a> 1", [{"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Ordinal.nat_lt_omega", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2463, 9], "def_end_pos": [2463, 21]}]], "state_before": "\u22a2 1 < \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Set.lean", "full_name": "Set.ext", "start": [64, 1], "end": [65, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.NormalWord.consRecOn_ofGroup", "start": [306, 1], "end": [312, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Galois/Examples.lean", "full_name": "CategoryTheory.FintypeCat.Action.pretransitive_of_isConnected", "start": [104, 1], "end": [124, 23], "traced_tactics": [{"tactic": "let T : Set X.V := MulAction.orbit G x", "annotated_tactic": ["let T : <a>Set</a> X.V := <a>MulAction.orbit</a> G x", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "MulAction.orbit", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "have : Fintype T := Fintype.ofFinite T", "annotated_tactic": ["have : <a>Fintype</a> T := <a>Fintype.ofFinite</a> T", [{"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [57, 7], "def_end_pos": [57, 14]}, {"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [449, 19], "def_end_pos": [449, 35]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis : Fintype \u2191T\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "letI : MulAction G (FintypeCat.of T) := inferInstanceAs <| MulAction G \u2191(MulAction.orbit G x)", "annotated_tactic": ["letI : <a>MulAction</a> G (<a>FintypeCat.of</a> T) := <a>inferInstanceAs</a> <| <a>MulAction</a> G \u2191(<a>MulAction.orbit</a> G x)", [{"full_name": "MulAction", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [110, 7], "def_end_pos": [110, 16]}, {"full_name": "FintypeCat.of", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [43, 5], "def_end_pos": [43, 7]}, {"full_name": "inferInstanceAs", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [113, 8], "def_end_pos": [113, 23]}, {"full_name": "MulAction", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [110, 7], "def_end_pos": [110, 16]}, {"full_name": "MulAction.orbit", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis : Fintype \u2191T\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "let T' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of T)", "annotated_tactic": ["let T' : <a>Action</a> <a>FintypeCat</a> (<a>MonCat.of</a> G) := <a>Action.FintypeCat.ofMulAction</a> G (<a>FintypeCat.of</a> T)", [{"full_name": "Action", "def_path": "Mathlib/RepresentationTheory/Action/Basic.lean", "def_pos": [39, 11], "def_end_pos": [39, 17]}, {"full_name": "FintypeCat", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [32, 5], "def_end_pos": [32, 15]}, {"full_name": "MonCat.of", "def_path": "Mathlib/Algebra/Category/MonCat/Basic.lean", "def_pos": [110, 5], "def_end_pos": [110, 7]}, {"full_name": "Action.FintypeCat.ofMulAction", "def_path": "Mathlib/RepresentationTheory/Action/Concrete.lean", "def_pos": [84, 5], "def_end_pos": [84, 16]}, {"full_name": "FintypeCat.of", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [43, 5], "def_end_pos": [43, 7]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "let i : T' \u27f6 X := \u27e8Subtype.val, fun _ \u21a6 rfl\u27e9", "annotated_tactic": ["let i : T' \u27f6 X := \u27e8<a>Subtype.val</a>, fun _ \u21a6 <a>rfl</a>\u27e9", [{"full_name": "Subtype.val", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [587, 3], "def_end_pos": [587, 6]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "have : Mono i := ConcreteCategory.mono_of_injective _ (Subtype.val_injective)", "annotated_tactic": ["have : <a>Mono</a> i := <a>ConcreteCategory.mono_of_injective</a> _ (<a>Subtype.val_injective</a>)", [{"full_name": "CategoryTheory.Mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [295, 7], "def_end_pos": [295, 11]}, {"full_name": "CategoryTheory.ConcreteCategory.mono_of_injective", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean", "def_pos": [38, 9], "def_end_pos": [38, 26]}, {"full_name": "Subtype.val_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [131, 9], "def_end_pos": [131, 22]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d : Fintype \u2191T\nthis : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "have : IsIso i := by\n  apply IsConnected.noTrivialComponent T' i\n  apply (not_initial_iff_fiber_nonempty (Action.forget _ _) T').mpr\n  exact Set.Nonempty.coe_sort (MulAction.orbit_nonempty x)", "annotated_tactic": ["have : <a>IsIso</a> i := by\n      apply <a>IsConnected.noTrivialComponent</a> T' i\n      apply (<a>not_initial_iff_fiber_nonempty</a> (<a>Action.forget</a> _ _) T').<a>mpr</a>\n      exact <a>Set.Nonempty.coe_sort</a> (<a>MulAction.orbit_nonempty</a> x)", [{"full_name": "CategoryTheory.IsIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [272, 7], "def_end_pos": [272, 12]}, {"full_name": "CategoryTheory.PreGaloisCategory.IsConnected.noTrivialComponent", "def_path": "Mathlib/CategoryTheory/Galois/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 21]}, {"full_name": "CategoryTheory.PreGaloisCategory.not_initial_iff_fiber_nonempty", "def_path": "Mathlib/CategoryTheory/Galois/Basic.lean", "def_pos": [183, 7], "def_end_pos": [183, 37]}, {"full_name": "Action.forget", "def_path": "Mathlib/RepresentationTheory/Action/Basic.lean", "def_pos": [270, 5], "def_end_pos": [270, 11]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "Set.Nonempty.coe_sort", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [421, 11], "def_end_pos": [421, 28]}, {"full_name": "MulAction.orbit_nonempty", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 23]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "have hb : Function.Bijective i.hom := by\n  apply (ConcreteCategory.isIso_iff_bijective i.hom).mp\n  exact map_isIso (forget\u2082 _ FintypeCat) i", "annotated_tactic": ["have hb : <a>Function.Bijective</a> i.hom := by\n      apply (<a>ConcreteCategory.isIso_iff_bijective</a> i.hom).<a>mp</a>\n      exact <a>map_isIso</a> (<a>forget\u2082</a> _ <a>FintypeCat</a>) i", [{"full_name": "Function.Bijective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [145, 5], "def_end_pos": [145, 14]}, {"full_name": "CategoryTheory.ConcreteCategory.isIso_iff_bijective", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "CategoryTheory.Functor.map_isIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [646, 10], "def_end_pos": [646, 19]}, {"full_name": "CategoryTheory.forget\u2082", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [168, 8], "def_end_pos": [168, 15]}, {"full_name": "FintypeCat", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [32, 5], "def_end_pos": [32, 15]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "obtain \u27e8\u27e8y', \u27e8g, (hg : g \u2022 x = y')\u27e9\u27e9, (hy' : y' = y)\u27e9 := hb.surjective y", "annotated_tactic": ["obtain \u27e8\u27e8y', \u27e8g, (hg : g \u2022 x = y')\u27e9\u27e9, (hy' : y' = y)\u27e9 := hb.surjective y", []], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "case intro.mk.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\ny' : \u2191X.V\ng : G\nhg : g \u2022 x = y'\nhy' : y' = y\n\u22a2 \u2203 g, g \u2022 x = y"}, {"tactic": "use g", "annotated_tactic": ["use g", []], "state_before": "case intro.mk.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\ny' : \u2191X.V\ng : G\nhg : g \u2022 x = y'\nhy' : y' = y\n\u22a2 \u2203 g, g \u2022 x = y", "state_after": "case h\nG : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\ny' : \u2191X.V\ng : G\nhg : g \u2022 x = y'\nhy' : y' = y\n\u22a2 g \u2022 x = y"}, {"tactic": "exact hg.trans hy'", "annotated_tactic": ["exact hg.trans hy'", []], "state_before": "case h\nG : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\nhb : Function.Bijective i.hom\ny' : \u2191X.V\ng : G\nhg : g \u2022 x = y'\nhy' : y' = y\n\u22a2 g \u2022 x = y", "state_after": "no goals"}, {"tactic": "apply IsConnected.noTrivialComponent T' i", "annotated_tactic": ["apply <a>IsConnected.noTrivialComponent</a> T' i", [{"full_name": "CategoryTheory.PreGaloisCategory.IsConnected.noTrivialComponent", "def_path": "Mathlib/CategoryTheory/Galois/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 21]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 IsIso i", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 IsInitial T' \u2192 False"}, {"tactic": "apply (not_initial_iff_fiber_nonempty (Action.forget _ _) T').mpr", "annotated_tactic": ["apply (<a>not_initial_iff_fiber_nonempty</a> (<a>Action.forget</a> _ _) T').<a>mpr</a>", [{"full_name": "CategoryTheory.PreGaloisCategory.not_initial_iff_fiber_nonempty", "def_path": "Mathlib/CategoryTheory/Galois/Basic.lean", "def_pos": [183, 7], "def_end_pos": [183, 37]}, {"full_name": "Action.forget", "def_path": "Mathlib/RepresentationTheory/Action/Basic.lean", "def_pos": [270, 5], "def_end_pos": [270, 11]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 IsInitial T' \u2192 False", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 Nonempty \u2191((Action.forget FintypeCat (MonCat.of G)).obj T')"}, {"tactic": "exact Set.Nonempty.coe_sort (MulAction.orbit_nonempty x)", "annotated_tactic": ["exact <a>Set.Nonempty.coe_sort</a> (<a>MulAction.orbit_nonempty</a> x)", [{"full_name": "Set.Nonempty.coe_sort", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [421, 11], "def_end_pos": [421, 28]}, {"full_name": "MulAction.orbit_nonempty", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 23]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b9 : Fintype \u2191T\nthis\u271d : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis : Mono i\n\u22a2 Nonempty \u2191((Action.forget FintypeCat (MonCat.of G)).obj T')", "state_after": "no goals"}, {"tactic": "apply (ConcreteCategory.isIso_iff_bijective i.hom).mp", "annotated_tactic": ["apply (<a>ConcreteCategory.isIso_iff_bijective</a> i.hom).<a>mp</a>", [{"full_name": "CategoryTheory.ConcreteCategory.isIso_iff_bijective", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean", "def_pos": [176, 9], "def_end_pos": [176, 28]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\n\u22a2 Function.Bijective i.hom", "state_after": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\n\u22a2 IsIso i.hom"}, {"tactic": "exact map_isIso (forget\u2082 _ FintypeCat) i", "annotated_tactic": ["exact <a>map_isIso</a> (<a>forget\u2082</a> _ <a>FintypeCat</a>) i", [{"full_name": "CategoryTheory.Functor.map_isIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [646, 10], "def_end_pos": [646, 19]}, {"full_name": "CategoryTheory.forget\u2082", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [168, 8], "def_end_pos": [168, 15]}, {"full_name": "FintypeCat", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [32, 5], "def_end_pos": [32, 15]}]], "state_before": "G : Type u\ninst\u271d\u00b9 : Group G\nX : Action FintypeCat (MonCat.of G)\ninst\u271d : IsConnected X\nx y : \u2191X.V\nT : Set \u2191X.V := MulAction.orbit G x\nthis\u271d\u00b2 : Fintype \u2191T\nthis\u271d\u00b9 : MulAction G \u2191(FintypeCat.of \u2191T) := inferInstanceAs (MulAction G \u2191(MulAction.orbit G x))\nT' : Action FintypeCat (MonCat.of G) := Action.FintypeCat.ofMulAction G (FintypeCat.of \u2191T)\ni : T' \u27f6 X := { hom := Subtype.val, comm := \u22ef }\nthis\u271d : Mono i\nthis : IsIso i\n\u22a2 IsIso i.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean", "full_name": "MeasureTheory.addContent_union_le", "start": [152, 1], "end": [157, 58], "traced_tactics": [{"tactic": "rw [\u2190 union_diff_self, addContent_union hC hs (hC.diff_mem ht hs)]", "annotated_tactic": ["rw [\u2190 <a>union_diff_self</a>, <a>addContent_union</a> hC hs (hC.diff_mem ht hs)]", [{"full_name": "Set.union_diff_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1985, 9], "def_end_pos": [1985, 24]}, {"full_name": "MeasureTheory.addContent_union", "def_path": "Mathlib/MeasureTheory/Measure/AddContent.lean", "def_pos": [147, 7], "def_end_pos": [147, 23]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetRing C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 m (s \u222a t) \u2264 m s + m t", "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetRing C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 m s + m (t \\ s) \u2264 m s + m t\n\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetRing C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 Disjoint s (t \\ s)"}, {"tactic": "exact add_le_add le_rfl\n  (addContent_mono hC.isSetSemiring (hC.diff_mem ht hs) ht diff_subset)", "annotated_tactic": ["exact <a>add_le_add</a> <a>le_rfl</a>\n      (<a>addContent_mono</a> hC.isSetSemiring (hC.diff_mem ht hs) ht <a>diff_subset</a>)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "MeasureTheory.addContent_mono", "def_path": "Mathlib/MeasureTheory/Measure/AddContent.lean", "def_pos": [134, 7], "def_end_pos": [134, 22]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1782, 9], "def_end_pos": [1782, 20]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetRing C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 m s + m (t \\ s) \u2264 m s + m t", "state_after": "no goals"}, {"tactic": "rw [Set.disjoint_iff_inter_eq_empty, inter_diff_self]", "annotated_tactic": ["rw [<a>Set.disjoint_iff_inter_eq_empty</a>, <a>inter_diff_self</a>]", [{"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1487, 9], "def_end_pos": [1487, 36]}, {"full_name": "Set.inter_diff_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1823, 9], "def_end_pos": [1823, 24]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetRing C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 Disjoint s (t \\ s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.mem_of_mem_erase", "start": [1137, 1], "end": [1138, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Coprod/Basic.lean", "full_name": "Monoid.Coprod.toProd_apply_inr", "start": [502, 1], "end": [502, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_pi", "start": [309, 1], "end": [311, 56], "traced_tactics": [{"tactic": "simpa only [\u2190 mem_univ_pi] using countable_univ_pi hs", "annotated_tactic": ["simpa only [\u2190 <a>mem_univ_pi</a>] using <a>countable_univ_pi</a> hs", [{"full_name": "Set.mem_univ_pi", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [257, 9], "def_end_pos": [257, 20]}, {"full_name": "Set.countable_univ_pi", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [304, 9], "def_end_pos": [304, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : Finite \u03b1\ns : (a : \u03b1) \u2192 Set (\u03c0 a)\nhs : \u2200 (a : \u03b1), (s a).Countable\n\u22a2 {f | \u2200 (a : \u03b1), f a \u2208 s a}.Countable", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Monotone.lean", "full_name": "strictAntiOn_Iic_of_succ_lt", "start": [256, 1], "end": [258, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Semisimple/Basic.lean", "full_name": "LieAlgebra.IsSemisimple.isSimple_of_isAtom", "start": [119, 1], "end": [182, 13], "traced_tactics": [{"tactic": "intro J", "annotated_tactic": ["intro J", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\n\u22a2 \u2200 (I_1 : LieIdeal R \u21a5\u2191I), I_1 = \u22a5 \u2228 I_1 = \u22a4", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\n\u22a2 J = \u22a5 \u2228 J = \u22a4"}, {"tactic": "rw [or_iff_not_imp_right]", "annotated_tactic": ["rw [<a>or_iff_not_imp_right</a>]", [{"full_name": "Classical.or_iff_not_imp_right", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [148, 9], "def_end_pos": [148, 29]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\n\u22a2 J = \u22a5 \u2228 J = \u22a4", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\n\u22a2 \u00acJ = \u22a4 \u2192 J = \u22a5"}, {"tactic": "intro hJ", "annotated_tactic": ["intro hJ", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\n\u22a2 \u00acJ = \u22a4 \u2192 J = \u22a5", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J = \u22a5"}, {"tactic": "suffices J' = \u22a5 by\n  rw [eq_bot_iff] at this \u22a2\n  intro x hx\n  suffices x \u2208 J \u2192 x = 0 from this hx\n  simpa [J'] using @this x.1", "annotated_tactic": ["suffices J' = \u22a5 by\n      rw [<a>eq_bot_iff</a>] at this \u22a2\n      intro x hx\n      suffices x \u2208 J \u2192 x = 0 from this hx\n      simpa [J'] using @this x.1", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J = \u22a5", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' = \u22a5"}, {"tactic": "apply hI.2", "annotated_tactic": ["apply hI.2", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' = \u22a5", "state_after": "case a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' < I"}, {"tactic": "rw [lt_iff_le_and_ne]", "annotated_tactic": ["rw [<a>lt_iff_le_and_ne</a>]", [{"full_name": "lt_iff_le_and_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}]], "state_before": "case a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' < I", "state_after": "case a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2264 I \u2227 J' \u2260 I"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2264 I \u2227 J' \u2260 I", "state_after": "case a.left\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2264 I\n\ncase a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2260 I"}, {"tactic": "contrapose! hJ", "annotated_tactic": ["contrapose! hJ", []], "state_before": "case a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2260 I", "state_after": "case a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\n\u22a2 J = \u22a4"}, {"tactic": "rw [eq_top_iff]", "annotated_tactic": ["rw [<a>eq_top_iff</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 19]}]], "state_before": "case a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\n\u22a2 J = \u22a4", "state_after": "case a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\n\u22a2 \u22a4 \u2264 J"}, {"tactic": "rintro \u27e8x, hx\u27e9 -", "annotated_tactic": ["rintro \u27e8x, hx\u27e9 -", []], "state_before": "case a.right\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\n\u22a2 \u22a4 \u2264 J", "state_after": "case a.right.mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\nx : L\nhx : x \u2208 \u2191I\n\u22a2 \u27e8x, hx\u27e9 \u2208 J"}, {"tactic": "rw [\u2190 hJ] at hx", "annotated_tactic": ["rw [\u2190 hJ] at hx", []], "state_before": "case a.right.mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\nx : L\nhx : x \u2208 \u2191I\n\u22a2 \u27e8x, hx\u27e9 \u2208 J", "state_after": "case a.right.mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\nx : L\nhx\u271d : x \u2208 \u2191I\nhx : x \u2208 \u2191J'\n\u22a2 \u27e8x, hx\u271d\u27e9 \u2208 J"}, {"tactic": "rcases hx with \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rcases hx with \u27e8y, hy, rfl\u27e9", []], "state_before": "case a.right.mk\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\nx : L\nhx\u271d : x \u2208 \u2191I\nhx : x \u2208 \u2191J'\n\u22a2 \u27e8x, hx\u271d\u27e9 \u2208 J", "state_after": "case a.right.mk.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\nhx : \u2191I.incl y \u2208 \u2191I\n\u22a2 \u27e8\u2191I.incl y, hx\u27e9 \u2208 J"}, {"tactic": "exact hy", "annotated_tactic": ["exact hy", []], "state_before": "case a.right.mk.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : J' = I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\nhx : \u2191I.incl y \u2208 \u2191I\n\u22a2 \u27e8\u2191I.incl y, hx\u27e9 \u2208 J", "state_after": "no goals"}, {"tactic": "rintro x _ \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro x _ \u27e8y, hy, rfl\u27e9", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\n\u22a2 \u2200 {x m : L}, m \u2208 __spread\u271d\u207b\u2070.carrier \u2192 \u2045x, m\u2046 \u2208 __spread\u271d\u207b\u2070.carrier", "state_after": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 \u2045x, \u2191I.incl y\u2046 \u2208 __spread\u271d\u207b\u2070.carrier"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 \u2045x, \u2191I.incl y\u2046 \u2208 __spread\u271d\u207b\u2070.carrier", "state_after": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 \u2045x, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier"}, {"tactic": "rw [LieSubmodule.mem_sup] at hx", "annotated_tactic": ["rw [<a>LieSubmodule.mem_sup</a>] at hx", [{"full_name": "LieSubmodule.mem_sup", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [591, 9], "def_end_pos": [591, 16]}]], "state_before": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\nhx : x \u2208 I \u2294 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045x, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier", "state_after": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\nhx : \u2203 y \u2208 I, \u2203 z \u2208 sSup ({I' | IsAtom I'} \\ {I}), y + z = x\n\u22a2 \u2045x, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier"}, {"tactic": "obtain \u27e8a, ha, b, hb, rfl\u27e9 := hx", "annotated_tactic": ["obtain \u27e8a, ha, b, hb, rfl\u27e9 := hx", []], "state_before": "case intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\nhx : \u2203 y \u2208 I, \u2203 z \u2208 sSup ({I' | IsAtom I'} \\ {I}), y + z = x\n\u22a2 \u2045x, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a + b, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier"}, {"tactic": "simp only [add_lie, AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,\n  Submodule.mem_toAddSubmonoid]", "annotated_tactic": ["simp only [<a>add_lie</a>, <a>AddSubsemigroup.mem_carrier</a>, <a>AddSubmonoid.mem_toSubsemigroup</a>,\n          <a>Submodule.mem_toAddSubmonoid</a>]", [{"full_name": "add_lie", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 16]}, {"full_name": "AddSubsemigroup.mem_carrier", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [111, 3], "def_end_pos": [111, 14]}, {"full_name": "AddSubmonoid.mem_toSubsemigroup", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [161, 3], "def_end_pos": [161, 14]}, {"full_name": "Submodule.mem_toAddSubmonoid", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 27]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a + b, \u2191y\u2046 \u2208 (Submodule.map (\u2191I).subtype \u2191J).carrier", "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a, \u2191y\u2046 + \u2045b, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J"}, {"tactic": "apply add_mem", "annotated_tactic": ["apply <a>add_mem</a>", [{"full_name": "AddMemClass.add_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a, \u2191y\u2046 + \u2045b, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J", "state_after": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J\n\ncase intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J"}, {"tactic": "nth_rewrite 1 [\u2190 sSup_singleton (a := I)]", "annotated_tactic": ["nth_rewrite 1 [\u2190 <a>sSup_singleton</a> (a := I)]", [{"full_name": "sSup_singleton", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 x \u2208 I \u2294 sSup ({I' | IsAtom I'} \\ {I})", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 x \u2208 sSup {I} \u2294 sSup ({I' | IsAtom I'} \\ {I})"}, {"tactic": "rw [\u2190 sSup_union, Set.union_diff_self, Set.union_eq_self_of_subset_left,\n  IsSemisimple.sSup_atoms_eq_top]", "annotated_tactic": ["rw [\u2190 <a>sSup_union</a>, <a>Set.union_diff_self</a>, <a>Set.union_eq_self_of_subset_left</a>,\n            <a>IsSemisimple.sSup_atoms_eq_top</a>]", [{"full_name": "sSup_union", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [402, 9], "def_end_pos": [402, 19]}, {"full_name": "Set.union_diff_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1985, 9], "def_end_pos": [1985, 24]}, {"full_name": "Set.union_eq_self_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 37]}, {"full_name": "LieAlgebra.IsSemisimple.sSup_atoms_eq_top", "def_path": "Mathlib/Algebra/Lie/Semisimple/Defs.lean", "def_pos": [76, 3], "def_end_pos": [76, 20]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 x \u2208 sSup {I} \u2294 sSup ({I' | IsAtom I'} \\ {I})", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 x \u2208 \u22a4\n\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 {I} \u2286 {I' | IsAtom I'}"}, {"tactic": "apply LieSubmodule.mem_top", "annotated_tactic": ["apply <a>LieSubmodule.mem_top</a>", [{"full_name": "LieSubmodule.mem_top", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [409, 9], "def_end_pos": [409, 16]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 x \u2208 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [Set.singleton_subset_iff, Set.mem_setOf_eq, hI]", "annotated_tactic": ["simp only [<a>Set.singleton_subset_iff</a>, <a>Set.mem_setOf_eq</a>, hI]", [{"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 29]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nx : L\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\n\u22a2 {I} \u2286 {I' | IsAtom I'}", "state_after": "no goals"}, {"tactic": "simp only [Submodule.mem_map, LieSubmodule.mem_coeSubmodule, Submodule.coeSubtype,\n  Subtype.exists, exists_and_right, exists_eq_right, ha, lie_mem_left, exists_true_left]", "annotated_tactic": ["simp only [<a>Submodule.mem_map</a>, <a>LieSubmodule.mem_coeSubmodule</a>, <a>Submodule.coeSubtype</a>,\n            <a>Subtype.exists</a>, <a>exists_and_right</a>, <a>exists_eq_right</a>, ha, <a>lie_mem_left</a>, <a>exists_true_left</a>]", [{"full_name": "Submodule.mem_map", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "LieSubmodule.mem_coeSubmodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [108, 9], "def_end_pos": [108, 25]}, {"full_name": "Submodule.coeSubtype", "def_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [63, 19], "def_end_pos": [63, 27]}, {"full_name": "exists_and_right", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [291, 17], "def_end_pos": [291, 33]}, {"full_name": "exists_eq_right", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 32]}, {"full_name": "lie_mem_left", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [239, 9], "def_end_pos": [239, 21]}, {"full_name": "exists_true_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [927, 17], "def_end_pos": [927, 33]}]], "state_before": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045a, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J", "state_after": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u27e8\u2045a, \u2191y\u2046, \u22ef\u27e9 \u2208 J"}, {"tactic": "exact lie_mem_right R I J \u27e8a, ha\u27e9 y hy", "annotated_tactic": ["exact <a>lie_mem_right</a> R I J \u27e8a, ha\u27e9 y hy", [{"full_name": "lie_mem_right", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [235, 9], "def_end_pos": [235, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u27e8\u2045a, \u2191y\u2046, \u22ef\u27e9 \u2208 J", "state_after": "no goals"}, {"tactic": "suffices \u2045b, y.val\u2046 = 0 by simp only [this, zero_mem]", "annotated_tactic": ["suffices \u2045b, y.val\u2046 = 0 by simp only [this, <a>zero_mem</a>]", [{"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}]], "state_before": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J", "state_after": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 = 0"}, {"tactic": "rw [\u2190 LieSubmodule.mem_bot (R := R) (L := L),\n    \u2190 (IsSemisimple.setIndependent_isAtom hI).eq_bot]", "annotated_tactic": ["rw [\u2190 <a>LieSubmodule.mem_bot</a> (R := R) (L := L),\n              \u2190 (<a>IsSemisimple.setIndependent_isAtom</a> hI).<a>eq_bot</a>]", [{"full_name": "LieSubmodule.mem_bot", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [383, 9], "def_end_pos": [383, 16]}, {"full_name": "LieAlgebra.IsSemisimple.setIndependent_isAtom", "def_path": "Mathlib/Algebra/Lie/Semisimple/Defs.lean", "def_pos": [78, 3], "def_end_pos": [78, 24]}, {"full_name": "Disjoint.eq_bot", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [143, 9], "def_end_pos": [143, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 = 0", "state_after": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 \u2208 I \u2293 sSup ({I | IsAtom I} \\ {I})"}, {"tactic": "exact \u27e8lie_mem_right R L I b y y.2, lie_mem_left _ _ _ _ _ hb\u27e9", "annotated_tactic": ["exact \u27e8<a>lie_mem_right</a> R L I b y y.2, <a>lie_mem_left</a> _ _ _ _ _ hb\u27e9", [{"full_name": "lie_mem_right", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [235, 9], "def_end_pos": [235, 22]}, {"full_name": "lie_mem_left", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [239, 9], "def_end_pos": [239, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\n\u22a2 \u2045b, \u2191y\u2046 \u2208 I \u2293 sSup ({I | IsAtom I} \\ {I})", "state_after": "no goals"}, {"tactic": "simp only [this, zero_mem]", "annotated_tactic": ["simp only [this, <a>zero_mem</a>]", [{"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\ny : \u21a5\u2191I\nhy : y \u2208 \u2191\u2191J\na : L\nha : a \u2208 I\nb : L\nhb : b \u2208 sSup ({I' | IsAtom I'} \\ {I})\nthis : \u2045b, \u2191y\u2046 = 0\n\u22a2 \u2045b, \u2191y\u2046 \u2208 Submodule.map (\u2191I).subtype \u2191J", "state_after": "no goals"}, {"tactic": "rw [eq_bot_iff] at this \u22a2", "annotated_tactic": ["rw [<a>eq_bot_iff</a>] at this \u22a2", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' = \u22a5\n\u22a2 J = \u22a5", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\n\u22a2 J \u2264 \u22a5"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\n\u22a2 J \u2264 \u22a5", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\nx : \u21a5\u2191I\nhx : x \u2208 J\n\u22a2 x \u2208 \u22a5"}, {"tactic": "suffices x \u2208 J \u2192 x = 0 from this hx", "annotated_tactic": ["suffices x \u2208 J \u2192 x = 0 from this hx", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\nx : \u21a5\u2191I\nhx : x \u2208 J\n\u22a2 x \u2208 \u22a5", "state_after": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\nx : \u21a5\u2191I\nhx : x \u2208 J\n\u22a2 x \u2208 J \u2192 x = 0"}, {"tactic": "simpa [J'] using @this x.1", "annotated_tactic": ["simpa [J'] using @this x.1", []], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nthis : J' \u2264 \u22a5\nx : \u21a5\u2191I\nhx : x \u2208 J\n\u22a2 x \u2208 J \u2192 x = 0", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8x, -, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, -, rfl\u27e9", []], "state_before": "case a.left\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\n\u22a2 J' \u2264 I", "state_after": "case a.left.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nx : \u21a5\u2191I\n\u22a2 \u2191I.incl x \u2208 I"}, {"tactic": "exact x.2", "annotated_tactic": ["exact x.2", []], "state_before": "case a.left.intro.intro\nR : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\ninst\u271d : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R \u21a5\u2191I\nJ' : LieIdeal R L :=\n  let __spread.0 := Submodule.map \u2191I.incl \u2191J;\n  { toSubmodule := __spread.0, lie_mem := \u22ef }\nhJ : \u00acJ = \u22a4\nx : \u21a5\u2191I\n\u22a2 \u2191I.incl x \u2208 I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.unsym_smul", "start": [231, 1], "end": [232, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.prod_pow", "start": [148, 1], "end": [150, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "full_name": "Projectivization.Subspace.span_eq_sInf", "start": [210, 1], "end": [216, 63], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "K : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\n\u22a2 span S = sInf {W | S \u2286 \u2191W}", "state_after": "case carrier.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\n\u22a2 x \u2208 (span S).carrier \u2194 x \u2208 (sInf {W | S \u2286 \u2191W}).carrier"}, {"tactic": "simp_rw [mem_carrier_iff, mem_span x]", "annotated_tactic": ["simp_rw [<a>mem_carrier_iff</a>, <a>mem_span</a> x]", [{"full_name": "Projectivization.Subspace.mem_carrier_iff", "def_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "def_pos": [65, 9], "def_end_pos": [65, 24]}, {"full_name": "Projectivization.Subspace.mem_span", "def_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "def_pos": [202, 9], "def_end_pos": [202, 17]}]], "state_before": "case carrier.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\n\u22a2 x \u2208 (span S).carrier \u2194 x \u2208 (sInf {W | S \u2286 \u2191W}).carrier", "state_after": "case carrier.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\n\u22a2 (\u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W) \u2194 x \u2208 sInf {W | S \u2286 \u2191W}"}, {"tactic": "refine \u27e8fun hx => ?_, fun hx W hW => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun hx => ?_, fun hx W hW => ?_\u27e9", []], "state_before": "case carrier.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\n\u22a2 (\u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W) \u2194 x \u2208 sInf {W | S \u2286 \u2191W}", "state_after": "case carrier.h.refine_1\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : \u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W\n\u22a2 x \u2208 sInf {W | S \u2286 \u2191W}\n\ncase carrier.h.refine_2\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : x \u2208 sInf {W | S \u2286 \u2191W}\nW : Subspace K V\nhW : S \u2286 \u2191W\n\u22a2 x \u2208 W"}, {"tactic": "rintro W \u27e8T, hT, rfl\u27e9", "annotated_tactic": ["rintro W \u27e8T, hT, rfl\u27e9", []], "state_before": "case carrier.h.refine_1\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : \u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W\n\u22a2 x \u2208 sInf {W | S \u2286 \u2191W}", "state_after": "case carrier.h.refine_1.intro.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : \u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W\nT : Subspace K V\nhT : T \u2208 {W | S \u2286 \u2191W}\n\u22a2 x \u2208 \u2191T"}, {"tactic": "exact hx T hT", "annotated_tactic": ["exact hx T hT", []], "state_before": "case carrier.h.refine_1.intro.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : \u2200 (W : Subspace K V), S \u2286 \u2191W \u2192 x \u2208 W\nT : Subspace K V\nhT : T \u2208 {W | S \u2286 \u2191W}\n\u22a2 x \u2208 \u2191T", "state_after": "no goals"}, {"tactic": "exact (@sInf_le _ _ { W : Subspace K V | S \u2286 \u2191W } W hW) hx", "annotated_tactic": ["exact (@<a>sInf_le</a> _ _ { W : <a>Subspace</a> K V | S \u2286 \u2191W } W hW) hx", [{"full_name": "sInf_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Projectivization.Subspace", "def_path": "Mathlib/LinearAlgebra/Projectivization/Subspace.lean", "def_pos": [45, 11], "def_end_pos": [45, 19]}]], "state_before": "case carrier.h.refine_2\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Set (\u2119 K V)\nx : \u2119 K V\nhx : x \u2208 sInf {W | S \u2286 \u2191W}\nW : Subspace K V\nhW : S \u2286 \u2191W\n\u22a2 x \u2208 W", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Conformal.lean", "full_name": "isConformalMap_complex_linear", "start": [50, 1], "end": [63, 33], "traced_tactics": [{"tactic": "have minor\u2081 : \u2016map 1\u2016 \u2260 0 := by simpa only [ext_ring_iff, Ne, norm_eq_zero] using nonzero", "annotated_tactic": ["have minor\u2081 : \u2016map 1\u2016 \u2260 0 := by simpa only [<a>ext_ring_iff</a>, <a>Ne</a>, <a>norm_eq_zero</a>] using nonzero", [{"full_name": "ContinuousLinearMap.ext_ring_iff", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [536, 9], "def_end_pos": [536, 21]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "norm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1524, 30], "def_end_pos": [1524, 42]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\n\u22a2 IsConformalMap (restrictScalars \u211d map)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 IsConformalMap (restrictScalars \u211d map)"}, {"tactic": "refine \u27e8\u2016map 1\u2016, minor\u2081, \u27e8\u2016map 1\u2016\u207b\u00b9 \u2022 ((map : \u2102 \u2192\u2097[\u2102] E) : \u2102 \u2192\u2097[\u211d] E), ?_\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u2016map 1\u2016, minor\u2081, \u27e8\u2016map 1\u2016\u207b\u00b9 \u2022 ((map : \u2102 \u2192\u2097[\u2102] E) : \u2102 \u2192\u2097[\u211d] E), ?_\u27e9, ?_\u27e9", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 IsConformalMap (restrictScalars \u211d map)", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 \u2200 (x : \u2102), \u2016(\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map) x\u2016 = \u2016x\u2016\n\ncase refine_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 restrictScalars \u211d map = \u2016map 1\u2016 \u2022 { toLinearMap := \u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map, norm_map' := ?refine_1 }.toContinuousLinearMap"}, {"tactic": "simpa only [ext_ring_iff, Ne, norm_eq_zero] using nonzero", "annotated_tactic": ["simpa only [<a>ext_ring_iff</a>, <a>Ne</a>, <a>norm_eq_zero</a>] using nonzero", [{"full_name": "ContinuousLinearMap.ext_ring_iff", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [536, 9], "def_end_pos": [536, 21]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "norm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1524, 30], "def_end_pos": [1524, 42]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\n\u22a2 \u2016map 1\u2016 \u2260 0", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 \u2200 (x : \u2102), \u2016(\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map) x\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\n\u22a2 \u2016(\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map) x\u2016 = \u2016x\u2016"}, {"tactic": "simp only [LinearMap.smul_apply]", "annotated_tactic": ["simp only [<a>LinearMap.smul_apply</a>]", [{"full_name": "LinearMap.smul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [831, 9], "def_end_pos": [831, 19]}]], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\n\u22a2 \u2016(\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map) x\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) x\u2016 = \u2016x\u2016"}, {"tactic": "have : x = x \u2022 (1 : \u2102) := by rw [smul_eq_mul, mul_one]", "annotated_tactic": ["have : x = x \u2022 (1 : \u2102) := by rw [<a>smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) x\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) x\u2016 = \u2016x\u2016"}, {"tactic": "nth_rw 1 [this]", "annotated_tactic": ["nth_rw 1 [this]", []], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) x\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) (x \u2022 1)\u2016 = \u2016x\u2016"}, {"tactic": "rw [LinearMap.coe_restrictScalars]", "annotated_tactic": ["rw [<a>LinearMap.coe_restrictScalars</a>]", [{"full_name": "LinearMap.coe_restrictScalars", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}]], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 (\u2191\u211d \u2191map) (x \u2022 1)\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191map (x \u2022 1)\u2016 = \u2016x\u2016"}, {"tactic": "simp only [map.coe_coe, map.map_smul, norm_smul, norm_inv, norm_norm]", "annotated_tactic": ["simp only [map.coe_coe, map.map_smul, <a>norm_smul</a>, <a>norm_inv</a>, <a>norm_norm</a>]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "norm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [754, 9], "def_end_pos": [754, 17]}, {"full_name": "norm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [1101, 9], "def_end_pos": [1101, 18]}]], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016\u2016map 1\u2016\u207b\u00b9 \u2022 \u2191map (x \u2022 1)\u2016 = \u2016x\u2016", "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016map 1\u2016\u207b\u00b9 * (\u2016x\u2016 * \u2016map 1\u2016) = \u2016x\u2016"}, {"tactic": "field_simp only [one_mul]", "annotated_tactic": ["field_simp only [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\nthis : x = x \u2022 1\n\u22a2 \u2016map 1\u2016\u207b\u00b9 * (\u2016x\u2016 * \u2016map 1\u2016) = \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [smul_eq_mul, mul_one]", "annotated_tactic": ["rw [<a>smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx : \u2102\n\u22a2 x = x \u2022 1", "state_after": "no goals"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case refine_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\n\u22a2 restrictScalars \u211d map = \u2016map 1\u2016 \u2022 { toLinearMap := \u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map, norm_map' := \u22ef }.toContinuousLinearMap", "state_after": "case refine_2.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx\u271d : \u2102\n\u22a2 (restrictScalars \u211d map) x\u271d =\n    (\u2016map 1\u2016 \u2022 { toLinearMap := \u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map, norm_map' := \u22ef }.toContinuousLinearMap) x\u271d"}, {"tactic": "simp [smul_inv_smul\u2080 minor\u2081]", "annotated_tactic": ["simp [<a>smul_inv_smul\u2080</a> minor\u2081]", [{"full_name": "smul_inv_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [212, 9], "def_end_pos": [212, 23]}]], "state_before": "case refine_2.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\ng : \u2102 \u2192L[\u211d] E\nf : \u2102 \u2192 E\nmap : \u2102 \u2192L[\u2102] E\nnonzero : map \u2260 0\nminor\u2081 : \u2016map 1\u2016 \u2260 0\nx\u271d : \u2102\n\u22a2 (restrictScalars \u211d map) x\u271d =\n    (\u2016map 1\u2016 \u2022 { toLinearMap := \u2016map 1\u2016\u207b\u00b9 \u2022 \u2191\u211d \u2191map, norm_map' := \u22ef }.toContinuousLinearMap) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LucasLehmer.lean", "full_name": "LucasLehmer.sMod_lt", "start": [148, 1], "end": [151, 47], "traced_tactics": [{"tactic": "rw [\u2190 sMod_mod]", "annotated_tactic": ["rw [\u2190 <a>sMod_mod</a>]", [{"full_name": "LucasLehmer.sMod_mod", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}]], "state_before": "p : \u2115\nhp : p \u2260 0\ni : \u2115\n\u22a2 sMod p i < 2 ^ p - 1", "state_after": "p : \u2115\nhp : p \u2260 0\ni : \u2115\n\u22a2 sMod p i % (2 ^ p - 1) < 2 ^ p - 1"}, {"tactic": "refine (Int.emod_lt _ (mersenne_int_ne_zero p hp)).trans_eq ?_", "annotated_tactic": ["refine (<a>Int.emod_lt</a> _ (<a>mersenne_int_ne_zero</a> p hp)).<a>trans_eq</a> ?_", [{"full_name": "Int.emod_lt", "def_path": "Mathlib/Algebra/Order/Group/Int.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}, {"full_name": "LucasLehmer.mersenne_int_ne_zero", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [134, 9], "def_end_pos": [134, 29]}, {"full_name": "LT.lt.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 21]}]], "state_before": "p : \u2115\nhp : p \u2260 0\ni : \u2115\n\u22a2 sMod p i % (2 ^ p - 1) < 2 ^ p - 1", "state_after": "p : \u2115\nhp : p \u2260 0\ni : \u2115\n\u22a2 |2 ^ p - 1| = 2 ^ p - 1"}, {"tactic": "exact abs_of_nonneg (mersenne_int_pos hp).le", "annotated_tactic": ["exact <a>abs_of_nonneg</a> (<a>mersenne_int_pos</a> hp).<a>le</a>", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "LucasLehmer.mersenne_int_pos", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [131, 9], "def_end_pos": [131, 25]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "p : \u2115\nhp : p \u2260 0\ni : \u2115\n\u22a2 |2 ^ p - 1| = 2 ^ p - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "ZFSet.toSet_sdiff", "start": [1120, 1], "end": [1123, 7], "traced_tactics": [{"tactic": "change (ZFSet.sep (fun z => z \u2209 y) x).toSet = _", "annotated_tactic": ["change (<a>ZFSet.sep</a> (fun z => z \u2209 y) x).<a>toSet</a> = _", [{"full_name": "ZFSet.sep", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [922, 15], "def_end_pos": [922, 18]}, {"full_name": "ZFSet.toSet", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [685, 5], "def_end_pos": [685, 10]}]], "state_before": "x y : ZFSet\n\u22a2 (x \\ y).toSet = x.toSet \\ y.toSet", "state_after": "x y : ZFSet\n\u22a2 (ZFSet.sep (fun z => z \u2209 y) x).toSet = x.toSet \\ y.toSet"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "x y : ZFSet\n\u22a2 (ZFSet.sep (fun z => z \u2209 y) x).toSet = x.toSet \\ y.toSet", "state_after": "case h\nx y x\u271d : ZFSet\n\u22a2 x\u271d \u2208 (ZFSet.sep (fun z => z \u2209 y) x).toSet \u2194 x\u271d \u2208 x.toSet \\ y.toSet"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nx y x\u271d : ZFSet\n\u22a2 x\u271d \u2208 (ZFSet.sep (fun z => z \u2209 y) x).toSet \u2194 x\u271d \u2208 x.toSet \\ y.toSet", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Sublattice.lean", "full_name": "Sublattice.map_equiv_eq_comap_symm", "start": [240, 1], "end": [242, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.zero_comp", "start": [820, 1], "end": [822, 7], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\u2081\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b2 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\u2081\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\nM\u2083 : Type u_7\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommMonoid M\u2083\nM\u2084 : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommMonoid M\u2084\ninst\u271d\u2074 : Module R\u2081 M\u2081\ninst\u271d\u00b3 : Module R\u2081 M'\u2081\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\nf : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\n\u22a2 comp 0 f = 0", "state_after": "case h\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\u2081\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b2 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\u2081\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\nM\u2083 : Type u_7\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommMonoid M\u2083\nM\u2084 : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommMonoid M\u2084\ninst\u271d\u2074 : Module R\u2081 M\u2081\ninst\u271d\u00b3 : Module R\u2081 M'\u2081\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\nf : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\u2081\n\u22a2 (comp 0 f) x\u271d = 0 x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\u2081\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b2 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid 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Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nA B : Matrix m n \u03b1\nx : n \u2192 \u03b1\n\u22a2 (A - B) *\u1d65 x = A *\u1d65 x - B *\u1d65 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Order.lean", "full_name": "IsSelfAdjoint.neg_algebraMap_norm_le_self", "start": [40, 1], "end": [45, 23], "traced_tactics": [{"tactic": "have : - a \u2264 algebraMap \u211d A \u2016a\u2016 := by\n  rw [\u2190 norm_neg]\n  exact IsSelfAdjoint.le_algebraMap_norm_self (neg ha)", "annotated_tactic": ["have : - a \u2264 <a>algebraMap</a> \u211d A \u2016a\u2016 := by\n    rw [\u2190 <a>norm_neg</a>]\n    exact <a>IsSelfAdjoint.le_algebraMap_norm_self</a> (<a>neg</a> ha)", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "norm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [435, 30], "def_end_pos": [435, 38]}, {"full_name": "IsSelfAdjoint.le_algebraMap_norm_self", "def_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Order.lean", "def_pos": [31, 7], "def_end_pos": [31, 44]}, {"full_name": "IsSelfAdjoint.neg", "def_path": "Mathlib/Algebra/Star/SelfAdjoint.lean", "def_pos": [140, 9], "def_end_pos": [140, 12]}]], "state_before": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\n\u22a2 -(algebraMap \u211d A) \u2016a\u2016 \u2264 a", "state_after": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\nthis : -a \u2264 (algebraMap \u211d A) \u2016a\u2016\n\u22a2 -(algebraMap \u211d A) \u2016a\u2016 \u2264 a"}, {"tactic": "exact neg_le.mp this", "annotated_tactic": ["exact neg_le.mp this", []], "state_before": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\nthis : -a \u2264 (algebraMap \u211d A) \u2016a\u2016\n\u22a2 -(algebraMap \u211d A) \u2016a\u2016 \u2264 a", "state_after": "no goals"}, {"tactic": "rw [\u2190 norm_neg]", "annotated_tactic": ["rw [\u2190 <a>norm_neg</a>]", [{"full_name": "norm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [435, 30], "def_end_pos": [435, 38]}]], "state_before": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\n\u22a2 -a \u2264 (algebraMap \u211d A) \u2016a\u2016", "state_after": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\n\u22a2 -a \u2264 (algebraMap \u211d A) \u2016-a\u2016"}, {"tactic": "exact IsSelfAdjoint.le_algebraMap_norm_self (neg ha)", "annotated_tactic": ["exact <a>IsSelfAdjoint.le_algebraMap_norm_self</a> (<a>neg</a> ha)", [{"full_name": "IsSelfAdjoint.le_algebraMap_norm_self", "def_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Order.lean", "def_pos": [31, 7], "def_end_pos": [31, 44]}, {"full_name": "IsSelfAdjoint.neg", "def_path": "Mathlib/Algebra/Star/SelfAdjoint.lean", "def_pos": [140, 9], "def_end_pos": [140, 12]}]], "state_before": "A : Type u_1\ninst\u271d\u2077 : NormedRing A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : CompleteSpace A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : StarModule \u2102 A\ninst\u271d\u00b9 : PartialOrder A\ninst\u271d : StarOrderedRing A\na : A\nha : autoParam (IsSelfAdjoint a) _auto\u271d\n\u22a2 -a \u2264 (algebraMap \u211d A) \u2016-a\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.countP_cons_of_neg", "start": [2296, 1], "end": [2297, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.range_single_subset", "start": [350, 1], "end": [351, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "full_name": "MeasureTheory.Measure.compProd_apply_prod", "start": [51, 1], "end": [58, 34], "traced_tactics": [{"tactic": "rw [compProd_apply (hs.prod ht), \u2190 lintegral_indicator _ hs]", "annotated_tactic": ["rw [<a>compProd_apply</a> (hs.prod ht), \u2190 <a>lintegral_indicator</a> _ hs]", [{"full_name": "MeasureTheory.Measure.compProd_apply", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [46, 7], "def_end_pos": [46, 21]}, {"full_name": "MeasureTheory.lintegral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [824, 9], "def_end_pos": [824, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 (\u03bc \u2297\u2098 \u03ba) (s \u00d7\u02e2 t) = \u222b\u207b (a : \u03b1) in s, (\u03ba a) t \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1), (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bc = \u222b\u207b (a : \u03b1), s.indicator (fun a => (\u03ba a) t) a \u2202\u03bc"}, {"tactic": "congr with a", "annotated_tactic": ["congr with a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1), (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bc = \u222b\u207b (a : \u03b1), s.indicator (fun a => (\u03ba a) t) a \u2202\u03bc", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) = s.indicator (fun a => (\u03ba a) t) a"}, {"tactic": "classical\nrw [Set.indicator_apply]\nsplit_ifs with ha <;> simp [ha]", "annotated_tactic": ["classical\n  rw [<a>Set.indicator_apply</a>]\n  split_ifs with ha <;> simp [ha]", [{"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [60, 3], "def_end_pos": [60, 14]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) = s.indicator (fun a => (\u03ba a) t) a", "state_after": "no goals"}, {"tactic": "rw [Set.indicator_apply]", "annotated_tactic": ["rw [<a>Set.indicator_apply</a>]", [{"full_name": "Set.indicator_apply", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [60, 3], "def_end_pos": [60, 14]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) = s.indicator (fun a => (\u03ba a) t) a", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) = if a \u2208 s then (\u03ba a) t else 0"}, {"tactic": "split_ifs with ha <;> simp [ha]", "annotated_tactic": ["split_ifs with ha <;> simp [ha]", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03ba \u03b7 : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : SFinite \u03bc\ninst\u271d : IsSFiniteKernel \u03ba\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\na : \u03b1\n\u22a2 (\u03ba a) (Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t) = if a \u2208 s then (\u03ba a) t else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "PSet.Resp.eval_val", "start": [576, 1], "end": [578, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_normalize", "start": [229, 1], "end": [230, 24], "traced_tactics": [{"tactic": "simp [normalize]", "annotated_tactic": ["simp [<a>normalize</a>]", [{"full_name": "Rat.normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [73, 15], "def_end_pos": [73, 28]}]], "state_before": "n : Int\nd : Nat\nz : d \u2260 0\n\u22a2 -normalize n d z = normalize (-n) d z", "state_after": "n : Int\nd : Nat\nz : d \u2260 0\n\u22a2 -{ num := n.div \u2191(n.natAbs.gcd d), den := d / n.natAbs.gcd d, den_nz := \u22ef, reduced := \u22ef } =\n    { num := -n.div \u2191(n.natAbs.gcd d), den := d / n.natAbs.gcd d, den_nz := \u22ef, reduced := \u22ef }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : Int\nd : Nat\nz : d \u2260 0\n\u22a2 -{ num := n.div \u2191(n.natAbs.gcd d), den := d / n.natAbs.gcd d, den_nz := \u22ef, reduced := \u22ef } =\n    { num := -n.div \u2191(n.natAbs.gcd d), den := d / n.natAbs.gcd d, den_nz := \u22ef, reduced := \u22ef }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.recF_eq_of_Wequiv", "start": [192, 1], "end": [198, 54], "traced_tactics": [{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "F : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : (P F).W\n\u22a2 Wequiv x y \u2192 recF u x = recF u y", "state_after": "F : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : (P F).W\nh : Wequiv x y\n\u22a2 recF u x = recF u y"}, {"tactic": "induction h with\n| ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]\n| abs a f a' f' h => simp only [recF_eq', abs_map, h]\n| trans x y z _ _ ih\u2081 ih\u2082 => exact Eq.trans ih\u2081 ih\u2082", "annotated_tactic": ["induction h with\n  | <a>ind</a> a f f' _ ih => simp only [<a>recF_eq'</a>, <a>PFunctor.map_eq</a>, <a>Function.comp</a>, ih]\n  | <a>abs</a> a f a' f' h => simp only [<a>recF_eq'</a>, <a>abs_map</a>, h]\n  | <a>trans</a> x y z _ _ ih\u2081 ih\u2082 => exact <a>Eq.trans</a> ih\u2081 ih\u2082", [{"full_name": "QPF.Wequiv.ind", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [184, 5], "def_end_pos": [184, 8]}, {"full_name": "QPF.recF_eq'", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 17]}, {"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [71, 19], "def_end_pos": [71, 25]}, {"full_name": "Function.comp", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "QPF.Wequiv.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [185, 5], "def_end_pos": [185, 8]}, {"full_name": "QPF.recF_eq'", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 17]}, {"full_name": "QPF.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}, {"full_name": "QPF.Wequiv.trans", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [187, 5], "def_end_pos": [187, 10]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "F : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : (P F).W\nh : Wequiv x y\n\u22a2 recF u x = recF u y", "state_after": "no goals"}, {"tactic": "simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]", "annotated_tactic": ["simp only [<a>recF_eq'</a>, <a>PFunctor.map_eq</a>, <a>Function.comp</a>, ih]", [{"full_name": "QPF.recF_eq'", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 17]}, {"full_name": "PFunctor.map_eq", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [71, 19], "def_end_pos": [71, 25]}, {"full_name": "Function.comp", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case ind\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : (P F).W\na : (P F).A\nf f' : (P F).B a \u2192 (P F).W\na\u271d : \u2200 (x : (P F).B a), Wequiv (f x) (f' x)\nih : \u2200 (x : (P F).B a), recF u (f x) = recF u (f' x)\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a f')", "state_after": "no goals"}, {"tactic": "simp only [recF_eq', abs_map, h]", "annotated_tactic": ["simp only [<a>recF_eq'</a>, <a>abs_map</a>, h]", [{"full_name": "QPF.recF_eq'", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 17]}, {"full_name": "QPF.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "state_before": "case abs\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : (P F).W\na : (P F).A\nf : (P F).B a \u2192 (P F).W\na' : (P F).A\nf' : (P F).B a' \u2192 (P F).W\nh : abs \u27e8a, f\u27e9 = abs \u27e8a', f'\u27e9\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a' f')", "state_after": "no goals"}, {"tactic": "exact Eq.trans ih\u2081 ih\u2082", "annotated_tactic": ["exact <a>Eq.trans</a> ih\u2081 ih\u2082", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "case trans\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx\u271d y\u271d x y z : (P F).W\na\u271d\u00b9 : Wequiv x y\na\u271d : Wequiv y z\nih\u2081 : recF u x = recF u y\nih\u2082 : recF u y = recF u z\n\u22a2 recF u x = recF u z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/HomEquiv.lean", "full_name": "CategoryTheory.LocalizerMorphism.id_homMap", "start": [86, 1], "end": [89, 63], "traced_tactics": [{"tactic": "simpa using (id W\u2081).homMap_apply L\u2081 L\u2081 (\ud835\udfed D\u2081) (Iso.refl _) f", "annotated_tactic": ["simpa using (<a>id</a> W\u2081).<a>homMap_apply</a> L\u2081 L\u2081 (\ud835\udfed D\u2081) (<a>Iso.refl</a> _) f", [{"full_name": "CategoryTheory.LocalizerMorphism.id", "def_path": "Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean", "def_pos": [51, 5], "def_end_pos": [51, 7]}, {"full_name": "CategoryTheory.LocalizerMorphism.homMap_apply", "def_path": "Mathlib/CategoryTheory/Localization/HomEquiv.lean", "def_pos": [70, 7], "def_end_pos": [70, 19]}, {"full_name": "CategoryTheory.Iso.refl", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [131, 5], "def_end_pos": [131, 9]}]], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.24763, u_1} C\ninst\u271d\u2078 : Category.{u_9, u_2} C\u2081\ninst\u271d\u2077 : Category.{?u.24771, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.24775, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{?u.24783, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.24787, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\n\u22a2 (id W\u2081).homMap L\u2081 L\u2081 f = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/IsPerfectClosure.lean", "full_name": "IsPerfectClosure.equiv_apply", "start": [453, 1], "end": [455, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean", "full_name": "Complex.tsum_exp_neg_mul_int_sq", "start": [124, 1], "end": [127, 86], "traced_tactics": [{"tactic": "simpa only [mul_zero, zero_mul, add_zero] using Complex.tsum_exp_neg_quadratic ha 0", "annotated_tactic": ["simpa only [<a>mul_zero</a>, <a>zero_mul</a>, <a>add_zero</a>] using <a>Complex.tsum_exp_neg_quadratic</a> ha 0", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "Complex.tsum_exp_neg_quadratic", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean", "def_pos": [88, 9], "def_end_pos": [88, 39]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u2102\nha : 0 < a.re\n\u22a2 \u2211' (n : \u2124), cexp (-\u2191\u03c0 * a * \u2191n ^ 2) = 1 / a ^ (1 / 2) * \u2211' (n : \u2124), cexp (-\u2191\u03c0 / a * \u2191n ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.gcd_add_mul_right_right", "start": [35, 1], "end": [36, 44], "traced_tactics": [{"tactic": "simp [gcd_rec m (n + k * m), gcd_rec m n]", "annotated_tactic": ["simp [<a>gcd_rec</a> m (n + k * m), <a>gcd_rec</a> m n]", [{"full_name": "Nat.gcd_rec", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [68, 9], "def_end_pos": [68, 16]}, {"full_name": "Nat.gcd_rec", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [68, 9], "def_end_pos": [68, 16]}]], "state_before": "m n k : \u2115\n\u22a2 m.gcd (n + k * m) = m.gcd n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "affineSpan_singleton_union_vadd_eq_top_of_span_eq_top", "start": [1241, 1], "end": [1252, 7], "traced_tactics": [{"tactic": "convert ext_of_direction_eq _\n    \u27e8p, mem_affineSpan k (Set.mem_union_left _ (Set.mem_singleton _)), mem_top k V p\u27e9", "annotated_tactic": ["convert <a>ext_of_direction_eq</a> _\n      \u27e8p, <a>mem_affineSpan</a> k (<a>Set.mem_union_left</a> _ (<a>Set.mem_singleton</a> _)), <a>mem_top</a> k V p\u27e9", [{"full_name": "AffineSubspace.ext_of_direction_eq", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [354, 9], "def_end_pos": [354, 28]}, {"full_name": "mem_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [571, 9], "def_end_pos": [571, 23]}, {"full_name": "Set.mem_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}, {"full_name": "AffineSubspace.mem_top", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [751, 9], "def_end_pos": [751, 16]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 affineSpan k ({p} \u222a (fun v => v +\u1d65 p) '' s) = \u22a4", "state_after": "case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 (affineSpan k ({p} \u222a (fun v => v +\u1d65 p) '' s)).direction = \u22a4.direction"}, {"tactic": "rw [direction_affineSpan, direction_top,\n  vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p \u2208 _),\n  eq_top_iff, \u2190 h]", "annotated_tactic": ["rw [<a>direction_affineSpan</a>, <a>direction_top</a>,\n    <a>vectorSpan_eq_span_vsub_set_right</a> k (<a>Set.mem_union_left</a> _ (<a>Set.mem_singleton</a> _) : p \u2208 _),\n    <a>eq_top_iff</a>, \u2190 h]", [{"full_name": "direction_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [559, 9], "def_end_pos": [559, 29]}, {"full_name": "AffineSubspace.direction_top", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [759, 9], "def_end_pos": [759, 22]}, {"full_name": "vectorSpan_eq_span_vsub_set_right", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1053, 9], "def_end_pos": [1053, 42]}, {"full_name": "Set.mem_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [716, 9], "def_end_pos": [716, 23]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}, {"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 19]}]], "state_before": "case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 (affineSpan k ({p} \u222a (fun v => v +\u1d65 p) '' s)).direction = \u22a4.direction", "state_after": "case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 Submodule.span k (range Subtype.val) \u2264 Submodule.span k ((fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s))"}, {"tactic": "apply Submodule.span_mono", "annotated_tactic": ["apply <a>Submodule.span_mono</a>", [{"full_name": "Submodule.span_mono", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [85, 9], "def_end_pos": [85, 18]}]], "state_before": "case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 Submodule.span k (range Subtype.val) \u2264 Submodule.span k ((fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s))", "state_after": "case convert_1.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 range Subtype.val \u2286 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)"}, {"tactic": "rintro v \u27e8v', rfl\u27e9", "annotated_tactic": ["rintro v \u27e8v', rfl\u27e9", []], "state_before": "case convert_1.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 range Subtype.val \u2286 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)", "state_after": "case convert_1.h.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' \u2208 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)"}, {"tactic": "use (v' : V) +\u1d65 p", "annotated_tactic": ["use (v' : V) +\u1d65 p", []], "state_before": "case convert_1.h.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' \u2208 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)", "state_after": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' +\u1d65 p \u2208 {p} \u222a (fun v => v +\u1d65 p) '' s \u2227 (fun x => x -\u1d65 p) (\u2191v' +\u1d65 p) = \u2191v'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' +\u1d65 p \u2208 {p} \u222a (fun v => v +\u1d65 p) '' s \u2227 (fun x => x -\u1d65 p) (\u2191v' +\u1d65 p) = \u2191v'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Density.lean", "full_name": "MeasureTheory.pdf.lintegral_eq_measure_univ", "start": [208, 1], "end": [211, 97], "traced_tactics": [{"tactic": "rw [\u2190 setLIntegral_univ, \u2190 map_eq_setLIntegral_pdf X \u2119 \u03bc MeasurableSet.univ,\n  map_apply_of_aemeasurable (HasPDF.aemeasurable X \u2119 \u03bc) MeasurableSet.univ, Set.preimage_univ]", "annotated_tactic": ["rw [\u2190 <a>setLIntegral_univ</a>, \u2190 <a>map_eq_setLIntegral_pdf</a> X \u2119 \u03bc <a>MeasurableSet.univ</a>,\n    <a>map_apply_of_aemeasurable</a> (<a>HasPDF.aemeasurable</a> X \u2119 \u03bc) <a>MeasurableSet.univ</a>, <a>Set.preimage_univ</a>]", [{"full_name": "MeasureTheory.setLIntegral_univ", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [684, 9], "def_end_pos": [684, 26]}, {"full_name": "MeasureTheory.map_eq_setLIntegral_pdf", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [192, 9], "def_end_pos": [192, 32]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [103, 19], "def_end_pos": [103, 37]}, {"full_name": "MeasureTheory.Measure.map_apply_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 34]}, {"full_name": "MeasureTheory.HasPDF.aemeasurable", "def_path": "Mathlib/Probability/Density.lean", "def_pos": [90, 9], "def_end_pos": [90, 28]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [103, 19], "def_end_pos": [103, 37]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [63, 9], "def_end_pos": [63, 22]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : MeasurableSpace E\nm : MeasurableSpace \u03a9\n\u2119 : Measure \u03a9\n\u03bc : Measure E\nX : \u03a9 \u2192 E\ninst\u271d : HasPDF X \u2119 \u03bc\n\u22a2 \u222b\u207b (x : E), pdf X \u2119 \u03bc x \u2202\u03bc = \u2119 Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/AddTorsor.lean", "full_name": "vsub_eq_zero_iff_eq", "start": [136, 1], "end": [137, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "full_name": "CochainComplex.HomComplex.Cochain.id_comp", "start": [332, 1], "end": [336, 77], "traced_tactics": [{"tactic": "ext p q hpq", "annotated_tactic": ["ext p q hpq", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn\u271d m n : \u2124\nz\u2082 : Cochain F G n\n\u22a2 (ofHom (\ud835\udfd9 F)).comp z\u2082 \u22ef = z\u2082", "state_after": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn\u271d m n : \u2124\nz\u2082 : Cochain F G n\np q : \u2124\nhpq : p + n = q\n\u22a2 ((ofHom (\ud835\udfd9 F)).comp z\u2082 \u22ef).v p q hpq = z\u2082.v p q hpq"}, {"tactic": "simp only [zero_cochain_comp_v, ofHom_v, HomologicalComplex.id_f, id_comp]", "annotated_tactic": ["simp only [<a>zero_cochain_comp_v</a>, <a>ofHom_v</a>, <a>HomologicalComplex.id_f</a>, <a>id_comp</a>]", [{"full_name": "CochainComplex.HomComplex.Cochain.zero_cochain_comp_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [248, 7], "def_end_pos": [248, 26]}, {"full_name": "CochainComplex.HomComplex.Cochain.ofHom_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [167, 7], "def_end_pos": [167, 14]}, {"full_name": "HomologicalComplex.id_f", "def_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "def_pos": [275, 9], "def_end_pos": [275, 13]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn\u271d m n : \u2124\nz\u2082 : Cochain F G n\np q : \u2124\nhpq : p + n = q\n\u22a2 ((ofHom (\ud835\udfd9 F)).comp z\u2082 \u22ef).v p q hpq = z\u2082.v p q hpq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.val_eq_singleton_iff", "start": [698, 1], "end": [700, 6], "traced_tactics": [{"tactic": "rw [\u2190 val_inj]", "annotated_tactic": ["rw [\u2190 <a>val_inj</a>]", [{"full_name": "Finset.val_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s.val = {a} \u2194 s = {a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s.val = {a} \u2194 s.val = {a}.val"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s.val = {a} \u2194 s.val = {a}.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "BooleanAlgebra.le_iff_atom_le_imp", "start": [371, 1], "end": [380, 71], "traced_tactics": [{"tactic": "have : x \u2293 y\u1d9c = \u22a5 := of_not_not fun hbot =>\n  have \u27e8a, ha, hle\u27e9 := (eq_bot_or_exists_atom_le _).resolve_left hbot\n  have \u27e8hx, hy'\u27e9 := le_inf_iff.1 hle\n  have hy := h a ha hx\n  have : a \u2264 y \u2293 y\u1d9c := le_inf_iff.2 \u27e8hy, hy'\u27e9\n  ha.1 (by simpa using this)", "annotated_tactic": ["have : x \u2293 y\u1d9c = \u22a5 := <a>of_not_not</a> fun hbot =>\n    have \u27e8a, ha, hle\u27e9 := (<a>eq_bot_or_exists_atom_le</a> _).<a>resolve_left</a> hbot\n    have \u27e8hx, hy'\u27e9 := <a>le_inf_iff</a>.1 hle\n    have hy := h a ha hx\n    have : a \u2264 y \u2293 y\u1d9c := <a>le_inf_iff</a>.2 \u27e8hy, hy'\u27e9\n    ha.1 (by simpa using this)", [{"full_name": "of_not_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 19]}, {"full_name": "IsAtomic.eq_bot_or_exists_atom_le", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [265, 3], "def_end_pos": [265, 27]}, {"full_name": "Or.resolve_left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [555, 9], "def_end_pos": [555, 24]}, {"full_name": "le_inf_iff", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [394, 9], "def_end_pos": [394, 19]}, {"full_name": "le_inf_iff", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [394, 9], "def_end_pos": [394, 19]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : PartialOrder \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nh : \u2200 (a : \u03b1), IsAtom a \u2192 a \u2264 x \u2192 a \u2264 y\n\u22a2 x \u2264 y", "state_after": "\u03b9 : Sort u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : PartialOrder \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nh : \u2200 (a : \u03b1), IsAtom a \u2192 a \u2264 x \u2192 a \u2264 y\nthis : x \u2293 y\u1d9c = \u22a5\n\u22a2 x \u2264 y"}, {"tactic": "exact (eq_compl_iff_isCompl.1 (by simp)).inf_right_eq_bot_iff.1 this", "annotated_tactic": ["exact (<a>eq_compl_iff_isCompl</a>.1 (by simp)).<a>inf_right_eq_bot_iff</a>.1 this", [{"full_name": "eq_compl_iff_isCompl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 29]}, {"full_name": "IsCompl.inf_right_eq_bot_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [544, 9], "def_end_pos": [544, 29]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : PartialOrder \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nh : \u2200 (a : \u03b1), IsAtom a \u2192 a \u2264 x \u2192 a \u2264 y\nthis : x \u2293 y\u1d9c = \u22a5\n\u22a2 x \u2264 y", "state_after": "no goals"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "\u03b9 : Sort u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : PartialOrder \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nh : \u2200 (a : \u03b1), IsAtom a \u2192 a \u2264 x \u2192 a \u2264 y\nhbot : \u00acx \u2293 y\u1d9c = \u22a5\na : \u03b1\nha : IsAtom a\nhle : a \u2264 x \u2293 y\u1d9c\nhx : a \u2264 x\nhy' : a \u2264 y\u1d9c\nhy : a \u2264 y\nthis : a \u2264 y \u2293 y\u1d9c\n\u22a2 a = \u22a5", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Sort u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : PartialOrder \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nh : \u2200 (a : \u03b1), IsAtom a \u2192 a \u2264 x \u2192 a \u2264 y\nthis : x \u2293 y\u1d9c = \u22a5\n\u22a2 y = y\u1d9c\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.dmuld", "start": [319, 1], "end": [319, 71], "traced_tactics": [{"tactic": "ext <;> simp", "annotated_tactic": ["ext <;> simp", []], "state_before": "d : \u2124\n\u22a2 sqrtd * sqrtd = \u2191d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "round_neg_two_inv", "start": [1593, 1], "end": [1594, 60], "traced_tactics": [{"tactic": "simp only [round_eq, \u2190 one_div, add_left_neg, floor_zero]", "annotated_tactic": ["simp only [<a>round_eq</a>, \u2190 <a>one_div</a>, <a>add_left_neg</a>, <a>floor_zero</a>]", [{"full_name": "round_eq", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1565, 9], "def_end_pos": [1565, 17]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "add_left_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1221, 3], "def_end_pos": [1221, 14]}, {"full_name": "Int.floor_zero", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [748, 9], "def_end_pos": [748, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round (-2\u207b\u00b9) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_ne_zero", "start": [1418, 1], "end": [1419, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/DirectLimit.lean", "full_name": "TensorProduct.toDirectLimit_tmul_of", "start": [65, 1], "end": [69, 6], "traced_tactics": [{"tactic": "rw [toDirectLimit, lift.tmul, lift_of]", "annotated_tactic": ["rw [<a>toDirectLimit</a>, <a>lift.tmul</a>, <a>lift_of</a>]", [{"full_name": "TensorProduct.toDirectLimit", "def_path": "Mathlib/LinearAlgebra/TensorProduct/DirectLimit.lean", "def_pos": [58, 19], "def_end_pos": [58, 32]}, {"full_name": "TensorProduct.lift.tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [556, 9], "def_end_pos": [556, 18]}, {"full_name": "Module.DirectLimit.lift_of", "def_path": "Mathlib/Algebra/DirectLimit.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (G i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192\u2097[R] G j\nM : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u03b9\ng : G i\nm : M\n\u22a2 (toDirectLimit f M) ((of R \u03b9 G f i) g \u2297\u209c[R] m) =\n    (of R \u03b9 (fun x => G x \u2297[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g \u2297\u209c[R] m)", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (G i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192\u2097[R] G j\nM : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u03b9\ng : G i\nm : M\n\u22a2 (((mk R (G i) M).compr\u2082 (of R \u03b9 (fun _i => G _i \u2297[R] M) (fun _i _j h => LinearMap.rTensor M (f _i _j h)) i)) g) m =\n    (of R \u03b9 (fun x => G x \u2297[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g \u2297\u209c[R] m)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type u_3\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (G i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192\u2097[R] G j\nM : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u03b9\ng : G i\nm : M\n\u22a2 (((mk R (G i) M).compr\u2082 (of R \u03b9 (fun _i => G _i \u2297[R] M) (fun _i _j h => LinearMap.rTensor M (f _i _j h)) i)) g) m =\n    (of R \u03b9 (fun x => G x \u2297[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g \u2297\u209c[R] m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean", "full_name": "CategoryTheory.Limits.\u03b9_colimitConstInitial_hom", "start": [520, 1], "end": [523, 35], "traced_tactics": [{"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C\u271d : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u271d\nJ : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} J\nC : Type u_2\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : HasInitial C\nj : J\n\u22a2 colimit.\u03b9 ((Functor.const J).obj (\u22a5_ C)) j \u226b colimitConstInitial.hom = initial.to (\u22a5_ C)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.map_op_mul", "start": [276, 1], "end": [288, 28], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 map (\u2191(opLinearEquiv R)) (M * N) = map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 map (\u2191(opLinearEquiv R)) (M * N) \u2264 map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M\n\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M \u2264 map (\u2191(opLinearEquiv R)) (M * N)"}, {"tactic": "simp_rw [map_le_iff_le_comap]", "annotated_tactic": ["simp_rw [<a>map_le_iff_le_comap</a>]", [{"full_name": "Submodule.map_le_iff_le_comap", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [224, 9], "def_end_pos": [224, 28]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 map (\u2191(opLinearEquiv R)) (M * N) \u2264 map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 M * N \u2264 comap (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M)"}, {"tactic": "refine mul_le.2 fun m hm n hn => ?_", "annotated_tactic": ["refine <a>mul_le</a>.2 fun m hm n hn => ?_", [{"full_name": "Submodule.mul_le", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [164, 9], "def_end_pos": [164, 15]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 M * N \u2264 comap (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M)", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 m * n \u2208 comap (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M)"}, {"tactic": "rw [mem_comap, map_equiv_eq_comap_symm, map_equiv_eq_comap_symm]", "annotated_tactic": ["rw [<a>mem_comap</a>, <a>map_equiv_eq_comap_symm</a>, <a>map_equiv_eq_comap_symm</a>]", [{"full_name": "Submodule.mem_comap", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [195, 9], "def_end_pos": [195, 18]}, {"full_name": "Submodule.map_equiv_eq_comap_symm", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [552, 9], "def_end_pos": [552, 32]}, {"full_name": "Submodule.map_equiv_eq_comap_symm", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [552, 9], "def_end_pos": [552, 32]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 m * n \u2208 comap (\u2191(opLinearEquiv R)) (map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M)", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 \u2191(opLinearEquiv R) (m * n) \u2208 comap (\u2191(opLinearEquiv R).symm) N * comap (\u2191(opLinearEquiv R).symm) M"}, {"tactic": "show op n * op m \u2208 _", "annotated_tactic": ["show <a>op</a> n * <a>op</a> m \u2208 _", [{"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}, {"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 \u2191(opLinearEquiv R) (m * n) \u2208 comap (\u2191(opLinearEquiv R).symm) N * comap (\u2191(opLinearEquiv R).symm) M", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 op n * op m \u2208 comap (\u2191(opLinearEquiv R).symm) N * comap (\u2191(opLinearEquiv R).symm) M"}, {"tactic": "exact mul_mem_mul hn hm", "annotated_tactic": ["exact <a>mul_mem_mul</a> hn hm", [{"full_name": "Submodule.mul_mem_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : m \u2208 M\nn : A\nhn : n \u2208 N\n\u22a2 op n * op m \u2208 comap (\u2191(opLinearEquiv R).symm) N * comap (\u2191(opLinearEquiv R).symm) M", "state_after": "no goals"}, {"tactic": "refine mul_le.2 (MulOpposite.rec' fun m hm => MulOpposite.rec' fun n hn => ?_)", "annotated_tactic": ["refine <a>mul_le</a>.2 (<a>MulOpposite.rec'</a> fun m hm => <a>MulOpposite.rec'</a> fun n hn => ?_)", [{"full_name": "Submodule.mul_le", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [164, 9], "def_end_pos": [164, 15]}, {"full_name": "MulOpposite.rec'", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [116, 15], "def_end_pos": [116, 19]}, {"full_name": "MulOpposite.rec'", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [116, 15], "def_end_pos": [116, 19]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 map (\u2191(opLinearEquiv R)) N * map (\u2191(opLinearEquiv R)) M \u2264 map (\u2191(opLinearEquiv R)) (M * N)", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : op m \u2208 map (\u2191(opLinearEquiv R)) N\nn : A\nhn : op n \u2208 map (\u2191(opLinearEquiv R)) M\n\u22a2 op m * op n \u2208 map (\u2191(opLinearEquiv R)) (M * N)"}, {"tactic": "rw [Submodule.mem_map_equiv] at hm hn \u22a2", "annotated_tactic": ["rw [<a>Submodule.mem_map_equiv</a>] at hm hn \u22a2", [{"full_name": "Submodule.mem_map_equiv", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [543, 9], "def_end_pos": [543, 22]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : op m \u2208 map (\u2191(opLinearEquiv R)) N\nn : A\nhn : op n \u2208 map (\u2191(opLinearEquiv R)) M\n\u22a2 op m * op n \u2208 map (\u2191(opLinearEquiv R)) (M * N)", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : (opLinearEquiv R).symm (op m) \u2208 N\nn : A\nhn : (opLinearEquiv R).symm (op n) \u2208 M\n\u22a2 (opLinearEquiv R).symm (op m * op n) \u2208 M * N"}, {"tactic": "exact mul_mem_mul hn hm", "annotated_tactic": ["exact <a>mul_mem_mul</a> hn hm", [{"full_name": "Submodule.mul_mem_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm\u271d n\u271d m : A\nhm : (opLinearEquiv R).symm (op m) \u2208 N\nn : A\nhn : (opLinearEquiv R).symm (op n) \u2208 M\n\u22a2 (opLinearEquiv R).symm (op m * op n) \u2208 M * N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.biprod.isoProd_hom", "start": [1736, 1], "end": [1738, 36], "traced_tactics": [{"tactic": "ext <;> simp [biprod.isoProd]", "annotated_tactic": ["ext <;> simp [<a>biprod.isoProd</a>]", [{"full_name": "CategoryTheory.Limits.biprod.isoProd", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [1731, 5], "def_end_pos": [1731, 19]}]], "state_before": "J : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nD : Type uD\ninst\u271d\u00b2 : Category.{uD', uD} D\ninst\u271d\u00b9 : HasZeroMorphisms D\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoProd X Y).hom = prod.lift fst snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Logic.lean", "full_name": "left_comm", "start": [539, 1], "end": [544, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.biproduct.bicone_\u03b9", "start": [495, 1], "end": [496, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Final.lean", "full_name": "CategoryTheory.Functor.initial_of_comp_full_faithful'", "start": [774, 1], "end": [776, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/DiscreteCategory.lean", "full_name": "CategoryTheory.Discrete.natIso_app", "start": [234, 1], "end": [235, 71], "traced_tactics": [{"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "\u03b1 : Type u\u2081\nC : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} C\nI : Type u\u2081\nF G : Discrete I \u2964 C\nf : (i : Discrete I) \u2192 F.obj i \u2245 G.obj i\ni : Discrete I\n\u22a2 (natIso f).app i = f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.inv_mem_inv", "start": [746, 1], "end": [747, 24], "traced_tactics": [{"tactic": "simp [inv_def]", "annotated_tactic": ["simp [<a>inv_def</a>]", [{"full_name": "Part.inv_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [702, 9], "def_end_pos": [702, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 ma\u207b\u00b9 \u2208 a\u207b\u00b9", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 \u2203 a_1 \u2208 a, a_1\u207b\u00b9 = ma\u207b\u00b9"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inv \u03b1\na : Part \u03b1\nma : \u03b1\nha : ma \u2208 a\n\u22a2 \u2203 a_1 \u2208 a, a_1\u207b\u00b9 = ma\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.conj_refl", "start": [588, 9], "end": [588, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.fract_sub_nat", "start": [936, 1], "end": [938, 7], "traced_tactics": [{"tactic": "rw [fract]", "annotated_tactic": ["rw [<a>fract</a>]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 fract (a - \u2191n) = fract a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 a - \u2191n - \u2191\u230aa - \u2191n\u230b = fract a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 a - \u2191n - \u2191\u230aa - \u2191n\u230b = fract a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Trunc.ind", "start": [492, 1], "end": [493, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean", "full_name": "pow_lt_one'", "start": [82, 1], "end": [83, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.le_of_forall_lt", "start": [617, 1], "end": [619, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "HasStrictFDerivAt.exists_lipschitzOnWith", "start": [481, 1], "end": [483, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "full_name": "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq", "start": [548, 1], "end": [549, 60], "traced_tactics": [{"tactic": "rw [\u2190 cancel_mono f, hl, lift_fac]", "annotated_tactic": ["rw [\u2190 <a>cancel_mono</a> f, hl, <a>lift_fac</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac", "def_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "def_pos": [544, 9], "def_end_pos": [544, 17]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X \u27f6 Z\nhf : IsOpenImmersion f\ng : Y \u27f6 Z\ns : PullbackCone f g\nH : Set.range \u21d1g.base \u2286 Set.range \u21d1f.base\nl : Y \u27f6 X\nhl : l \u226b f = g\n\u22a2 l = lift f g H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.LeftInvOn.mono", "start": [1208, 1], "end": [1209, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.Tendsto.of_tendsto_comp", "start": [3151, 1], "end": [3156, 64], "traced_tactics": [{"tactic": "rw [tendsto_iff_comap] at hfg \u22a2", "annotated_tactic": ["rw [<a>tendsto_iff_comap</a>] at hfg \u22a2", [{"full_name": "Filter.tendsto_iff_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3058, 9], "def_end_pos": [3058, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na : Filter \u03b1\nb : Filter \u03b2\nc : Filter \u03b3\nhfg : Tendsto (g \u2218 f) a c\nhg : comap g c \u2264 b\n\u22a2 Tendsto f a b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na : Filter \u03b1\nb : Filter \u03b2\nc : Filter \u03b3\nhfg : a \u2264 comap (g \u2218 f) c\nhg : comap g c \u2264 b\n\u22a2 a \u2264 comap f b"}, {"tactic": "calc\n  a \u2264 comap (g \u2218 f) c := hfg\n  _ \u2264 comap f b := by simpa [comap_comap] using comap_mono hg", "annotated_tactic": ["calc\n    a \u2264 <a>comap</a> (g \u2218 f) c := hfg\n    _ \u2264 <a>comap</a> f b := by simpa [<a>comap_comap</a>] using <a>comap_mono</a> hg", [{"full_name": "Filter.comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1997, 5], "def_end_pos": [1997, 10]}, {"full_name": "Filter.comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1997, 5], "def_end_pos": [1997, 10]}, {"full_name": "Filter.comap_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2245, 9], "def_end_pos": [2245, 20]}, {"full_name": "Filter.comap_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2366, 9], "def_end_pos": [2366, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na : Filter \u03b1\nb : Filter \u03b2\nc : Filter \u03b3\nhfg : a \u2264 comap (g \u2218 f) c\nhg : comap g c \u2264 b\n\u22a2 a \u2264 comap f b", "state_after": "no goals"}, {"tactic": "simpa [comap_comap] using comap_mono hg", "annotated_tactic": ["simpa [<a>comap_comap</a>] using <a>comap_mono</a> hg", [{"full_name": "Filter.comap_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2245, 9], "def_end_pos": [2245, 20]}, {"full_name": "Filter.comap_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2366, 9], "def_end_pos": [2366, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na : Filter \u03b1\nb : Filter \u03b2\nc : Filter \u03b3\nhfg : a \u2264 comap (g \u2218 f) c\nhg : comap g c \u2264 b\n\u22a2 comap (g \u2218 f) c \u2264 comap f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Deriv.lean", "full_name": "ConvexOn.slope_le_deriv", "start": [508, 1], "end": [512, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.eval_comp", "start": [1130, 1], "end": [1135, 9], "traced_tactics": [{"tactic": "induction p using Polynomial.induction_on' with\n| h_add r s hr hs =>\n  simp [add_comp, hr, hs]\n| h_monomial n a =>\n  simp", "annotated_tactic": ["induction p using <a>Polynomial.induction_on'</a> with\n  | h_add r s hr hs =>\n    simp [<a>add_comp</a>, hr, hs]\n  | h_monomial n a =>\n    simp", [{"full_name": "Polynomial.induction_on'", "def_path": "Mathlib/Algebra/Polynomial/Induction.lean", "def_pos": [63, 19], "def_end_pos": [63, 32]}, {"full_name": "Polynomial.add_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [589, 9], "def_end_pos": [589, 17]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\n\u22a2 eval x (p.comp q) = eval (eval x q) p", "state_after": "no goals"}, {"tactic": "simp [add_comp, hr, hs]", "annotated_tactic": ["simp [<a>add_comp</a>, hr, hs]", [{"full_name": "Polynomial.add_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [589, 9], "def_end_pos": [589, 17]}]], "state_before": "case h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nr s : R[X]\nhr : eval x (r.comp q) = eval (eval x q) r\nhs : eval x (s.comp q) = eval (eval x q) s\n\u22a2 eval x ((r + s).comp q) = eval (eval x q) (r + s)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 eval x (((monomial n) a).comp q) = eval (eval x q) ((monomial n) a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "LinearMap.adjoint_adjoint", "start": [398, 1], "end": [401, 47], "traced_tactics": [{"tactic": "ext v", "annotated_tactic": ["ext v", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : RCLike \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c F\ninst\u271d : FiniteDimensional \ud835\udd5c G\nA : E \u2192\u2097[\ud835\udd5c] F\n\u22a2 adjoint (adjoint A) = A", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : RCLike \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c F\ninst\u271d : FiniteDimensional \ud835\udd5c G\nA : E \u2192\u2097[\ud835\udd5c] F\nv : E\n\u22a2 (adjoint (adjoint A)) v = A v"}, {"tactic": "refine ext_inner_left \ud835\udd5c fun w => ?_", "annotated_tactic": ["refine <a>ext_inner_left</a> \ud835\udd5c fun w => ?_", [{"full_name": "ext_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 23]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : RCLike \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c F\ninst\u271d : FiniteDimensional \ud835\udd5c G\nA : E \u2192\u2097[\ud835\udd5c] F\nv : E\n\u22a2 (adjoint (adjoint A)) v = A v", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : RCLike \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c F\ninst\u271d : FiniteDimensional \ud835\udd5c G\nA : E \u2192\u2097[\ud835\udd5c] F\nv : E\nw : F\n\u22a2 \u27eaw, (adjoint (adjoint A)) v\u27eb_\ud835\udd5c = \u27eaw, A v\u27eb_\ud835\udd5c"}, {"tactic": "rw [adjoint_inner_right, adjoint_inner_left]", "annotated_tactic": ["rw [<a>adjoint_inner_right</a>, <a>adjoint_inner_left</a>]", [{"full_name": "LinearMap.adjoint_inner_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [389, 9], "def_end_pos": [389, 28]}, {"full_name": "LinearMap.adjoint_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [381, 9], "def_end_pos": [381, 27]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : RCLike \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c F\ninst\u271d : FiniteDimensional \ud835\udd5c G\nA : E \u2192\u2097[\ud835\udd5c] F\nv : E\nw : F\n\u22a2 \u27eaw, (adjoint (adjoint A)) v\u27eb_\ud835\udd5c = \u27eaw, A v\u27eb_\ud835\udd5c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measurePreserving_prod_inv_mul", "start": [127, 1], "end": [129, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Ergodic/Function.lean", "full_name": "Ergodic.ae_eq_const_of_ae_eq_comp_ae", "start": [99, 1], "end": [101, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Interval.lean", "full_name": "Finset.Iic_eq_powerset", "start": [70, 1], "end": [71, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "contDiff_top_iff_deriv", "start": [2104, 1], "end": [2107, 56], "traced_tactics": [{"tactic": "simp only [\u2190 contDiffOn_univ, \u2190 differentiableOn_univ, \u2190 derivWithin_univ]", "annotated_tactic": ["simp only [\u2190 <a>contDiffOn_univ</a>, \u2190 <a>differentiableOn_univ</a>, \u2190 <a>derivWithin_univ</a>]", [{"full_name": "contDiffOn_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1455, 9], "def_end_pos": [1455, 24]}, {"full_name": "differentiableOn_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [665, 9], "def_end_pos": [665, 30]}, {"full_name": "derivWithin_univ", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u2082 : \ud835\udd5c \u2192 F\ns\u2082 : Set \ud835\udd5c\n\u22a2 ContDiff \ud835\udd5c \u22a4 f\u2082 \u2194 Differentiable \ud835\udd5c f\u2082 \u2227 ContDiff \ud835\udd5c \u22a4 (deriv f\u2082)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u2082 : \ud835\udd5c \u2192 F\ns\u2082 : Set \ud835\udd5c\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f\u2082 univ \u2194 DifferentiableOn \ud835\udd5c f\u2082 univ \u2227 ContDiffOn \ud835\udd5c \u22a4 (derivWithin f\u2082 univ) univ"}, {"tactic": "rw [contDiffOn_top_iff_derivWithin uniqueDiffOn_univ]", "annotated_tactic": ["rw [<a>contDiffOn_top_iff_derivWithin</a> <a>uniqueDiffOn_univ</a>]", [{"full_name": "contDiffOn_top_iff_derivWithin", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 39]}, {"full_name": "uniqueDiffOn_univ", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [244, 9], "def_end_pos": [244, 26]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u2082 : \ud835\udd5c \u2192 F\ns\u2082 : Set \ud835\udd5c\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f\u2082 univ \u2194 DifferentiableOn \ud835\udd5c f\u2082 univ \u2227 ContDiffOn 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R\nx\u271d : M\nb' : Basis \u03b9' R M'\ne : \u03b9 \u2243 \u03b9'\nx : M\n\u22a2 (b.reindex e).sumCoords x = b.sumCoords x"}, {"tactic": "simp only [coe_sumCoords, repr_reindex]", "annotated_tactic": ["simp only [<a>coe_sumCoords</a>, <a>repr_reindex</a>]", [{"full_name": "Basis.coe_sumCoords", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [227, 9], "def_end_pos": [227, 22]}, {"full_name": "Basis.repr_reindex", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [433, 9], "def_end_pos": [433, 21]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb b\u2081 : Basis \u03b9 R M\ni : \u03b9\nc : R\nx\u271d : M\nb' : Basis \u03b9' R M'\ne : \u03b9 \u2243 \u03b9'\nx : M\n\u22a2 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"full_name": "LieModuleHom.congr_fun", "start": [819, 1], "end": [820, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Exponent.lean", "full_name": "Monoid.exponent_dvd", "start": [240, 1], "end": [241, 75], "traced_tactics": [{"tactic": "simp_rw [exponent_dvd_iff_forall_pow_eq_one, orderOf_dvd_iff_pow_eq_one]", "annotated_tactic": ["simp_rw [<a>exponent_dvd_iff_forall_pow_eq_one</a>, <a>orderOf_dvd_iff_pow_eq_one</a>]", [{"full_name": "Monoid.exponent_dvd_iff_forall_pow_eq_one", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [217, 9], "def_end_pos": [217, 43]}, {"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [272, 9], "def_end_pos": [272, 35]}]], "state_before": "G : Type u\ninst\u271d : Monoid G\nn : \u2115\n\u22a2 exponent G \u2223 n \u2194 \u2200 (g : G), orderOf g \u2223 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_lt_Icc", "start": [223, 1], "end": [224, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/JordanHolder.lean", "full_name": "JordanHolderLattice.isMaximal_inf_right_of_isMaximal_sup", "start": [102, 1], "end": [106, 52], "traced_tactics": [{"tactic": "rw [inf_comm]", "annotated_tactic": ["rw [<a>inf_comm</a>]", [{"full_name": "inf_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [461, 9], "def_end_pos": [461, 17]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (x \u2294 y)\nhyz : IsMaximal y (x \u2294 y)\n\u22a2 IsMaximal (x \u2293 y) y", "state_after": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (x \u2294 y)\nhyz : IsMaximal y (x \u2294 y)\n\u22a2 IsMaximal (y \u2293 x) y"}, {"tactic": "rw [sup_comm] at hxz hyz", "annotated_tactic": ["rw [<a>sup_comm</a>] at hxz hyz", [{"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (x \u2294 y)\nhyz : IsMaximal y (x \u2294 y)\n\u22a2 IsMaximal (y \u2293 x) y", "state_after": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (y \u2294 x)\nhyz : IsMaximal y (y \u2294 x)\n\u22a2 IsMaximal (y \u2293 x) y"}, {"tactic": "exact isMaximal_inf_left_of_isMaximal_sup hyz hxz", "annotated_tactic": ["exact <a>isMaximal_inf_left_of_isMaximal_sup</a> hyz hxz", [{"full_name": "JordanHolderLattice.isMaximal_inf_left_of_isMaximal_sup", "def_path": "Mathlib/Order/JordanHolder.lean", "def_pos": [90, 3], "def_end_pos": [90, 38]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (y \u2294 x)\nhyz : IsMaximal y (y \u2294 x)\n\u22a2 IsMaximal (y \u2293 x) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.LHom.setOf_realize_onFormula", "start": [706, 1], "end": [709, 7], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : L.Structure M\ninst\u271d\u00b3 : L.Structure N\ninst\u271d\u00b2 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ninst\u271d\u00b9 : L'.Structure M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : \u03c6.IsExpansionOn M\n\u03c8 : L.Formula \u03b1\n\u22a2 setOf (\u03c6.onFormula \u03c8).Realize = setOf \u03c8.Realize", "state_after": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : L.Structure M\ninst\u271d\u00b3 : L.Structure N\ninst\u271d\u00b2 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ninst\u271d\u00b9 : L'.Structure M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : \u03c6.IsExpansionOn M\n\u03c8 : L.Formula \u03b1\nx\u271d : \u03b1 \u2192 M\n\u22a2 x\u271d \u2208 setOf (\u03c6.onFormula \u03c8).Realize \u2194 x\u271d \u2208 setOf \u03c8.Realize"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : L.Structure M\ninst\u271d\u00b3 : L.Structure N\ninst\u271d\u00b2 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ninst\u271d\u00b9 : L'.Structure M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : \u03c6.IsExpansionOn M\n\u03c8 : L.Formula \u03b1\nx\u271d : \u03b1 \u2192 M\n\u22a2 x\u271d \u2208 setOf (\u03c6.onFormula \u03c8).Realize \u2194 x\u271d \u2208 setOf \u03c8.Realize", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BoundedOrder.lean", "full_name": "Pi.bot_apply", "start": [601, 1], "end": [602, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "full_name": "Collinear.oangle_sign_of_sameRay_vsub", "start": [687, 1], "end": [763, 72], "traced_tactics": [{"tactic": "by_cases hc\u2085\u2081\u2082 : Collinear \u211d ({p\u2085, p\u2081, p\u2082} : Set P)", "annotated_tactic": ["by_cases hc\u2085\u2081\u2082 : <a>Collinear</a> \u211d ({p\u2085, p\u2081, p\u2082} : <a>Set</a> P)", [{"full_name": "Collinear", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d ({p\u2085, p\u2081, p\u2082, p\u2083, p\u2084} : Set P) :=\n  (hc.collinear_insert_iff_of_ne (Set.mem_insert _ _)\n    (Set.mem_insert_of_mem _ (Set.mem_insert _ _)) hp\u2081p\u2082).2 hc\u2085\u2081\u2082", "annotated_tactic": ["have hc\u2085\u2081\u2082\u2083\u2084 : <a>Collinear</a> \u211d ({p\u2085, p\u2081, p\u2082, p\u2083, p\u2084} : <a>Set</a> P) :=\n      (hc.collinear_insert_iff_of_ne (<a>Set.mem_insert</a> _ _)\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _)) hp\u2081p\u2082).2 hc\u2085\u2081\u2082", [{"full_name": "Collinear", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hc\u2085\u2083\u2084 : Collinear \u211d ({p\u2085, p\u2083, p\u2084} : Set P) :=\n  (hc.collinear_insert_iff_of_ne\n    (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _)))\n    (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _\n      (Set.mem_singleton _)))) hp\u2083p\u2084).1 hc\u2085\u2081\u2082\u2083\u2084", "annotated_tactic": ["have hc\u2085\u2083\u2084 : <a>Collinear</a> \u211d ({p\u2085, p\u2083, p\u2084} : <a>Set</a> P) :=\n      (hc.collinear_insert_iff_of_ne\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _)))\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert_of_mem</a> _\n          (<a>Set.mem_singleton</a> _)))) hp\u2083p\u2084).1 hc\u2085\u2081\u2082\u2083\u2084", [{"full_name": "Collinear", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2085, p\u2083, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "rw [Set.insert_comm] at hc\u2085\u2081\u2082 hc\u2085\u2083\u2084", "annotated_tactic": ["rw [<a>Set.insert_comm</a>] at hc\u2085\u2081\u2082 hc\u2085\u2083\u2084", [{"full_name": "Set.insert_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1153, 9], "def_end_pos": [1153, 20]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2085, p\u2083, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hs\u2081\u2085\u2082 := oangle_eq_zero_or_eq_pi_iff_collinear.2 hc\u2085\u2081\u2082", "annotated_tactic": ["have hs\u2081\u2085\u2082 := <a>oangle_eq_zero_or_eq_pi_iff_collinear</a>.2 hc\u2085\u2081\u2082", [{"full_name": "EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [229, 9], "def_end_pos": [229, 46]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hs\u2083\u2085\u2084 := oangle_eq_zero_or_eq_pi_iff_collinear.2 hc\u2085\u2083\u2084", "annotated_tactic": ["have hs\u2083\u2085\u2084 := <a>oangle_eq_zero_or_eq_pi_iff_collinear</a>.2 hc\u2085\u2083\u2084", [{"full_name": "EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [229, 9], "def_end_pos": [229, 46]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\nhs\u2083\u2085\u2084 : \u2221 p\u2083 p\u2085 p\u2084 = 0 \u2228 \u2221 p\u2083 p\u2085 p\u2084 = \u2191\u03c0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "rw [\u2190 Real.Angle.sign_eq_zero_iff] at hs\u2081\u2085\u2082 hs\u2083\u2085\u2084", "annotated_tactic": ["rw [\u2190 <a>Real.Angle.sign_eq_zero_iff</a>] at hs\u2081\u2085\u2082 hs\u2083\u2085\u2084", [{"full_name": "Real.Angle.sign_eq_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [893, 9], "def_end_pos": [893, 25]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\nhs\u2083\u2085\u2084 : \u2221 p\u2083 p\u2085 p\u2084 = 0 \u2228 \u2221 p\u2083 p\u2085 p\u2084 = \u2191\u03c0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : (\u2221 p\u2081 p\u2085 p\u2082).sign = 0\nhs\u2083\u2085\u2084 : (\u2221 p\u2083 p\u2085 p\u2084).sign = 0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "rw [hs\u2081\u2085\u2082, hs\u2083\u2085\u2084]", "annotated_tactic": ["rw [hs\u2081\u2085\u2082, hs\u2083\u2085\u2084]", []], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : (\u2221 p\u2081 p\u2085 p\u2082).sign = 0\nhs\u2083\u2085\u2084 : (\u2221 p\u2083 p\u2085 p\u2084).sign = 0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "no goals"}, {"tactic": "let s : Set (P \u00d7 P \u00d7 P) :=\n  (fun x : line[\u211d, p\u2081, p\u2082] \u00d7 V => (x.1, p\u2085, x.2 +\u1d65 (x.1 : P))) ''\n    Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}", "annotated_tactic": ["let s : <a>Set</a> (P \u00d7 P \u00d7 P) :=\n      (fun x : line[\u211d, p\u2081, p\u2082] \u00d7 V => (x.1, p\u2085, x.2 +\u1d65 (x.1 : P))) ''\n        <a>Set.univ</a> \u00d7\u02e2 {v | <a>SameRay</a> \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "SameRay", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [40, 5], "def_end_pos": [40, 12]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hco : IsConnected s :=\n  haveI : ConnectedSpace line[\u211d, p\u2081, p\u2082] := AddTorsor.connectedSpace _ _\n  (isConnected_univ.prod (isConnected_setOf_sameRay_and_ne_zero\n    (vsub_ne_zero.2 hp\u2081p\u2082.symm))).image _\n    (continuous_fst.subtype_val.prod_mk (continuous_const.prod_mk\n      (continuous_snd.vadd continuous_fst.subtype_val))).continuousOn", "annotated_tactic": ["have hco : <a>IsConnected</a> s :=\n      haveI : <a>ConnectedSpace</a> line[\u211d, p\u2081, p\u2082] := <a>AddTorsor.connectedSpace</a> _ _\n      (isConnected_univ.prod (<a>isConnected_setOf_sameRay_and_ne_zero</a>\n        (<a>vsub_ne_zero</a>.2 hp\u2081p\u2082.symm))).<a>image</a> _\n        (continuous_fst.subtype_val.prod_mk (continuous_const.prod_mk\n          (continuous_snd.vadd continuous_fst.subtype_val))).<a>continuousOn</a>", [{"full_name": "IsConnected", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [58, 5], "def_end_pos": [58, 16]}, {"full_name": "ConnectedSpace", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [754, 7], "def_end_pos": [754, 21]}, {"full_name": "AddTorsor.connectedSpace", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [293, 19], "def_end_pos": [293, 43]}, {"full_name": "isConnected_setOf_sameRay_and_ne_zero", "def_path": "Mathlib/Analysis/Convex/Normed.lean", "def_pos": [151, 9], "def_end_pos": [151, 46]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}, {"full_name": "IsConnected.image", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [333, 19], "def_end_pos": [333, 36]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [968, 9], "def_end_pos": [968, 32]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s := by\n  simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,\n    Prod.ext_iff]\n  refine \u27e8\u27e8\u27e8p\u2081, left_mem_affineSpan_pair \u211d _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9,\n    \u27e8SameRay.rfl, vsub_ne_zero.2 hp\u2081p\u2082.symm\u27e9, ?_\u27e9\n  simp", "annotated_tactic": ["have hp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s := by\n      simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_setOf</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>,\n        <a>Prod.ext_iff</a>]\n      refine \u27e8\u27e8\u27e8p\u2081, <a>left_mem_affineSpan_pair</a> \u211d _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9,\n        \u27e8<a>SameRay.rfl</a>, <a>vsub_ne_zero</a>.2 hp\u2081p\u2082.symm\u27e9, ?_\u27e9\n      simp", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "left_mem_affineSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1313, 9], "def_end_pos": [1313, 33]}, {"full_name": "SameRay.rfl", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [79, 19], "def_end_pos": [79, 22]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "have hp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s := by\n  simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,\n    Prod.ext_iff]\n  refine \u27e8\u27e8\u27e8p\u2083, hc.mem_affineSpan_of_mem_of_ne (Set.mem_insert _ _)\n    (Set.mem_insert_of_mem _ (Set.mem_insert _ _))\n    (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _))) hp\u2081p\u2082\u27e9, p\u2084 -\u1d65 p\u2083\u27e9,\n    \u27e8hr, vsub_ne_zero.2 hp\u2083p\u2084.symm\u27e9, ?_\u27e9\n  simp", "annotated_tactic": ["have hp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s := by\n      simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_setOf</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>,\n        <a>Prod.ext_iff</a>]\n      refine \u27e8\u27e8\u27e8p\u2083, hc.mem_affineSpan_of_mem_of_ne (<a>Set.mem_insert</a> _ _)\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _))\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _))) hp\u2081p\u2082\u27e9, p\u2084 -\u1d65 p\u2083\u27e9,\n        \u27e8hr, <a>vsub_ne_zero</a>.2 hp\u2083p\u2084.symm\u27e9, ?_\u27e9\n      simp", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\nhp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign"}, {"tactic": "convert Real.Angle.sign_eq_of_continuousOn hco hf hsp hp\u2083p\u2084s hp\u2081p\u2082s", "annotated_tactic": ["convert <a>Real.Angle.sign_eq_of_continuousOn</a> hco hf hsp hp\u2083p\u2084s hp\u2081p\u2082s", [{"full_name": "Real.Angle.sign_eq_of_continuousOn", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [1052, 9], "def_end_pos": [1052, 32]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\nhp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s\n\u22a2 (\u2221 p\u2081 p\u2085 p\u2082).sign = (\u2221 p\u2083 p\u2085 p\u2084).sign", "state_after": "no goals"}, {"tactic": "refine ContinuousAt.continuousOn fun p hp => continuousAt_oangle ?_ ?_", "annotated_tactic": ["refine <a>ContinuousAt.continuousOn</a> fun p hp => <a>continuousAt_oangle</a> ?_ ?_", [{"full_name": "ContinuousAt.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [922, 9], "def_end_pos": [922, 34]}, {"full_name": "EuclideanGeometry.continuousAt_oangle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [49, 9], "def_end_pos": [49, 28]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s", "state_after": "case refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.1 \u2260 p.2.1\n\ncase refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.2.2 \u2260 p.2.1"}, {"tactic": "all_goals\n  simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp\n  obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n  dsimp only at hp \u22a2\n  obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n  dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n  obtain \u27e8hvr, -\u27e9 := hv\n  rintro rfl\n  refine hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan ?_).2 (collinear_pair _ _ _))", "annotated_tactic": ["all_goals\n        simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>, <a>Prod.ext_iff</a>] at hp\n        obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n        dsimp only at hp \u22a2\n        obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n        dsimp only [<a>Subtype.coe_mk</a>, <a>Set.mem_setOf</a>] at hv \u22a2\n        obtain \u27e8hvr, -\u27e9 := hv\n        rintro rfl\n        refine hc\u2085\u2081\u2082 ((<a>collinear_insert_iff_of_mem_affineSpan</a> ?_).2 (<a>collinear_pair</a> _ _ _))", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "collinear_insert_iff_of_mem_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [614, 9], "def_end_pos": [614, 47]}, {"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [493, 9], "def_end_pos": [493, 23]}]], "state_before": "case refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.1 \u2260 p.2.1\n\ncase refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.2.2 \u2260 p.2.1", "state_after": "case refine_1.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\ncase refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}"}, {"tactic": "simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp", "annotated_tactic": ["simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>, <a>Prod.ext_iff</a>] at hp", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "case refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.2.2 \u2260 p.2.1", "state_after": "case refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.1 = p.1 \u2227 p\u2085 = p.2.1 \u2227 x.2 +\u1d65 \u2191x.1 = p.2.2\n\u22a2 p.2.2 \u2260 p.2.1"}, {"tactic": "obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p", "annotated_tactic": ["obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p", []], "state_before": "case refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.1 = p.1 \u2227 p\u2085 = p.2.1 \u2227 x.2 +\u1d65 \u2191x.1 = p.2.2\n\u22a2 p.2.2 \u2260 p.2.1", "state_after": "case refine_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227\n      \u2191x.1 = (q\u2081, q\u2085, q\u2082).1 \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).2.1 \u2227 x.2 +\u1d65 \u2191x.1 = (q\u2081, q\u2085, q\u2082).2.2\n\u22a2 (q\u2081, q\u2085, q\u2082).2.2 \u2260 (q\u2081, q\u2085, q\u2082).2.1"}, {"tactic": "dsimp only at hp \u22a2", "annotated_tactic": ["dsimp only at hp \u22a2", []], "state_before": "case refine_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227\n      \u2191x.1 = (q\u2081, q\u2085, q\u2082).1 \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).2.1 \u2227 x.2 +\u1d65 \u2191x.1 = (q\u2081, q\u2085, q\u2082).2.2\n\u22a2 (q\u2081, q\u2085, q\u2082).2.2 \u2260 (q\u2081, q\u2085, q\u2082).2.1", "state_after": "case refine_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.1 = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.2 +\u1d65 \u2191x.1 = q\u2082\n\u22a2 q\u2082 \u2260 q\u2085"}, {"tactic": "obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp", "annotated_tactic": ["obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp", []], "state_before": "case refine_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, x.2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.1 = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.2 +\u1d65 \u2191x.1 = q\u2082\n\u22a2 q\u2082 \u2260 q\u2085", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : (\u27e8q, hq\u27e9, v).2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 (\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1 \u2260 p\u2085"}, {"tactic": "dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2", "annotated_tactic": ["dsimp only [<a>Subtype.coe_mk</a>, <a>Set.mem_setOf</a>] at hv \u22a2", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : (\u27e8q, hq\u27e9, v).2 \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 (\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1 \u2260 p\u2085", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : v \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 v +\u1d65 q \u2260 p\u2085"}, {"tactic": "obtain \u27e8hvr, -\u27e9 := hv", "annotated_tactic": ["obtain \u27e8hvr, -\u27e9 := hv", []], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : v \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 v +\u1d65 q \u2260 p\u2085", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\n\u22a2 v +\u1d65 q \u2260 p\u2085"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\n\u22a2 v +\u1d65 q \u2260 p\u2085", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 False"}, {"tactic": "refine hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan ?_).2 (collinear_pair _ _ _))", "annotated_tactic": ["refine hc\u2085\u2081\u2082 ((<a>collinear_insert_iff_of_mem_affineSpan</a> ?_).2 (<a>collinear_pair</a> _ _ _))", [{"full_name": "collinear_insert_iff_of_mem_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [614, 9], "def_end_pos": [614, 47]}, {"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [493, 9], "def_end_pos": [493, 23]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 False", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}"}, {"tactic": "exact hq", "annotated_tactic": ["exact hq", []], "state_before": "case refine_1.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 q \u2208 affineSpan \u211d {p\u2081, p\u2082}", "state_after": "no goals"}, {"tactic": "refine vadd_mem_of_mem_direction ?_ hq", "annotated_tactic": ["refine <a>vadd_mem_of_mem_direction</a> ?_ hq", [{"full_name": "AffineSubspace.vadd_mem_of_mem_direction", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [249, 9], "def_end_pos": [249, 34]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction"}, {"tactic": "rw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr", "annotated_tactic": ["rw [\u2190 <a>exists_nonneg_left_iff_sameRay</a> (<a>vsub_ne_zero</a>.2 hp\u2081p\u2082.symm)] at hvr", [{"full_name": "exists_nonneg_left_iff_sameRay", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [717, 9], "def_end_pos": [717, 39]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction"}, {"tactic": "obtain \u27e8r, -, rfl\u27e9 := hvr", "annotated_tactic": ["obtain \u27e8r, -, rfl\u27e9 := hvr", []], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, v +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.1, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction"}, {"tactic": "rw [direction_affineSpan]", "annotated_tactic": ["rw [<a>direction_affineSpan</a>]", [{"full_name": "direction_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [559, 9], "def_end_pos": [559, 29]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.1, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction", "state_after": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.1, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}"}, {"tactic": "exact smul_vsub_rev_mem_vectorSpan_pair _ _ _", "annotated_tactic": ["exact <a>smul_vsub_rev_mem_vectorSpan_pair</a> _ _ _", [{"full_name": "smul_vsub_rev_mem_vectorSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 42]}]], "state_before": "case refine_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.1, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\n\u22a2 \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0"}, {"tactic": "simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,\n  Prod.ext_iff] at hp", "annotated_tactic": ["simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_setOf</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>,\n        <a>Prod.ext_iff</a>] at hp", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p.1 \u2227 p\u2085 = p.2.1 \u2227 x.2 +\u1d65 \u2191x.1 = p.2.2\n\u22a2 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0"}, {"tactic": "obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p", "annotated_tactic": ["obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p.1 \u2227 p\u2085 = p.2.1 \u2227 x.2 +\u1d65 \u2191x.1 = p.2.2\n\u22a2 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0", "state_after": "case mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227\n      \u2191x.1 = (q\u2081, q\u2085, q\u2082).1 \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).2.1 \u2227 x.2 +\u1d65 \u2191x.1 = (q\u2081, q\u2085, q\u2082).2.2\n\u22a2 \u2221 (q\u2081, q\u2085, q\u2082).1 (q\u2081, q\u2085, q\u2082).2.1 (q\u2081, q\u2085, q\u2082).2.2 \u2260 0 \u2227 \u2221 (q\u2081, q\u2085, q\u2082).1 (q\u2081, q\u2085, q\u2082).2.1 (q\u2081, q\u2085, q\u2082).2.2 \u2260 \u2191\u03c0"}, {"tactic": "dsimp only at hp \u22a2", "annotated_tactic": ["dsimp only at hp \u22a2", []], "state_before": "case mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227\n      \u2191x.1 = (q\u2081, q\u2085, q\u2082).1 \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).2.1 \u2227 x.2 +\u1d65 \u2191x.1 = (q\u2081, q\u2085, q\u2082).2.2\n\u22a2 \u2221 (q\u2081, q\u2085, q\u2082).1 (q\u2081, q\u2085, q\u2082).2.1 (q\u2081, q\u2085, q\u2082).2.2 \u2260 0 \u2227 \u2221 (q\u2081, q\u2085, q\u2082).1 (q\u2081, q\u2085, q\u2082).2.1 (q\u2081, q\u2085, q\u2082).2.2 \u2260 \u2191\u03c0", "state_after": "case mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.2 +\u1d65 \u2191x.1 = q\u2082\n\u22a2 \u2221 q\u2081 q\u2085 q\u2082 \u2260 0 \u2227 \u2221 q\u2081 q\u2085 q\u2082 \u2260 \u2191\u03c0"}, {"tactic": "obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp", "annotated_tactic": ["obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp", []], "state_before": "case mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.2 +\u1d65 \u2191x.1 = q\u2082\n\u22a2 \u2221 q\u2081 q\u2085 q\u2082 \u2260 0 \u2227 \u2221 q\u2081 q\u2085 q\u2082 \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (\u27e8q, hq\u27e9, v).2 \u2227 (\u27e8q, hq\u27e9, v).2 \u2260 0\n\u22a2 \u2221 (\u2191(\u27e8q, hq\u27e9, v).1) p\u2085 ((\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1) \u2260 0 \u2227\n    \u2221 (\u2191(\u27e8q, hq\u27e9, v).1) p\u2085 ((\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1) \u2260 \u2191\u03c0"}, {"tactic": "dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2", "annotated_tactic": ["dsimp only [<a>Subtype.coe_mk</a>, <a>Set.mem_setOf</a>] at hv \u22a2", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (\u27e8q, hq\u27e9, v).2 \u2227 (\u27e8q, hq\u27e9, v).2 \u2260 0\n\u22a2 \u2221 (\u2191(\u27e8q, hq\u27e9, v).1) p\u2085 ((\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1) \u2260 0 \u2227\n    \u2221 (\u2191(\u27e8q, hq\u27e9, v).1) p\u2085 ((\u27e8q, hq\u27e9, v).2 +\u1d65 \u2191(\u27e8q, hq\u27e9, v).1) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0"}, {"tactic": "obtain \u27e8hvr, hv0\u27e9 := hv", "annotated_tactic": ["obtain \u27e8hvr, hv0\u27e9 := hv", []], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0"}, {"tactic": "rw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr", "annotated_tactic": ["rw [\u2190 <a>exists_nonneg_left_iff_sameRay</a> (<a>vsub_ne_zero</a>.2 hp\u2081p\u2082.symm)] at hvr", [{"full_name": "exists_nonneg_left_iff_sameRay", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [717, 9], "def_end_pos": [717, 39]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0"}, {"tactic": "obtain \u27e8r, -, rfl\u27e9 := hvr", "annotated_tactic": ["obtain \u27e8r, -, rfl\u27e9 := hvr", []], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0"}, {"tactic": "change q \u2208 line[\u211d, p\u2081, p\u2082] at hq", "annotated_tactic": ["change q \u2208 line[\u211d, p\u2081, p\u2082] at hq", []], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0"}, {"tactic": "rw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]", "annotated_tactic": ["rw [<a>oangle_ne_zero_and_ne_pi_iff_affineIndependent</a>]", [{"full_name": "EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [218, 9], "def_end_pos": [218, 55]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 AffineIndependent \u211d ![q, p\u2085, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q]"}, {"tactic": "refine affineIndependent_of_ne_of_mem_of_not_mem_of_mem ?_ hq\n    (fun h => hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan h).2 (collinear_pair _ _ _))) ?_", "annotated_tactic": ["refine <a>affineIndependent_of_ne_of_mem_of_not_mem_of_mem</a> ?_ hq\n          (fun h => hc\u2085\u2081\u2082 ((<a>collinear_insert_iff_of_mem_affineSpan</a> h).2 (<a>collinear_pair</a> _ _ _))) ?_", [{"full_name": "affineIndependent_of_ne_of_mem_of_not_mem_of_mem", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [718, 9], "def_end_pos": [718, 57]}, {"full_name": "collinear_insert_iff_of_mem_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [614, 9], "def_end_pos": [614, 47]}, {"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [493, 9], "def_end_pos": [493, 23]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 AffineIndependent \u211d ![q, p\u2085, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q]", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 q \u2260 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q\n\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}"}, {"tactic": "rwa [\u2190 @vsub_ne_zero V, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, neg_ne_zero]", "annotated_tactic": ["rwa [\u2190 @<a>vsub_ne_zero</a> V, <a>vsub_vadd_eq_vsub_sub</a>, <a>vsub_self</a>, <a>zero_sub</a>, <a>neg_ne_zero</a>]", [{"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}, {"full_name": "vsub_vadd_eq_vsub_sub", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [165, 9], "def_end_pos": [165, 30]}, {"full_name": "vsub_self", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [124, 9], "def_end_pos": [124, 18]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "neg_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [646, 3], "def_end_pos": [646, 14]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 q \u2260 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q", "state_after": "no goals"}, {"tactic": "refine vadd_mem_of_mem_direction ?_ hq", "annotated_tactic": ["refine <a>vadd_mem_of_mem_direction</a> ?_ hq", [{"full_name": "AffineSubspace.vadd_mem_of_mem_direction", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [249, 9], "def_end_pos": [249, 34]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction"}, {"tactic": "rw [direction_affineSpan]", "annotated_tactic": ["rw [<a>direction_affineSpan</a>]", [{"full_name": "direction_affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [559, 9], "def_end_pos": [559, 29]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 (affineSpan \u211d {p\u2081, p\u2082}).direction", "state_after": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}"}, {"tactic": "exact smul_vsub_rev_mem_vectorSpan_pair _ _ _", "annotated_tactic": ["exact <a>smul_vsub_rev_mem_vectorSpan_pair</a> _ _ _", [{"full_name": "smul_vsub_rev_mem_vectorSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 42]}]], "state_before": "case mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}", "state_after": "no goals"}, {"tactic": "simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,\n  Prod.ext_iff]", "annotated_tactic": ["simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_setOf</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>,\n        <a>Prod.ext_iff</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 (p\u2081, p\u2085, p\u2082) \u2208 s", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p\u2081 \u2227 True \u2227 x.2 +\u1d65 \u2191x.1 = p\u2082"}, {"tactic": "refine \u27e8\u27e8\u27e8p\u2081, left_mem_affineSpan_pair \u211d _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9,\n  \u27e8SameRay.rfl, vsub_ne_zero.2 hp\u2081p\u2082.symm\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8\u27e8p\u2081, <a>left_mem_affineSpan_pair</a> \u211d _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9,\n        \u27e8<a>SameRay.rfl</a>, <a>vsub_ne_zero</a>.2 hp\u2081p\u2082.symm\u27e9, ?_\u27e9", [{"full_name": "left_mem_affineSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1313, 9], "def_end_pos": [1313, 33]}, {"full_name": "SameRay.rfl", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [79, 19], "def_end_pos": [79, 22]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p\u2081 \u2227 True \u2227 x.2 +\u1d65 \u2191x.1 = p\u2082", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 \u2191(\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).1 = p\u2081 \u2227 True \u2227 (\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).2 +\u1d65 \u2191(\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).1 = p\u2082"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\n\u22a2 \u2191(\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).1 = p\u2081 \u2227 True \u2227 (\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).2 +\u1d65 \u2191(\u27e8p\u2081, \u22ef\u27e9, p\u2082 -\u1d65 p\u2081).1 = p\u2082", "state_after": "no goals"}, {"tactic": "simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,\n  Prod.ext_iff]", "annotated_tactic": ["simp_rw [s, <a>Set.mem_image</a>, <a>Set.mem_prod</a>, <a>Set.mem_setOf</a>, <a>Set.mem_univ</a>, <a>true_and_iff</a>,\n        <a>Prod.ext_iff</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 18]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}, {"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 (p\u2083, p\u2085, p\u2084) \u2208 s", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p\u2083 \u2227 True \u2227 x.2 +\u1d65 \u2191x.1 = p\u2084"}, {"tactic": "refine \u27e8\u27e8\u27e8p\u2083, hc.mem_affineSpan_of_mem_of_ne (Set.mem_insert _ _)\n  (Set.mem_insert_of_mem _ (Set.mem_insert _ _))\n  (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _))) hp\u2081p\u2082\u27e9, p\u2084 -\u1d65 p\u2083\u27e9,\n  \u27e8hr, vsub_ne_zero.2 hp\u2083p\u2084.symm\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8\u27e8p\u2083, hc.mem_affineSpan_of_mem_of_ne (<a>Set.mem_insert</a> _ _)\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _))\n        (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_insert</a> _ _))) hp\u2081p\u2082\u27e9, p\u2084 -\u1d65 p\u2083\u27e9,\n        \u27e8hr, <a>vsub_ne_zero</a>.2 hp\u2083p\u2084.symm\u27e9, ?_\u27e9", [{"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}, {"full_name": "vsub_ne_zero", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [140, 9], "def_end_pos": [140, 21]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.2 \u2227 x.2 \u2260 0) \u2227 \u2191x.1 = p\u2083 \u2227 True \u2227 x.2 +\u1d65 \u2191x.1 = p\u2084", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2191(\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).1 = p\u2083 \u2227 True \u2227 (\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).2 +\u1d65 \u2191(\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).1 = p\u2084"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.1, p\u2085, x.2 +\u1d65 \u2191x.1)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.1 p.2.1 p.2.2) s\nhsp : \u2200 p \u2208 s, \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2191(\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).1 = p\u2083 \u2227 True \u2227 (\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).2 +\u1d65 \u2191(\u27e8p\u2083, \u22ef\u27e9, p\u2084 -\u1d65 p\u2083).1 = p\u2084", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "Substring.ValidFor.stopPos", "start": [785, 1], "end": [786, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Nonneg/Ring.lean", "full_name": "Nonneg.coe_mul", "start": [206, 11], "end": [208, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasureTheory.Measure.InnerRegularWRT.of_restrict", "start": [558, 1], "end": [573, 58], "traced_tactics": [{"tactic": "intro F hF r hr", "annotated_tactic": ["intro F hF r hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\n\u22a2 \u03bc.InnerRegularWRT p MeasurableSet", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "have hBU : \u22c3 n, F \u2229 s n = F := by  rw [\u2190 inter_iUnion, univ_subset_iff.mp hs, inter_univ]", "annotated_tactic": ["have hBU : \u22c3 n, F \u2229 s n = F := by  rw [\u2190 <a>inter_iUnion</a>, univ_subset_iff.mp hs, <a>inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [482, 9], "def_end_pos": [482, 21]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 9], "def_end_pos": [981, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "have : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n) := by\n  rw [\u2190 measure_iUnion_eq_iSup, hBU]\n  exact Monotone.directed_le fun m n h \u21a6 inter_subset_inter_right _ (hmono h)", "annotated_tactic": ["have : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n) := by\n    rw [\u2190 <a>measure_iUnion_eq_iSup</a>, hBU]\n    exact <a>Monotone.directed_le</a> fun m n h \u21a6 <a>inter_subset_inter_right</a> _ (hmono h)", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [470, 9], "def_end_pos": [470, 31]}, {"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [206, 9], "def_end_pos": [206, 29]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "rw [this] at hr", "annotated_tactic": ["rw [this] at hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "rcases lt_iSup_iff.1 hr with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>lt_iSup_iff</a>.1 hr with \u27e8n, hn\u27e9", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [544, 9], "def_end_pos": [544, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "rw [\u2190 restrict_apply hF] at hn", "annotated_tactic": ["rw [\u2190 <a>restrict_apply</a> hF] at hn", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < \u03bc (F \u2229 s n)\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < (\u03bc.restrict (s n)) F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "rcases h n hF _ hn with \u27e8K, KF, hKp, hK\u27e9", "annotated_tactic": ["rcases h n hF _ hn with \u27e8K, KF, hKp, hK\u27e9", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < (\u03bc.restrict (s n)) F\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < (\u03bc.restrict (s n)) F\nK : Set \u03b1\nKF : K \u2286 F\nhKp : p K\nhK : r < (\u03bc.restrict (s n)) K\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K"}, {"tactic": "exact \u27e8K, KF, hKp, hK.trans_le (restrict_apply_le _ _)\u27e9", "annotated_tactic": ["exact \u27e8K, KF, hKp, hK.trans_le (<a>restrict_apply_le</a> _ _)\u27e9", [{"full_name": "MeasureTheory.Measure.restrict_apply_le", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [150, 9], "def_end_pos": [150, 26]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u2a06 n, \u03bc (F \u2229 s n)\nhBU : \u22c3 n, F \u2229 s n = F\nthis : \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)\nn : \u2115\nhn : r < (\u03bc.restrict (s n)) F\nK : Set \u03b1\nKF : K \u2286 F\nhKp : p K\nhK : r < (\u03bc.restrict (s n)) K\n\u22a2 \u2203 K \u2286 F, p K \u2227 r < \u03bc K", "state_after": "no goals"}, {"tactic": "rw [\u2190 inter_iUnion, univ_subset_iff.mp hs, inter_univ]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>, univ_subset_iff.mp hs, <a>inter_univ</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [482, 9], "def_end_pos": [482, 21]}, {"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 9], "def_end_pos": [981, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\n\u22a2 \u22c3 n, F \u2229 s n = F", "state_after": "no goals"}, {"tactic": "rw [\u2190 measure_iUnion_eq_iSup, hBU]", "annotated_tactic": ["rw [\u2190 <a>measure_iUnion_eq_iSup</a>, hBU]", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [470, 9], "def_end_pos": [470, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\n\u22a2 \u03bc F = \u2a06 n, \u03bc (F \u2229 s n)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 s n"}, {"tactic": "exact Monotone.directed_le fun m n h \u21a6 inter_subset_inter_right _ (hmono h)", "annotated_tactic": ["exact <a>Monotone.directed_le</a> fun m n h \u21a6 <a>inter_subset_inter_right</a> _ (hmono h)", [{"full_name": "Monotone.directed_le", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [206, 9], "def_end_pos": [206, 29]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (n : \u2115), (\u03bc.restrict (s n)).InnerRegularWRT p MeasurableSet\nhs : univ \u2286 \u22c3 n, s n\nhmono : Monotone s\nF : Set \u03b1\nhF : MeasurableSet F\nr : \u211d\u22650\u221e\nhr : r < \u03bc F\nhBU : \u22c3 n, F \u2229 s n = F\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 s n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalSubalgebra.coe_toNonUnitalSubsemiring", "start": [89, 1], "end": [91, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Multiset.lean", "full_name": "Multiset.toFinsupp_toMultiset", "start": [177, 1], "end": [178, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "upperClosure_empty", "start": [1529, 1], "end": [1530, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Finset.sum_nat_mod", "start": [2219, 1], "end": [2221, 80], "traced_tactics": [{"tactic": "rw [Finset.sum, Multiset.map_map]", "annotated_tactic": ["rw [<a>Finset.sum</a>, <a>Multiset.map_map</a>]", [{"full_name": "Finset.sum", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [52, 3], "def_end_pos": [52, 14]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nn : \u2115\nf : \u03b1 \u2192 \u2115\n\u22a2 (Multiset.map (fun x => x % n) (Multiset.map (fun i => f i) s.val)).sum % n = (\u2211 i \u2208 s, f i % n) % n", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nn : \u2115\nf : \u03b1 \u2192 \u2115\n\u22a2 (Multiset.map ((fun x => x % n) \u2218 fun i => f i) s.val).sum % n = (Multiset.map (fun i => f i % n) s.val).sum % n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nn : \u2115\nf : \u03b1 \u2192 \u2115\n\u22a2 (Multiset.map ((fun x => x % n) \u2218 fun i => f i) s.val).sum % n = (Multiset.map (fun i => f i % n) s.val).sum % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Bounded.mem_gt", "start": [974, 1], "end": [976, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Associates.dvdNotUnit_of_lt", "start": [1159, 1], "end": [1168, 7], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 DvdNotUnit a b", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 a \u2260 0\n\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 \u2203 x, \u00acIsUnit x \u2227 b = a * x"}, {"tactic": "rcases hlt with \u27e8\u27e8x, rfl\u27e9, ndvd\u27e9", "annotated_tactic": ["rcases hlt with \u27e8\u27e8x, rfl\u27e9, ndvd\u27e9", []], "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 \u2203 x, \u00acIsUnit x \u2227 b = a * x", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u2203 x_1, \u00acIsUnit x_1 \u2227 a * x = a * x_1"}, {"tactic": "refine \u27e8x, ?_, rfl\u27e9", "annotated_tactic": ["refine \u27e8x, ?_, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u2203 x_1, \u00acIsUnit x_1 \u2227 a * x = a * x_1", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u00acIsUnit x"}, {"tactic": "contrapose! ndvd", "annotated_tactic": ["contrapose! ndvd", []], "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u00acIsUnit x", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : IsUnit x\n\u22a2 a * x \u2223 a"}, {"tactic": "rcases ndvd with \u27e8u, rfl\u27e9", "annotated_tactic": ["rcases ndvd with \u27e8u, rfl\u27e9", []], "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : IsUnit x\n\u22a2 a * x \u2223 a", "state_after": "case right.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nu : (Associates \u03b1)\u02e3\n\u22a2 a * \u2191u \u2223 a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case right.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nu : (Associates \u03b1)\u02e3\n\u22a2 a * \u2191u \u2223 a", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 a \u2260 0", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 False"}, {"tactic": "apply not_lt_of_le _ hlt", "annotated_tactic": ["apply <a>not_lt_of_le</a> _ hlt", [{"full_name": "not_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [343, 9], "def_end_pos": [343, 21]}]], "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 b \u2264 0"}, {"tactic": "apply dvd_zero", "annotated_tactic": ["apply <a>dvd_zero</a>", [{"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 b \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "exists_between_of_forall_le", "start": [661, 1], "end": [663, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Basic.lean", "full_name": "update_le_iff", "start": [981, 1], "end": [982, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Process/Adapted.lean", "full_name": "MeasureTheory.progMeasurable_const", "start": [118, 1], "end": [120, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "exists_maximal_orthonormal", "start": [945, 1], "end": [954, 46], "traced_tactics": [{"tactic": "have := zorn_subset_nonempty { b | Orthonormal \ud835\udd5c (Subtype.val : b \u2192 E) } ?_ _ hs", "annotated_tactic": ["have := <a>zorn_subset_nonempty</a> { b | <a>Orthonormal</a> \ud835\udd5c (<a>Subtype.val</a> : b \u2192 E) } ?_ _ hs", [{"full_name": "zorn_subset_nonempty", "def_path": "Mathlib/Order/Zorn.lean", "def_pos": [192, 9], "def_end_pos": [192, 29]}, {"full_name": "Orthonormal", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [735, 5], "def_end_pos": [735, 16]}, {"full_name": "Subtype.val", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [587, 3], "def_end_pos": [587, 6]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2203 w \u2287 s, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 u \u2287 w, Orthonormal \ud835\udd5c Subtype.val \u2192 u = w", "state_after": "case refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nthis : \u2203 m \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, s \u2286 m \u2227 \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 w \u2287 s, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 u \u2287 w, Orthonormal \ud835\udd5c Subtype.val \u2192 u = w\n\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2200 c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val},\n    IsChain (fun x x_1 => x \u2286 x_1) c \u2192 c.Nonempty \u2192 \u2203 ub \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, \u2200 s \u2208 c, s \u2286 ub"}, {"tactic": "obtain \u27e8b, bi, sb, h\u27e9 := this", "annotated_tactic": ["obtain \u27e8b, bi, sb, h\u27e9 := this", []], "state_before": "case refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nthis : \u2203 m \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, s \u2286 m \u2227 \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, m \u2286 a \u2192 a = m\n\u22a2 \u2203 w \u2287 s, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 u \u2287 w, Orthonormal \ud835\udd5c Subtype.val \u2192 u = w", "state_after": "case refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, b \u2286 a \u2192 a = b\n\u22a2 \u2203 w \u2287 s, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 u \u2287 w, Orthonormal \ud835\udd5c Subtype.val \u2192 u = w"}, {"tactic": "refine \u27e8b, sb, bi, ?_\u27e9", "annotated_tactic": ["refine \u27e8b, sb, bi, ?_\u27e9", []], "state_before": "case refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, b \u2286 a \u2192 a = b\n\u22a2 \u2203 w \u2287 s, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 u \u2287 w, Orthonormal \ud835\udd5c Subtype.val \u2192 u = w", "state_after": "case refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, b \u2286 a \u2192 a = b\n\u22a2 \u2200 u \u2287 b, Orthonormal \ud835\udd5c Subtype.val \u2192 u = b"}, {"tactic": "exact fun u hus hu => h u hu hus", "annotated_tactic": ["exact fun u hus hu => h u hu hus", []], "state_before": "case refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, b \u2286 a \u2192 a = b\n\u22a2 \u2200 u \u2287 b, Orthonormal \ud835\udd5c Subtype.val \u2192 u = b", "state_after": "no goals"}, {"tactic": "refine fun c hc cc _c0 => \u27e8\u22c3\u2080 c, ?_, ?_\u27e9", "annotated_tactic": ["refine fun c hc cc _c0 => \u27e8\u22c3\u2080 c, ?_, ?_\u27e9", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2200 c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val},\n    IsChain (fun x x_1 => x \u2286 x_1) c \u2192 c.Nonempty \u2192 \u2203 ub \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}, \u2200 s \u2208 c, s \u2286 ub", "state_after": "case refine_1.refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : c.Nonempty\n\u22a2 \u22c3\u2080 c \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\n\ncase refine_1.refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : c.Nonempty\n\u22a2 \u2200 s \u2208 c, s \u2286 \u22c3\u2080 c"}, {"tactic": "exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc", "annotated_tactic": ["exact <a>orthonormal_sUnion_of_directed</a> cc.directedOn fun x xc => hc xc", [{"full_name": "orthonormal_sUnion_of_directed", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [937, 9], "def_end_pos": [937, 39]}]], "state_before": "case refine_1.refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : c.Nonempty\n\u22a2 \u22c3\u2080 c \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}", "state_after": "no goals"}, {"tactic": "exact fun _ => Set.subset_sUnion_of_mem", "annotated_tactic": ["exact fun _ => <a>Set.subset_sUnion_of_mem</a>", [{"full_name": "Set.subset_sUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 29]}]], "state_before": "case refine_1.refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : c.Nonempty\n\u22a2 \u2200 s \u2208 c, s \u2286 \u22c3\u2080 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.coe_coe_fst", "start": [598, 1], "end": [599, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean", "full_name": "RingSubgroupsBasis.mem_addGroupFilterBasis", "start": [132, 1], "end": [133, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Inseparable.lean", "full_name": "IsClosed.not_specializes", "start": [118, 1], "end": [119, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.floor_le_floor", "start": [201, 1], "end": [201, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Segment.lean", "full_name": "mem_segment_translate", "start": [260, 1], "end": [261, 65], "traced_tactics": [{"tactic": "simp_rw [\u2190 vadd_eq_add, \u2190 vadd_segment, vadd_mem_vadd_set_iff]", "annotated_tactic": ["simp_rw [\u2190 <a>vadd_eq_add</a>, \u2190 <a>vadd_segment</a>, <a>vadd_mem_vadd_set_iff</a>]", [{"full_name": "vadd_eq_add", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [88, 3], "def_end_pos": [88, 14]}, {"full_name": "vadd_segment", "def_path": "Mathlib/Analysis/Convex/Segment.lean", "def_pos": [248, 9], "def_end_pos": [248, 21]}, {"full_name": "Set.vadd_mem_vadd_set_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [893, 3], "def_end_pos": [893, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\na x b c : E\n\u22a2 a + x \u2208 [a + b-[\ud835\udd5c]a + c] \u2194 x \u2208 [b-[\ud835\udd5c]c]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Bounded.lean", "full_name": "BotHom.dual_id", "start": [789, 1], "end": [790, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.parts_map", "start": [139, 1], "end": [140, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/StarAlgHom.lean", "full_name": "StarAlgEquiv.to_ringEquiv_symm", "start": [930, 1], "end": [931, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean", "full_name": "IsBoundedLinearMap.snd", "start": [121, 1], "end": [124, 25], "traced_tactics": [{"tactic": "refine (LinearMap.snd \ud835\udd5c E F).isLinear.with_bound 1 fun x => ?_", "annotated_tactic": ["refine (<a>LinearMap.snd</a> \ud835\udd5c E F).isLinear.with_bound 1 fun x => ?_", [{"full_name": "LinearMap.snd", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [66, 5], "def_end_pos": [66, 8]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun x => x.2", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 1 * \u2016x\u2016"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 1 * \u2016x\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 \u2016x\u2016"}, {"tactic": "exact le_max_right _ _", "annotated_tactic": ["exact <a>le_max_right</a> _ _", [{"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.denseRange_denseSeq", "start": [354, 1], "end": [355, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean", "full_name": "CategoryTheory.monoidalOfHasFiniteCoproducts.tensorHom", "start": [212, 1], "end": [213, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.measure_le_lt_top", "start": [888, 1], "end": [895, 59], "traced_tactics": [{"tactic": "refine lt_of_le_of_lt (measure_mono ?_) (hf.measure_norm_ge_lt_top (show 0 < -c by linarith))", "annotated_tactic": ["refine <a>lt_of_le_of_lt</a> (<a>measure_mono</a> ?_) (hf.measure_norm_ge_lt_top (show 0 < -c by linarith))", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\n\u22a2 \u03bc {a | f a \u2264 c} < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\n\u22a2 {a | f a \u2264 c} \u2286 {x | -c \u2264 \u2016f x\u2016}"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\n\u22a2 {a | f a \u2264 c} \u2286 {x | -c \u2264 \u2016f x\u2016}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\nx : \u03b1\nhx : x \u2208 {a | f a \u2264 c}\n\u22a2 x \u2208 {x | -c \u2264 \u2016f x\u2016}"}, {"tactic": "simp only [Real.norm_eq_abs, Set.mem_setOf_eq] at hx \u22a2", "annotated_tactic": ["simp only [<a>Real.norm_eq_abs</a>, <a>Set.mem_setOf_eq</a>] at hx \u22a2", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\nx : \u03b1\nhx : x \u2208 {a | f a \u2264 c}\n\u22a2 x \u2208 {x | -c \u2264 \u2016f x\u2016}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\nx : \u03b1\nhx : f x \u2264 c\n\u22a2 -c \u2264 |f x|"}, {"tactic": "exact (show -c \u2264 - f x by linarith).trans (neg_le_abs _)", "annotated_tactic": ["exact (show -c \u2264 - f x by linarith).<a>trans</a> (<a>neg_le_abs</a> _)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "neg_le_abs", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [78, 3], "def_end_pos": [78, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\nx : \u03b1\nhx : f x \u2264 c\n\u22a2 -c \u2264 |f x|", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\n\u22a2 0 < -c", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\nc : \u211d\nc_neg : c < 0\nx : \u03b1\nhx : f x \u2264 c\n\u22a2 -c \u2264 -f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Calculus.lean", "full_name": "HasDerivWithinAt.norm_sq", "start": [238, 1], "end": [241, 60], "traced_tactics": [{"tactic": "simpa using hf.hasFDerivWithinAt.norm_sq.hasDerivWithinAt", "annotated_tactic": ["simpa using hf.hasFDerivWithinAt.norm_sq.hasDerivWithinAt", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g : G \u2192 E\nf'\u271d g' : G \u2192L[\u211d] E\ns\u271d : Set G\nx\u271d : G\nn : \u2115\u221e\nf : \u211d \u2192 F\nf' : F\ns : Set \u211d\nx : \u211d\nhf : HasDerivWithinAt f f' s x\n\u22a2 HasDerivWithinAt (fun x => \u2016f x\u2016 ^ 2) (2 * \u27eaf x, f'\u27eb_\u211d) s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dimension/Constructions.lean", "full_name": "FiniteDimensional.finrank_pi", "start": [282, 1], "end": [284, 17], "traced_tactics": [{"tactic": "simp [finrank]", "annotated_tactic": ["simp [<a>finrank</a>]", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}]], "state_before": "R S : Type u\nM : Type v\nM' : Type v'\nM\u2081 : Type v\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d\u00b9\u00b3 : Ring R\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup M'\ninst\u271d\u2079 : AddCommGroup M\u2081\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : StrongRankCondition R\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : (i : \u03b7) \u2192 AddCommGroup (\u03c6 i)\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Module R (\u03c6 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b7), Module.Free R (\u03c6 i)\n\u03b9 : Type v\ninst\u271d : Fintype \u03b9\n\u22a2 finrank R (\u03b9 \u2192 R) = Fintype.card \u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "isOpenMap_eval", "start": [623, 1], "end": [633, 22], "traced_tactics": [{"tactic": "refine (isTopologicalBasis_pi fun _ \u21a6 isTopologicalBasis_opens).isOpenMap_iff.2 ?_", "annotated_tactic": ["refine (<a>isTopologicalBasis_pi</a> fun _ \u21a6 <a>isTopologicalBasis_opens</a>).<a>isOpenMap_iff</a>.2 ?_", [{"full_name": "isTopologicalBasis_pi", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [601, 9], "def_end_pos": [601, 30]}, {"full_name": "TopologicalSpace.isTopologicalBasis_opens", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [244, 9], "def_end_pos": [244, 33]}, {"full_name": "TopologicalSpace.IsTopologicalBasis.isOpenMap_iff", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [230, 9], "def_end_pos": [230, 41]}]], "state_before": "\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\n\u22a2 IsOpenMap (eval i)", "state_after": "\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\n\u22a2 \u2200 s \u2208 {S | \u2203 U F, (\u2200 i \u2208 F, U i \u2208 {U | IsOpen U}) \u2227 S = (\u2191F).pi U}, IsOpen (eval i '' s)"}, {"tactic": "rintro _ \u27e8U, s, hU, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8U, s, hU, rfl\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\n\u22a2 \u2200 s \u2208 {S | \u2203 U F, (\u2200 i \u2208 F, U i \u2208 {U | IsOpen U}) \u2227 S = (\u2191F).pi U}, IsOpen (eval i '' s)", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)"}, {"tactic": "obtain h | h := ((s : Set \u03b9).pi U).eq_empty_or_nonempty", "annotated_tactic": ["obtain h | h := ((s : <a>Set</a> \u03b9).<a>pi</a> U).<a>eq_empty_or_nonempty</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [249, 5], "def_end_pos": [249, 7]}, {"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 29]}]], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)", "state_after": "case intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : (\u2191s).pi U = \u2205\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)\n\ncase intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)"}, {"tactic": "by_cases hi : i \u2208 s", "annotated_tactic": ["by_cases hi : i \u2208 s", []], "state_before": "case intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)", "state_after": "case pos\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2208 s\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)\n\ncase neg\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2209 s\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : (\u2191s).pi U = \u2205\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)", "state_after": "no goals"}, {"tactic": "rw [eval_image_pi (mod_cast hi) h]", "annotated_tactic": ["rw [<a>eval_image_pi</a> (mod_cast hi) h]", [{"full_name": "Set.eval_image_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [915, 9], "def_end_pos": [915, 22]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2208 s\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)", "state_after": "case pos\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2208 s\n\u22a2 IsOpen (U i)"}, {"tactic": "exact hU _ hi", "annotated_tactic": ["exact hU _ hi", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2208 s\n\u22a2 IsOpen (U i)", "state_after": "no goals"}, {"tactic": "rw [eval_image_pi_of_not_mem (mod_cast hi), if_pos h]", "annotated_tactic": ["rw [<a>eval_image_pi_of_not_mem</a> (mod_cast hi), <a>if_pos</a> h]", [{"full_name": "Set.eval_image_pi_of_not_mem", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [919, 7], "def_end_pos": [919, 31]}, {"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2209 s\n\u22a2 IsOpen (eval i '' (\u2191s).pi U)", "state_after": "case neg\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2209 s\n\u22a2 IsOpen univ"}, {"tactic": "exact isOpen_univ", "annotated_tactic": ["exact <a>isOpen_univ</a>", [{"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [88, 17], "def_end_pos": [88, 28]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nU : (i : \u03b9) \u2192 Set (\u03c0 i)\ns : Finset \u03b9\nhU : \u2200 i \u2208 s, U i \u2208 {U | IsOpen U}\nh : ((\u2191s).pi U).Nonempty\nhi : i \u2209 s\n\u22a2 IsOpen univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "div_div_cancel_left'", "start": [542, 1], "end": [542, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.face_mono", "start": [413, 1], "end": [415, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/TensorProduct/Basic.lean", "full_name": "Algebra.TensorProduct.map_comp", "start": [890, 1], "end": [892, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "full_name": "tsum_sq_fourierCoeff", "start": [428, 1], "end": [439, 34], "traced_tactics": [{"tactic": "simp_rw [\u2190 fourierBasis_repr]", "annotated_tactic": ["simp_rw [\u2190 <a>fourierBasis_repr</a>]", [{"full_name": "fourierBasis_repr", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [408, 9], "def_end_pos": [408, 26]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 \u2211' (i : \u2124), \u2016fourierCoeff (\u2191\u2191f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle"}, {"tactic": "have H\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' i, \u2016fourierBasis.repr f i\u2016 ^ 2 := by\n  apply_mod_cast lp.norm_rpow_eq_tsum ?_ (fourierBasis.repr f)\n  norm_num", "annotated_tactic": ["have H\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' i, \u2016fourierBasis.repr f i\u2016 ^ 2 := by\n    apply_mod_cast <a>lp.norm_rpow_eq_tsum</a> ?_ (fourierBasis.repr f)\n    norm_num", [{"full_name": "lp.norm_rpow_eq_tsum", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [413, 9], "def_end_pos": [413, 26]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle"}, {"tactic": "have H\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2 := by simp", "annotated_tactic": ["have H\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2 := by simp", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle"}, {"tactic": "have H\u2083 := congr_arg RCLike.re (@L2.inner_def (AddCircle T) \u2102 \u2102 _ _ _ _ _ f f)", "annotated_tactic": ["have H\u2083 := <a>congr_arg</a> <a>RCLike.re</a> (@<a>L2.inner_def</a> (<a>AddCircle</a> T) \u2102 \u2102 _ _ _ _ _ f f)", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "RCLike.re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [55, 3], "def_end_pos": [55, 5]}, {"full_name": "MeasureTheory.L2.inner_def", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [150, 9], "def_end_pos": [150, 18]}, {"full_name": "AddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [122, 8], "def_end_pos": [122, 17]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = RCLike.re (\u222b (a : AddCircle T), \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102 \u2202haarAddCircle)\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle"}, {"tactic": "rw [\u2190 integral_re] at H\u2083", "annotated_tactic": ["rw [\u2190 <a>integral_re</a>] at H\u2083", [{"full_name": "integral_re", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1388, 9], "def_end_pos": [1388, 20]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = RCLike.re (\u222b (a : AddCircle T), \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102 \u2202haarAddCircle)\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = \u222b (x : AddCircle T), RCLike.re \u27ea\u2191\u2191f x, \u2191\u2191f x\u27eb_\u2102 \u2202haarAddCircle\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle\n\nT : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = RCLike.re (\u222b (a : AddCircle T), \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102 \u2202haarAddCircle)\n\u22a2 Integrable (fun a => \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102) haarAddCircle"}, {"tactic": "apply_mod_cast lp.norm_rpow_eq_tsum ?_ (fourierBasis.repr f)", "annotated_tactic": ["apply_mod_cast <a>lp.norm_rpow_eq_tsum</a> ?_ (fourierBasis.repr f)", [{"full_name": "lp.norm_rpow_eq_tsum", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [413, 9], "def_end_pos": [413, 26]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 0 < ENNReal.toReal 2"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\n\u22a2 0 < ENNReal.toReal 2", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\n\u22a2 \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2", "state_after": "no goals"}, {"tactic": "simp only [\u2190 norm_sq_eq_inner] at H\u2083", "annotated_tactic": ["simp only [\u2190 <a>norm_sq_eq_inner</a>] at H\u2083", [{"full_name": "InnerProductSpace.norm_sq_eq_inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [109, 3], "def_end_pos": [109, 19]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = \u222b (x : AddCircle T), RCLike.re \u27ea\u2191\u2191f x, \u2191\u2191f x\u27eb_\u2102 \u2202haarAddCircle\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : \u2016f\u2016 ^ 2 = \u222b (x : AddCircle T), \u2016\u2191\u2191f x\u2016 ^ 2 \u2202haarAddCircle\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle"}, {"tactic": "rw [\u2190 H\u2081, H\u2082, H\u2083]", "annotated_tactic": ["rw [\u2190 H\u2081, H\u2082, H\u2083]", []], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : \u2016f\u2016 ^ 2 = \u222b (x : AddCircle T), \u2016\u2191\u2191f x\u2016 ^ 2 \u2202haarAddCircle\n\u22a2 \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2 = \u222b (t : AddCircle T), \u2016\u2191\u2191f t\u2016 ^ 2 \u2202haarAddCircle", "state_after": "no goals"}, {"tactic": "exact L2.integrable_inner f f", "annotated_tactic": ["exact <a>L2.integrable_inner</a> f f", [{"full_name": "MeasureTheory.L2.integrable_inner", "def_path": "Mathlib/MeasureTheory/Function/L2Space.lean", "def_pos": [188, 9], "def_end_pos": [188, 25]}]], "state_before": "T : \u211d\nhT : Fact (0 < T)\nf : \u21a5(Lp \u2102 2 haarAddCircle)\nH\u2081 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2211' (i : \u2124), \u2016\u2191(fourierBasis.repr f) i\u2016 ^ 2\nH\u2082 : \u2016fourierBasis.repr f\u2016 ^ 2 = \u2016f\u2016 ^ 2\nH\u2083 : RCLike.re \u27eaf, f\u27eb_\u2102 = RCLike.re (\u222b (a : AddCircle T), \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102 \u2202haarAddCircle)\n\u22a2 Integrable (fun a => \u27ea\u2191\u2191f a, \u2191\u2191f a\u27eb_\u2102) haarAddCircle", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/IntervalCases.lean", "full_name": "Mathlib.Tactic.IntervalCases.of_not_le_right", "start": [129, 1], "end": [129, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "full_name": "GaussianInt.natCast_natAbs_norm", "start": [188, 1], "end": [189, 44], "traced_tactics": [{"tactic": "rw [\u2190 Int.cast_natCast, abs_natCast_norm]", "annotated_tactic": ["rw [\u2190 <a>Int.cast_natCast</a>, <a>abs_natCast_norm</a>]", [{"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "GaussianInt.abs_natCast_norm", "def_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "def_pos": [181, 9], "def_end_pos": [181, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nx : \u2124[i]\n\u22a2 \u2191(norm x).natAbs = \u2191(norm x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.mem_submodule_iff'", "start": [988, 1], "end": [992, 84], "traced_tactics": [{"tactic": "simp [Finsupp.sum_fintype, Finsupp.equivFunOnFinite]", "annotated_tactic": ["simp [<a>Finsupp.sum_fintype</a>, <a>Finsupp.equivFunOnFinite</a>]", [{"full_name": "Finsupp.sum_fintype", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [61, 3], "def_end_pos": [61, 14]}, {"full_name": "Finsupp.equivFunOnFinite", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [228, 5], "def_end_pos": [228, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nS : Type u_10\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : Module S M'\ninst\u271d\u00b9 : SMulCommClass R S M'\ninst\u271d : Fintype \u03b9\nP : Submodule R M\nb : Basis \u03b9 R \u21a5P\nx : M\nc : \u03b9 \u2192 R\n\u22a2 (x = (Finsupp.equivFunOnFinite.symm c).sum fun i x => x \u2022 \u2191(b i)) \u2194 x = \u2211 i : \u03b9, c i \u2022 \u2191(b i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoundedVariation.lean", "full_name": "variationOnFromTo.monotoneOn", "start": [735, 11], "end": [739, 71], "traced_tactics": [{"tactic": "rintro b bs c cs bc", "annotated_tactic": ["rintro b bs c cs bc", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\n\u22a2 MonotoneOn (variationOnFromTo f s a) s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a c"}, {"tactic": "rw [\u2190 variationOnFromTo.add hf as bs cs]", "annotated_tactic": ["rw [\u2190 <a>variationOnFromTo.add</a> hf as bs cs]", [{"full_name": "variationOnFromTo.add", "def_path": "Mathlib/Analysis/BoundedVariation.lean", "def_pos": [672, 19], "def_end_pos": [672, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a b + variationOnFromTo f s b c"}, {"tactic": "exact le_add_of_nonneg_right (variationOnFromTo.nonneg_of_le f s bc)", "annotated_tactic": ["exact <a>le_add_of_nonneg_right</a> (<a>variationOnFromTo.nonneg_of_le</a> f s bc)", [{"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}, {"full_name": "variationOnFromTo.nonneg_of_le", "def_path": "Mathlib/Analysis/BoundedVariation.lean", "def_pos": [645, 19], "def_end_pos": [645, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a b + variationOnFromTo f s b c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/WellKnown.lean", "full_name": "PowerSeries.invOneSubPow_val_eq_mk_choose_add", "start": [120, 1], "end": [121, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/DiscreteQuotient.lean", "full_name": "DiscreteQuotient.ofLE_comp_proj", "start": [241, 1], "end": [242, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "List.length_sigma", "start": [711, 1], "end": [712, 89], "traced_tactics": [{"tactic": "simp [length_sigma']", "annotated_tactic": ["simp [<a>length_sigma'</a>]", [{"full_name": "List.length_sigma'", "def_path": "Mathlib/Data/List/ProdSigma.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\n\u03c3 : \u03b1 \u2192 Type u_8\nl\u2081 : List \u03b1\nl\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)\n\u22a2 (l\u2081.sigma l\u2082).length = (map (fun a => (l\u2082 a).length) l\u2081).sum", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "full_name": "Polynomial.smeval_comp", "start": [277, 1], "end": [282, 53], "traced_tactics": [{"tactic": "induction p using Polynomial.induction_on' with\n| h_add r s hr hs =>\n  simp [add_comp, hr, hs, smeval_add]\n| h_monomial n a =>\n  simp [smeval_monomial, smeval_C_mul, smeval_pow]", "annotated_tactic": ["induction p using <a>Polynomial.induction_on'</a> with\n  | h_add r s hr hs =>\n    simp [<a>add_comp</a>, hr, hs, <a>smeval_add</a>]\n  | h_monomial n a =>\n    simp [<a>smeval_monomial</a>, <a>smeval_C_mul</a>, <a>smeval_pow</a>]", [{"full_name": "Polynomial.induction_on'", "def_path": "Mathlib/Algebra/Polynomial/Induction.lean", "def_pos": [63, 19], "def_end_pos": [63, 32]}, {"full_name": "Polynomial.add_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [589, 9], "def_end_pos": [589, 17]}, {"full_name": "Polynomial.smeval_add", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [105, 9], "def_end_pos": [105, 19]}, {"full_name": "Polynomial.smeval_monomial", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}, {"full_name": "Polynomial.smeval_C_mul", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [248, 9], "def_end_pos": [248, 21]}, {"full_name": "Polynomial.smeval_pow", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [271, 9], "def_end_pos": [271, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d : R[X]\nr : R\np q : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\n\u22a2 (p.comp q).smeval x = p.smeval (q.smeval x)", "state_after": "no goals"}, {"tactic": "simp [add_comp, hr, hs, smeval_add]", "annotated_tactic": ["simp [<a>add_comp</a>, hr, hs, <a>smeval_add</a>]", [{"full_name": "Polynomial.add_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [589, 9], "def_end_pos": [589, 17]}, {"full_name": "Polynomial.smeval_add", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [105, 9], "def_end_pos": [105, 19]}]], "state_before": "case h_add\nR : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d : R[X]\nr\u271d : R\np q : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\nr s : R[X]\nhr : (r.comp q).smeval x = r.smeval (q.smeval x)\nhs : (s.comp q).smeval x = s.smeval (q.smeval x)\n\u22a2 ((r + s).comp q).smeval x = (r + s).smeval (q.smeval x)", "state_after": "no goals"}, {"tactic": "simp [smeval_monomial, smeval_C_mul, smeval_pow]", "annotated_tactic": ["simp [<a>smeval_monomial</a>, <a>smeval_C_mul</a>, <a>smeval_pow</a>]", [{"full_name": "Polynomial.smeval_monomial", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}, {"full_name": "Polynomial.smeval_C_mul", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [248, 9], "def_end_pos": [248, 21]}, {"full_name": "Polynomial.smeval_pow", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [271, 9], "def_end_pos": [271, 19]}]], "state_before": "case h_monomial\nR : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d : R[X]\nr : R\np q : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\nn : \u2115\na : R\n\u22a2 (((monomial n) a).comp q).smeval x = ((monomial n) a).smeval (q.smeval x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.toFun_eq_coe", "start": [92, 1], "end": [93, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.map_bind'", "start": [171, 1], "end": [172, 87], "traced_tactics": [{"tactic": "cases x <;> simp", "annotated_tactic": ["cases x <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\np : \u03b1 \u2192 Prop\nf\u271d : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx\u271d : Option \u03b1\nf : \u03b2 \u2192 \u03b3\nx : Option \u03b1\ng : \u03b1 \u2192 Option \u03b2\n\u22a2 Option.map f (x.bind g) = x.bind fun a => Option.map f (g a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.cliqueFinset_map_of_equiv", "start": [664, 1], "end": [667, 68], "traced_tactics": [{"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG H : SimpleGraph \u03b1\ninst\u271d\u2075 : Fintype \u03b1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableRel G.Adj\nn\u271d : \u2115\na b c : \u03b1\ns : Finset \u03b1\ninst\u271d\u00b2 : DecidableRel H.Adj\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : DecidableEq \u03b2\ne : \u03b1 \u2243 \u03b2\nn : \u2115\n\u22a2 \u2191((SimpleGraph.map e.toEmbedding G).cliqueFinset n) =\n    \u2191(map { toFun := map e.toEmbedding, inj' := \u22ef } (G.cliqueFinset n))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG H : SimpleGraph \u03b1\ninst\u271d\u2075 : Fintype \u03b1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableRel G.Adj\nn\u271d : \u2115\na b c : \u03b1\ns : Finset \u03b1\ninst\u271d\u00b2 : DecidableRel H.Adj\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : DecidableEq \u03b2\ne : \u03b1 \u2243 \u03b2\nn : \u2115\n\u22a2 (SimpleGraph.map e.toEmbedding G).cliqueSet n = \u21d1{ toFun := map e.toEmbedding, inj' := \u22ef } '' G.cliqueSet n"}, {"tactic": "exact cliqueSet_map_of_equiv _ _ _", "annotated_tactic": ["exact <a>cliqueSet_map_of_equiv</a> _ _ _", [{"full_name": "SimpleGraph.cliqueSet_map_of_equiv", "def_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "def_pos": [598, 9], "def_end_pos": [598, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG H : SimpleGraph \u03b1\ninst\u271d\u2075 : Fintype \u03b1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableRel G.Adj\nn\u271d : \u2115\na b c : \u03b1\ns : Finset \u03b1\ninst\u271d\u00b2 : DecidableRel H.Adj\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : DecidableEq \u03b2\ne : \u03b1 \u2243 \u03b2\nn : \u2115\n\u22a2 (SimpleGraph.map e.toEmbedding G).cliqueSet n = \u21d1{ toFun := map e.toEmbedding, inj' := \u22ef } '' G.cliqueSet n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.eq_iff_exists", "start": [605, 1], "end": [610, 45], "traced_tactics": [{"tactic": "replace h := congr_arg f.toMap h", "annotated_tactic": ["replace h := <a>congr_arg</a> f.toMap h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : \u21a5S\nh : \u2191c * x = \u2191c * y\n\u22a2 f.toMap x = f.toMap y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : \u21a5S\nh : f.toMap (\u2191c * x) = f.toMap (\u2191c * y)\n\u22a2 f.toMap x = f.toMap y"}, {"tactic": "rw [map_mul, map_mul] at h", "annotated_tactic": ["rw [<a>map_mul</a>, <a>map_mul</a>] at h", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : \u21a5S\nh : f.toMap (\u2191c * x) = f.toMap (\u2191c * y)\n\u22a2 f.toMap x = f.toMap y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : \u21a5S\nh : f.toMap \u2191c * f.toMap x = f.toMap \u2191c * f.toMap y\n\u22a2 f.toMap x = f.toMap y"}, {"tactic": "exact (f.map_units c).mul_right_inj.mp h", "annotated_tactic": ["exact (f.map_units c).mul_right_inj.mp h", []], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : \u21a5S\nh : f.toMap \u2191c * f.toMap x = f.toMap \u2191c * f.toMap y\n\u22a2 f.toMap x = f.toMap y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/Intersecting.lean", "full_name": "Set.Intersecting.exists_card_eq", "start": [195, 1], "end": [208, 19], "traced_tactics": [{"tactic": "have := hs.card_le", "annotated_tactic": ["have := hs.card_le", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : (\u2191s).Intersecting\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : (\u2191s).Intersecting\nthis : 2 * s.card \u2264 Fintype.card \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "rw [mul_comm, \u2190 Nat.le_div_iff_mul_le' Nat.two_pos] at this", "annotated_tactic": ["rw [<a>mul_comm</a>, \u2190 <a>Nat.le_div_iff_mul_le'</a> <a>Nat.two_pos</a>] at this", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.le_div_iff_mul_le'", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [529, 7], "def_end_pos": [529, 25]}, {"full_name": "Nat.two_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [597, 19], "def_end_pos": [597, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : (\u2191s).Intersecting\nthis : 2 * s.card \u2264 Fintype.card \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : (\u2191s).Intersecting\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "revert hs", "annotated_tactic": ["revert hs", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : (\u2191s).Intersecting\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 (\u2191s).Intersecting \u2192 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "refine s.strongDownwardInductionOn ?_ this", "annotated_tactic": ["refine s.strongDownwardInductionOn ?_ this", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 (\u2191s).Intersecting \u2192 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2200 (t\u2081 : Finset \u03b1),\n    (\u2200 {t\u2082 : Finset \u03b1},\n        t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n          t\u2081 \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting) \u2192\n      t\u2081.card \u2264 Fintype.card \u03b1 / 2 \u2192 (\u2191t\u2081).Intersecting \u2192 \u2203 t, t\u2081 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "rintro s ih _hcard hs", "annotated_tactic": ["rintro s ih _hcard hs", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : s.card \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2200 (t\u2081 : Finset \u03b1),\n    (\u2200 {t\u2082 : Finset \u03b1},\n        t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n          t\u2081 \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting) \u2192\n      t\u2081.card \u2264 Fintype.card \u03b1 / 2 \u2192 (\u2191t\u2081).Intersecting \u2192 \u2203 t, t\u2081 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "by_cases h : \u2200 t : Finset \u03b1, (t : Set \u03b1).Intersecting \u2192 s \u2286 t \u2192 s = t", "annotated_tactic": ["by_cases h : \u2200 t : <a>Finset</a> \u03b1, (t : <a>Set</a> \u03b1).<a>Intersecting</a> \u2192 s \u2286 t \u2192 s = t", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Set.Intersecting", "def_path": "Mathlib/Combinatorics/SetFamily/Intersecting.lean", "def_pos": [41, 5], "def_end_pos": [41, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u2200 (t : Finset \u03b1), (\u2191t).Intersecting \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u00ac\u2200 (t : Finset \u03b1), (\u2191t).Intersecting \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "push_neg at h", "annotated_tactic": ["push_neg at h", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u00ac\u2200 (t : Finset \u03b1), (\u2191t).Intersecting \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u2203 t, (\u2191t).Intersecting \u2227 s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "obtain \u27e8t, ht, hst\u27e9 := h", "annotated_tactic": ["obtain \u27e8t, ht, hst\u27e9 := h", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u2203 t, (\u2191t).Intersecting \u2227 s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting"}, {"tactic": "refine (ih ?_ (_root_.ssubset_iff_subset_ne.2 hst) ht).imp fun u => And.imp_left hst.1.trans", "annotated_tactic": ["refine (ih ?_ (<a>_root_.ssubset_iff_subset_ne</a>.2 hst) ht).<a>imp</a> fun u => <a>And.imp_left</a> hst.1.<a>trans</a>", [{"full_name": "ssubset_iff_subset_ne", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [811, 9], "def_end_pos": [811, 30]}, {"full_name": "Exists.imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [193, 9], "def_end_pos": [193, 19]}, {"full_name": "And.imp_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [655, 7], "def_end_pos": [655, 29]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 t.card \u2264 Fintype.card \u03b1 / 2"}, {"tactic": "rw [Nat.le_div_iff_mul_le' Nat.two_pos, mul_comm]", "annotated_tactic": ["rw [<a>Nat.le_div_iff_mul_le'</a> <a>Nat.two_pos</a>, <a>mul_comm</a>]", [{"full_name": "Nat.le_div_iff_mul_le'", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [529, 7], "def_end_pos": [529, 25]}, {"full_name": "Nat.two_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [597, 19], "def_end_pos": [597, 26]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 t.card \u2264 Fintype.card \u03b1 / 2", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 2 * t.card \u2264 Fintype.card \u03b1"}, {"tactic": "exact ht.card_le", "annotated_tactic": ["exact ht.card_le", []], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nt : Finset \u03b1\nht : (\u2191t).Intersecting\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 2 * t.card \u2264 Fintype.card \u03b1", "state_after": "no goals"}, {"tactic": "exact \u27e8s, Subset.rfl, hs.is_max_iff_card_eq.1 h, hs\u27e9", "annotated_tactic": ["exact \u27e8s, <a>Subset.rfl</a>, hs.is_max_iff_card_eq.1 h, hs\u27e9", [{"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [334, 9], "def_end_pos": [334, 19]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : s\u271d.card \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n  \u2200 {t\u2082 : Finset \u03b1},\n    t\u2082.card \u2264 Fintype.card \u03b1 / 2 \u2192\n      s \u2282 t\u2082 \u2192 (\u2191t\u2082).Intersecting \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting\n_hcard : s.card \u2264 Fintype.card \u03b1 / 2\nhs : (\u2191s).Intersecting\nh : \u2200 (t : Finset \u03b1), (\u2191t).Intersecting \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * t.card = Fintype.card \u03b1 \u2227 (\u2191t).Intersecting", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/NNReal.lean", "full_name": "NNReal.summable_nat_add", "start": [235, 1], "end": [236, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "full_name": "Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint", "start": [194, 1], "end": [196, 81], "traced_tactics": [{"tactic": "rw [sum_smul_vsub_eq_weightedVSubOfPoint_sub, weightedVSubOfPoint_apply_const]", "annotated_tactic": ["rw [<a>sum_smul_vsub_eq_weightedVSubOfPoint_sub</a>, <a>weightedVSubOfPoint_apply_const</a>]", [{"full_name": "Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "def_pos": [179, 9], "def_end_pos": [179, 49]}, {"full_name": "Finset.weightedVSubOfPoint_apply_const", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "def_pos": [79, 9], "def_end_pos": [79, 40]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\nw : \u03b9 \u2192 k\np\u2082 : \u03b9 \u2192 P\np\u2081 b : P\n\u22a2 \u2211 i \u2208 s, w i \u2022 (p\u2081 -\u1d65 p\u2082 i) = (\u2211 i \u2208 s, w i) \u2022 (p\u2081 -\u1d65 b) - (s.weightedVSubOfPoint p\u2082 b) w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order.lean", "full_name": "setOf_isOpen_sSup", "start": [954, 1], "end": [956, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Card.lean", "full_name": "MeasurableSpace.cardinal_generateMeasurable_le_continuum", "start": [176, 1], "end": [181, 79], "traced_tactics": [{"tactic": "rw [\u2190 continuum_power_aleph0]", "annotated_tactic": ["rw [\u2190 <a>continuum_power_aleph0</a>]", [{"full_name": "Cardinal.continuum_power_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [208, 9], "def_end_pos": [208, 31]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20", "state_after": "\u03b1 : Type u\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ^ \u2135\u2080"}, {"tactic": "exact mod_cast power_le_power_right (max_le hs (nat_lt_continuum 2).le)", "annotated_tactic": ["exact mod_cast <a>power_le_power_right</a> (<a>max_le</a> hs (<a>nat_lt_continuum</a> 2).<a>le</a>)", [{"full_name": "Cardinal.power_le_power_right", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [759, 9], "def_end_pos": [759, 29]}, {"full_name": "max_le", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [68, 9], "def_end_pos": [68, 15]}, {"full_name": "Cardinal.nat_lt_continuum", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [82, 9], "def_end_pos": [82, 25]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "\u03b1 : Type u\ns : Set (Set \u03b1)\nhs : #\u2191s \u2264 \ud835\udd20\n\u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ^ \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/MulOpposite.lean", "full_name": "Subring.op_iInf", "start": [116, 1], "end": [116, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean", "full_name": "measurableSet_eball", "start": [113, 1], "end": [114, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.coe_of_injective_castLE_symm", "start": [792, 1], "end": [795, 61], "traced_tactics": [{"tactic": "rw [\u2190 coe_castLE h]", "annotated_tactic": ["rw [\u2190 <a>coe_castLE</a> h]", [{"full_name": "Fin.coe_castLE", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [304, 17], "def_end_pos": [304, 27]}]], "state_before": "n\u271d m n k : \u2115\nh : n \u2264 k\ni : Fin k\nhi : i \u2208 Set.range (castLE h)\n\u22a2 \u2191((Equiv.ofInjective (castLE h) \u22ef).symm \u27e8i, hi\u27e9) = \u2191i", "state_after": "n\u271d m n k : \u2115\nh : n \u2264 k\ni : Fin k\nhi : i \u2208 Set.range (castLE h)\n\u22a2 \u2191(castLE h ((Equiv.ofInjective (castLE h) \u22ef).symm \u27e8i, hi\u27e9)) = \u2191i"}, {"tactic": "exact congr_arg Fin.val (Equiv.apply_ofInjective_symm _ _)", "annotated_tactic": ["exact <a>congr_arg</a> <a>Fin.val</a> (<a>Equiv.apply_ofInjective_symm</a> _ _)", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Fin.val", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1836, 3], "def_end_pos": [1836, 6]}, {"full_name": "Equiv.apply_ofInjective_symm", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [617, 9], "def_end_pos": [617, 31]}]], "state_before": "n\u271d m n k : \u2115\nh : n \u2264 k\ni : Fin k\nhi : i \u2208 Set.range (castLE h)\n\u22a2 \u2191(castLE h ((Equiv.ofInjective (castLE h) \u22ef).symm \u27e8i, hi\u27e9)) = \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "full_name": "cfc_inv_id", "start": [617, 1], "end": [623, 46], "traced_tactics": [{"tactic": "rw [\u2190 Ring.inverse_unit]", "annotated_tactic": ["rw [\u2190 <a>Ring.inverse_unit</a>]", [{"full_name": "Ring.inverse_unit", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [98, 9], "def_end_pos": [98, 21]}]], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 cfc (fun x => x\u207b\u00b9) \u2191a = \u2191a\u207b\u00b9", "state_after": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 cfc (fun x => x\u207b\u00b9) \u2191a = Ring.inverse \u2191a"}, {"tactic": "convert cfc_inv (id : R \u2192 R) (a : A) ?_", "annotated_tactic": ["convert <a>cfc_inv</a> (<a>id</a> : R \u2192 R) (a : A) ?_", [{"full_name": "cfc_inv", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [612, 7], "def_end_pos": [612, 14]}, {"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 cfc (fun x => x\u207b\u00b9) \u2191a = Ring.inverse \u2191a", "state_after": "case h.e'_3.h.e'_3\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 \u2191a = cfc id \u2191a\n\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 \u2200 x \u2208 spectrum R \u2191a, id x \u2260 0"}, {"tactic": "exact (cfc_id R (a : A)).symm", "annotated_tactic": ["exact (<a>cfc_id</a> R (a : A)).<a>symm</a>", [{"full_name": "cfc_id", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [336, 7], "def_end_pos": [336, 13]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_3.h.e'_3\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 \u2191a = cfc id \u2191a", "state_after": "no goals"}, {"tactic": "rintro x hx rfl", "annotated_tactic": ["rintro x hx rfl", []], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\n\u22a2 \u2200 x \u2208 spectrum R \u2191a, id x \u2260 0", "state_after": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\nhx : 0 \u2208 spectrum R \u2191a\n\u22a2 False"}, {"tactic": "exact spectrum.zero_not_mem R a.isUnit hx", "annotated_tactic": ["exact <a>spectrum.zero_not_mem</a> R a.isUnit hx", [{"full_name": "spectrum.zero_not_mem", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [126, 32], "def_end_pos": [126, 44]}]], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u00b9\u2070 : Semifield R\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : MetricSpace R\ninst\u271d\u2077 : TopologicalSemiring R\ninst\u271d\u2076 : ContinuousStar R\ninst\u271d\u2075 : HasContinuousInv\u2080 R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf : R \u2192 R\na\u271d : A\na : A\u02e3\nha : autoParam (p \u2191a) _auto\u271d\nhx : 0 \u2208 spectrum R \u2191a\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "summable_of_ratio_norm_eventually_le", "start": [574, 1], "end": [592, 88], "traced_tactics": [{"tactic": "by_cases hr\u2080 : 0 \u2264 r", "annotated_tactic": ["by_cases hr\u2080 : 0 \u2264 r", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\n\u22a2 Summable f", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : 0 \u2264 r\n\u22a2 Summable f\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : \u00ac0 \u2264 r\n\u22a2 Summable f"}, {"tactic": "rw [eventually_atTop] at h", "annotated_tactic": ["rw [<a>eventually_atTop</a>] at h", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : 0 \u2264 r\n\u22a2 Summable f", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2203 a, \u2200 b \u2265 a, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nhr\u2080 : 0 \u2264 r\n\u22a2 Summable f"}, {"tactic": "rcases h with \u27e8N, hN\u27e9", "annotated_tactic": ["rcases h with \u27e8N, hN\u27e9", []], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2203 a, \u2200 b \u2265 a, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nhr\u2080 : 0 \u2264 r\n\u22a2 Summable f", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\n\u22a2 Summable f"}, {"tactic": "rw [\u2190 @summable_nat_add_iff \u03b1 _ _ _ _ N]", "annotated_tactic": ["rw [\u2190 @<a>summable_nat_add_iff</a> \u03b1 _ _ _ _ N]", [{"full_name": "summable_nat_add_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}]], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\n\u22a2 Summable f", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\n\u22a2 Summable fun n => f (n + N)"}, {"tactic": "refine .of_norm_bounded (fun n \u21a6 \u2016f N\u2016 * r ^ n)\n  (Summable.mul_left _ <| summable_geometric_of_lt_one hr\u2080 hr\u2081) fun n \u21a6 ?_", "annotated_tactic": ["refine .of_norm_bounded (fun n \u21a6 \u2016f N\u2016 * r ^ n)\n      (<a>Summable.mul_left</a> _ <| <a>summable_geometric_of_lt_one</a> hr\u2080 hr\u2081) fun n \u21a6 ?_", [{"full_name": "Summable.mul_left", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}, {"full_name": "summable_geometric_of_lt_one", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [273, 9], "def_end_pos": [273, 37]}]], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\n\u22a2 Summable fun n => f (n + N)", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 (fun n => \u2016f N\u2016 * r ^ n) n"}, {"tactic": "simp only", "annotated_tactic": ["simp only", []], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 (fun n => \u2016f N\u2016 * r ^ n) n", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 \u2016f N\u2016 * r ^ n"}, {"tactic": "conv_rhs => rw [mul_comm, \u2190 zero_add N]", "annotated_tactic": ["conv_rhs => rw [<a>mul_comm</a>, \u2190 <a>zero_add</a> N]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 \u2016f N\u2016 * r ^ n", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 r ^ n * \u2016f (0 + N)\u2016"}, {"tactic": "refine le_geom (u := fun n \u21a6 \u2016f (n + N)\u2016) hr\u2080 n fun i _ \u21a6 ?_", "annotated_tactic": ["refine <a>le_geom</a> (u := fun n \u21a6 \u2016f (n + N)\u2016) hr\u2080 n fun i _ \u21a6 ?_", [{"full_name": "le_geom", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}]], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn : \u2115\n\u22a2 \u2016f (n + N)\u2016 \u2264 r ^ n * \u2016f (0 + N)\u2016", "state_after": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn i : \u2115\nx\u271d : i < n\n\u22a2 (fun n => \u2016f (n + N)\u2016) (i + 1) \u2264 r * (fun n => \u2016f (n + N)\u2016) i"}, {"tactic": "convert hN (i + N) (N.le_add_left i) using 3", "annotated_tactic": ["convert hN (i + N) (N.le_add_left i) using 3", []], "state_before": "case pos.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn i : \u2115\nx\u271d : i < n\n\u22a2 (fun n => \u2016f (n + N)\u2016) (i + 1) \u2264 r * (fun n => \u2016f (n + N)\u2016) i", "state_after": "case h.e'_3.h.e'_3.h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn i : \u2115\nx\u271d : i < n\n\u22a2 i + 1 + N = i + N + 1"}, {"tactic": "ac_rfl", "annotated_tactic": ["ac_rfl", []], "state_before": "case h.e'_3.h.e'_3.h.e'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nhr\u2080 : 0 \u2264 r\nN : \u2115\nhN : \u2200 b \u2265 N, \u2016f (b + 1)\u2016 \u2264 r * \u2016f b\u2016\nn i : \u2115\nx\u271d : i < n\n\u22a2 i + 1 + N = i + N + 1", "state_after": "no goals"}, {"tactic": "push_neg at hr\u2080", "annotated_tactic": ["push_neg at hr\u2080", []], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : \u00ac0 \u2264 r\n\u22a2 Summable f", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\n\u22a2 Summable f"}, {"tactic": "refine .of_norm_bounded_eventually_nat 0 summable_zero ?_", "annotated_tactic": ["refine .of_norm_bounded_eventually_nat 0 <a>summable_zero</a> ?_", [{"full_name": "summable_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 14]}]], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\n\u22a2 Summable f", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, \u2016f i\u2016 \u2264 0 i"}, {"tactic": "filter_upwards [h] with _ hn", "annotated_tactic": ["filter_upwards [h] with _ hn", []], "state_before": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, \u2016f i\u2016 \u2264 0 i", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\na\u271d : \u2115\nhn : \u2016f (a\u271d + 1)\u2016 \u2264 r * \u2016f a\u271d\u2016\n\u22a2 \u2016f a\u271d\u2016 \u2264 0 a\u271d"}, {"tactic": "by_contra! h", "annotated_tactic": ["by_contra! h", []], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\na\u271d : \u2115\nhn : \u2016f (a\u271d + 1)\u2016 \u2264 r * \u2016f a\u271d\u2016\n\u22a2 \u2016f a\u271d\u2016 \u2264 0 a\u271d", "state_after": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\na\u271d : \u2115\nhn : \u2016f (a\u271d + 1)\u2016 \u2264 r * \u2016f a\u271d\u2016\nh : 0 a\u271d < \u2016f a\u271d\u2016\n\u22a2 False"}, {"tactic": "exact not_lt.mpr (norm_nonneg _) (lt_of_le_of_lt hn <| mul_neg_of_neg_of_pos hr\u2080 h)", "annotated_tactic": ["exact not_lt.mpr (<a>norm_nonneg</a> _) (<a>lt_of_le_of_lt</a> hn <| <a>mul_neg_of_neg_of_pos</a> hr\u2080 h)", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "mul_neg_of_neg_of_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [423, 9], "def_end_pos": [423, 30]}]], "state_before": "case h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 \u03b1\nr : \u211d\nhr\u2081 : r < 1\nh\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2016f (n + 1)\u2016 \u2264 r * \u2016f n\u2016\nhr\u2080 : r < 0\na\u271d : \u2115\nhn : \u2016f (a\u271d + 1)\u2016 \u2264 r * \u2016f a\u271d\u2016\nh : 0 a\u271d < \u2016f a\u271d\u2016\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Category/Basic.lean", "full_name": "CategoryTheory.whisker_eq", "start": [217, 1], "end": [217, 86], "traced_tactics": [{"tactic": "rw [w]", "annotated_tactic": ["rw [w]", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : g = h\n\u22a2 f \u226b g = f \u226b h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Norm.lean", "full_name": "IntermediateField.AdjoinSimple.norm_gen_eq_one", "start": [214, 1], "end": [221, 68], "traced_tactics": [{"tactic": "rw [norm_eq_one_of_not_exists_basis]", "annotated_tactic": ["rw [<a>norm_eq_one_of_not_exists_basis</a>]", [{"full_name": "Algebra.norm_eq_one_of_not_exists_basis", "def_path": "Mathlib/RingTheory/Norm.lean", "def_pos": [72, 9], "def_end_pos": [72, 40]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 (norm K) (AdjoinSimple.gen K x) = 1", "state_after": "case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u00ac\u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef)"}, {"tactic": "contrapose! hx", "annotated_tactic": ["contrapose! hx", []], "state_before": "case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u00ac\u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef)", "state_after": "case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef)\n\u22a2 IsIntegral K x"}, {"tactic": "obtain \u27e8s, \u27e8b\u27e9\u27e9 := hx", "annotated_tactic": ["obtain \u27e8s, \u27e8b\u27e9\u27e9 := hx", []], "state_before": "case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef)\n\u22a2 IsIntegral K x", "state_after": "case h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 IsIntegral K x"}, {"tactic": "refine .of_mem_of_fg K\u27eex\u27ef.toSubalgebra ?_ x ?_", "annotated_tactic": ["refine .of_mem_of_fg K\u27eex\u27ef.<a>toSubalgebra</a> ?_ x ?_", [{"full_name": "IntermediateField.toSubalgebra", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [54, 14], "def_end_pos": [54, 44]}]], "state_before": "case h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 IsIntegral K x", "state_after": "case h.intro.intro.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 (Subalgebra.toSubmodule K\u27eex\u27ef.toSubalgebra).FG\n\ncase h.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 x \u2208 K\u27eex\u27ef.toSubalgebra"}, {"tactic": "exact (Submodule.fg_iff_finiteDimensional _).mpr (of_fintype_basis b)", "annotated_tactic": ["exact (<a>Submodule.fg_iff_finiteDimensional</a> _).<a>mpr</a> (<a>of_fintype_basis</a> b)", [{"full_name": "Submodule.fg_iff_finiteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [405, 9], "def_end_pos": [405, 33]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "FiniteDimensional.of_fintype_basis", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "case h.intro.intro.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 (Subalgebra.toSubmodule K\u27eex\u27ef.toSubalgebra).FG", "state_after": "no goals"}, {"tactic": "exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)", "annotated_tactic": ["exact <a>IntermediateField.subset_adjoin</a> K _ (<a>Set.mem_singleton</a> x)", [{"full_name": "IntermediateField.subset_adjoin", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [361, 9], "def_end_pos": [361, 22]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case h.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset \u21a5K\u27eex\u27ef\nb : Basis { x_1 // x_1 \u2208 s } K \u21a5K\u27eex\u27ef\n\u22a2 x \u2208 K\u27eex\u27ef.toSubalgebra", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/PartialSups.lean", "full_name": "partialSups_disjoint_of_disjoint", "start": [149, 1], "end": [151, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Permutation.lean", "full_name": "List.permutationsAux2_fst", "start": [56, 1], "end": [59, 76], "traced_tactics": [{"tactic": "simp [permutationsAux2, permutationsAux2_fst t _ _ ys]", "annotated_tactic": ["simp [<a>permutationsAux2</a>, permutationsAux2_fst t _ _ ys]", [{"full_name": "List.permutationsAux2", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [210, 5], "def_end_pos": [210, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\ny : \u03b1\nys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts r (y :: ys) f).1 = y :: ys ++ ts", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.image_iUnion\u2082", "start": [1659, 1], "end": [1660, 81], "traced_tactics": [{"tactic": "simp_rw [image_iUnion]", "annotated_tactic": ["simp_rw [<a>image_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 f '' \u22c3 i, \u22c3 j, s i j = \u22c3 i, \u22c3 j, f '' s i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/NeZero.lean", "full_name": "eq_zero_or_neZero", "start": [48, 1], "end": [49, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.preimage_Ioc", "start": [1399, 1], "end": [1401, 28], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ioc a b = Ioc \u230aa\u230b \u230ab\u230b", "state_after": "case h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ioc a b \u2194 x\u271d \u2208 Ioc \u230aa\u230b \u230ab\u230b"}, {"tactic": "simp [floor_lt, le_floor]", "annotated_tactic": ["simp [<a>floor_lt</a>, <a>le_floor</a>]", [{"full_name": "Int.floor_lt", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [698, 9], "def_end_pos": [698, 17]}, {"full_name": "Int.le_floor", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [694, 9], "def_end_pos": [694, 17]}]], "state_before": "case h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ioc a b \u2194 x\u271d \u2208 Ioc \u230aa\u230b \u230ab\u230b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "full_name": "EuclideanGeometry.two_zsmul_oangle_of_vectorSpan_eq", "start": [258, 1], "end": [263, 61], "traced_tactics": [{"tactic": "simp_rw [vectorSpan_pair] at h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085", "annotated_tactic": ["simp_rw [<a>vectorSpan_pair</a>] at h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085", [{"full_name": "vectorSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 24]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : vectorSpan \u211d {p\u2081, p\u2082} = vectorSpan \u211d {p\u2084, p\u2085}\nh\u2083\u2082\u2086\u2085 : vectorSpan \u211d {p\u2083, p\u2082} = vectorSpan \u211d {p\u2086, p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : Submodule.span \u211d {p\u2081 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2084 -\u1d65 p\u2085}\nh\u2083\u2082\u2086\u2085 : Submodule.span \u211d {p\u2083 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2086 -\u1d65 p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086"}, {"tactic": "exact o.two_zsmul_oangle_of_span_eq_of_span_eq h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085", "annotated_tactic": ["exact o.two_zsmul_oangle_of_span_eq_of_span_eq h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : Submodule.span \u211d {p\u2081 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2084 -\u1d65 p\u2085}\nh\u2083\u2082\u2086\u2085 : Submodule.span \u211d {p\u2083 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2086 -\u1d65 p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.mk_mem_graph", "start": [83, 1], "end": [84, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Maps.lean", "full_name": "SimpleGraph.Iso.toEmbedding_completeGraph", "start": [632, 1], "end": [634, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Matrix.lean", "full_name": "Matrix.frobenius_nnnorm_diagonal", "start": [634, 1], "end": [645, 77], "traced_tactics": [{"tactic": "simp_rw [frobenius_nnnorm_def, \u2190 Finset.sum_product', Finset.univ_product_univ,\n  PiLp.nnnorm_eq_of_L2]", "annotated_tactic": ["simp_rw [<a>frobenius_nnnorm_def</a>, \u2190 <a>Finset.sum_product'</a>, <a>Finset.univ_product_univ</a>,\n    <a>PiLp.nnnorm_eq_of_L2</a>]", [{"full_name": "Matrix.frobenius_nnnorm_def", "def_path": "Mathlib/Analysis/Matrix.lean", "def_pos": [559, 9], "def_end_pos": [559, 29]}, {"full_name": "Finset.sum_product'", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [895, 3], "def_end_pos": [895, 14]}, {"full_name": "Finset.univ_product_univ", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [50, 15], "def_end_pos": [50, 32]}, {"full_name": "PiLp.nnnorm_eq_of_L2", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [629, 9], "def_end_pos": [629, 24]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 \u2016diagonal v\u2016\u208a = \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v\u2016\u208a", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)"}, {"tactic": "let s := (Finset.univ : Finset n).map \u27e8fun i : n => (i, i), fun i j h => congr_arg Prod.fst h\u27e9", "annotated_tactic": ["let s := (<a>Finset.univ</a> : <a>Finset</a> n).<a>map</a> \u27e8fun i : n => (i, i), fun i j h => <a>congr_arg</a> <a>Prod.fst</a> h\u27e9", [{"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [70, 5], "def_end_pos": [70, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [58, 5], "def_end_pos": [58, 8]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Prod.fst", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [483, 3], "def_end_pos": [483, 6]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)"}, {"tactic": "rw [\u2190 Finset.sum_subset (Finset.subset_univ s) fun i _hi his => ?_]", "annotated_tactic": ["rw [\u2190 <a>Finset.sum_subset</a> (<a>Finset.subset_univ</a> s) fun i _hi his => ?_]", [{"full_name": "Finset.sum_subset", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [594, 3], "def_end_pos": [594, 14]}, {"full_name": "Finset.subset_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 20]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x \u2208 s, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)\n\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : i \u2209 s\n\u22a2 \u2016diagonal v i.1 i.2\u2016\u208a ^ 2 = 0"}, {"tactic": "rw [Finset.sum_map, NNReal.sqrt_eq_rpow]", "annotated_tactic": ["rw [<a>Finset.sum_map</a>, <a>NNReal.sqrt_eq_rpow</a>]", [{"full_name": "Finset.sum_map", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [407, 3], "def_end_pos": [407, 14]}, {"full_name": "NNReal.sqrt_eq_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [128, 9], "def_end_pos": [128, 21]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x \u2208 s, \u2016diagonal v x.1 x.2\u2016\u208a ^ 2) ^ (1 / 2) = NNReal.sqrt (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2)", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n,\n        \u2016diagonal v ({ toFun := fun i => (i, i), inj' := \u22ef } x).1 ({ toFun := fun i => (i, i), inj' := \u22ef } x).2\u2016\u208a ^ 2) ^\n      (1 / 2) =\n    (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2) ^ (1 / 2)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n,\n        \u2016diagonal v ({ toFun := fun i => (i, i), inj' := \u22ef } x).1 ({ toFun := fun i => (i, i), inj' := \u22ef } x).2\u2016\u208a ^ 2) ^\n      (1 / 2) =\n    (\u2211 i : n, \u2016(WithLp.equiv 2 (n \u2192 \u03b1)).symm v i\u2016\u208a ^ 2) ^ (1 / 2)", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n, \u2016diagonal v x x\u2016\u208a ^ 2) ^ (1 / 2) = (\u2211 i : n, \u2016v i\u2016\u208a ^ 2) ^ (1 / 2)"}, {"tactic": "simp_rw [diagonal_apply_eq, NNReal.rpow_two]", "annotated_tactic": ["simp_rw [<a>diagonal_apply_eq</a>, <a>NNReal.rpow_two</a>]", [{"full_name": "Matrix.diagonal_apply_eq", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [397, 9], "def_end_pos": [397, 26]}, {"full_name": "NNReal.rpow_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [147, 9], "def_end_pos": [147, 17]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\n\u22a2 (\u2211 x : n, \u2016diagonal v x x\u2016\u208a ^ 2) ^ (1 / 2) = (\u2211 i : n, \u2016v i\u2016\u208a ^ 2) ^ (1 / 2)", "state_after": "no goals"}, {"tactic": "suffices i.1 \u2260 i.2 by rw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero]", "annotated_tactic": ["suffices i.1 \u2260 i.2 by rw [<a>diagonal_apply_ne</a> _ this, <a>nnnorm_zero</a>, <a>NNReal.zero_rpow</a> <a>two_ne_zero</a>]", [{"full_name": "Matrix.diagonal_apply_ne", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [402, 9], "def_end_pos": [402, 26]}, {"full_name": "nnnorm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [791, 30], "def_end_pos": [791, 41]}, {"full_name": "NNReal.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : i \u2209 s\n\u22a2 \u2016diagonal v i.1 i.2\u2016\u208a ^ 2 = 0", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : i \u2209 s\n\u22a2 i.1 \u2260 i.2"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : i \u2209 s\n\u22a2 i.1 \u2260 i.2", "state_after": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b9 : Type u_7\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Unique \u03b9\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) := Finset.map { toFun := fun i => (i, i), inj' := \u22ef } Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : i \u2209 s\nh : i.1 = i.2\n\u22a2 False"}, 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u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\n\u22a2 NoZeroSMulDivisors R M", "state_after": "R\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r \u2022 m = 0\n\u22a2 r = 0 \u2228 m = 0"}, {"tactic": "rw [algebra_compatible_smul A r m] at h", "annotated_tactic": ["rw [<a>algebra_compatible_smul</a> A r m] at h", [{"full_name": "algebra_compatible_smul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 32]}]], "state_before": "R\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r \u2022 m = 0\n\u22a2 r = 0 \u2228 m = 0", "state_after": "R\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\n\u22a2 r = 0 \u2228 m = 0"}, {"tactic": "cases' smul_eq_zero.1 h with H H", "annotated_tactic": ["cases' <a>smul_eq_zero</a>.1 h with H H", [{"full_name": "smul_eq_zero", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}]], "state_before": "R\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\n\u22a2 r = 0 \u2228 m = 0", "state_after": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\n\u22a2 r = 0 \u2228 m = 0\n\ncase inr\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : m = 0\n\u22a2 r = 0 \u2228 m = 0"}, {"tactic": "have : Function.Injective (algebraMap R A) :=\n  NoZeroSMulDivisors.iff_algebraMap_injective.1 inferInstance", "annotated_tactic": ["have : <a>Function.Injective</a> (<a>algebraMap</a> R A) :=\n      <a>NoZeroSMulDivisors.iff_algebraMap_injective</a>.1 <a>inferInstance</a>", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "NoZeroSMulDivisors.iff_algebraMap_injective", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [365, 9], "def_end_pos": [365, 33]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\n\u22a2 r = 0 \u2228 m = 0", "state_after": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\nthis : Function.Injective \u21d1(algebraMap R A)\n\u22a2 r = 0 \u2228 m = 0"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\nthis : Function.Injective \u21d1(algebraMap R A)\n\u22a2 r = 0 \u2228 m = 0", "state_after": "case inl.h\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\nthis : Function.Injective \u21d1(algebraMap R A)\n\u22a2 r = 0"}, {"tactic": "exact (injective_iff_map_eq_zero _).1 this _ H", "annotated_tactic": ["exact (<a>injective_iff_map_eq_zero</a> _).1 this _ H", [{"full_name": "injective_iff_map_eq_zero", "def_path": "Mathlib/Algebra/Group/Hom/Basic.lean", "def_pos": [125, 3], "def_end_pos": [125, 14]}]], "state_before": "case inl.h\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : (algebraMap R A) r = 0\nthis : Function.Injective \u21d1(algebraMap R A)\n\u22a2 r = 0", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case inr\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : m = 0\n\u22a2 r = 0 \u2228 m = 0", "state_after": "case inr.h\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : m = 0\n\u22a2 m = 0"}, {"tactic": "exact H", "annotated_tactic": ["exact H", []], "state_before": "case inr.h\nR\u271d : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring A\u271d\ninst\u271d\u00b9\u2078 : Algebra R\u271d A\u271d\nM\u271d : Type u_3\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module A\u271d M\u271d\ninst\u271d\u00b9\u2075 : Module R\u271d M\u271d\ninst\u271d\u00b9\u2074 : IsScalarTower R\u271d A\u271d M\u271d\nN : Type u_4\ninst\u271d\u00b9\u00b3 : AddCommMonoid N\ninst\u271d\u00b9\u00b2 : Module A\u271d N\ninst\u271d\u00b9\u00b9 : Module R\u271d N\ninst\u271d\u00b9\u2070 : IsScalarTower R\u271d A\u271d N\nR : Type u_5\nA : Type u_6\nM : Type u_7\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module A M\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : NoZeroSMulDivisors A M\nr : R\nm : M\nh : (algebraMap R A) r \u2022 m = 0\nH : m = 0\n\u22a2 m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.not_lt_zero", "start": [408, 11], "end": [409, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "full_name": "CategoryTheory.Subobject.ofLEMk_comp", "start": [349, 1], "end": [350, 51], "traced_tactics": [{"tactic": "simp [ofLEMk]", "annotated_tactic": ["simp [<a>ofLEMk</a>]", [{"full_name": "CategoryTheory.Subobject.ofLEMk", "def_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "def_pos": [339, 5], "def_end_pos": [339, 11]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh : X \u2264 mk f\n\u22a2 X.ofLEMk f h \u226b f = X.arrow", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Closeds.lean", "full_name": "EMetric.isClosed_subsets_of_isClosed", "start": [74, 1], "end": [84, 30], "traced_tactics": [{"tactic": "refine isClosed_of_closure_subset fun\n  (t : Closeds \u03b1) (ht : t \u2208 closure {t : Closeds \u03b1 | (t : Set \u03b1) \u2286 s}) (x : \u03b1) (hx : x \u2208 t) => ?_", "annotated_tactic": ["refine <a>isClosed_of_closure_subset</a> fun\n    (t : <a>Closeds</a> \u03b1) (ht : t \u2208 <a>closure</a> {t : <a>Closeds</a> \u03b1 | (t : <a>Set</a> \u03b1) \u2286 s}) (x : \u03b1) (hx : x \u2208 t) => ?_", [{"full_name": "isClosed_of_closure_subset", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [460, 9], "def_end_pos": [460, 35]}, {"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 IsClosed {t | \u2191t \u2286 s}", "state_after": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s"}, {"tactic": "have : x \u2208 closure s := by\n  refine mem_closure_iff.2 fun \u03b5 \u03b5pos => ?_\n  obtain \u27e8u : Closeds \u03b1, hu : u \u2208 {t : Closeds \u03b1 | (t : Set \u03b1) \u2286 s}, Dtu : edist t u < \u03b5\u27e9 :=\n    mem_closure_iff.1 ht \u03b5 \u03b5pos\n  obtain \u27e8y : \u03b1, hy : y \u2208 u, Dxy : edist x y < \u03b5\u27e9 := exists_edist_lt_of_hausdorffEdist_lt hx Dtu\n  exact \u27e8y, hu hy, Dxy\u27e9", "annotated_tactic": ["have : x \u2208 <a>closure</a> s := by\n    refine <a>mem_closure_iff</a>.2 fun \u03b5 \u03b5pos => ?_\n    obtain \u27e8u : <a>Closeds</a> \u03b1, hu : u \u2208 {t : <a>Closeds</a> \u03b1 | (t : <a>Set</a> \u03b1) \u2286 s}, Dtu : <a>edist</a> t u < \u03b5\u27e9 :=\n      <a>mem_closure_iff</a>.1 ht \u03b5 \u03b5pos\n    obtain \u27e8y : \u03b1, hy : y \u2208 u, Dxy : <a>edist</a> x y < \u03b5\u27e9 := <a>exists_edist_lt_of_hausdorffEdist_lt</a> hx Dtu\n    exact \u27e8y, hu hy, Dxy\u27e9", [{"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}, {"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [52, 3], "def_end_pos": [52, 8]}, {"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}, {"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [52, 3], "def_end_pos": [52, 8]}, {"full_name": "EMetric.exists_edist_lt_of_hausdorffEdist_lt", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [326, 9], "def_end_pos": [326, 45]}]], "state_before": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s", "state_after": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\nthis : x \u2208 closure s\n\u22a2 x \u2208 s"}, {"tactic": "rwa [hs.closure_eq] at this", "annotated_tactic": ["rwa [hs.closure_eq] at this", []], "state_before": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\nthis : x \u2208 closure s\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "refine mem_closure_iff.2 fun \u03b5 \u03b5pos => ?_", "annotated_tactic": ["refine <a>mem_closure_iff</a>.2 fun \u03b5 \u03b5pos => ?_", [{"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}]], "state_before": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 closure s", "state_after": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5"}, {"tactic": "obtain \u27e8u : Closeds \u03b1, hu : u \u2208 {t : Closeds \u03b1 | (t : Set \u03b1) \u2286 s}, Dtu : edist t u < \u03b5\u27e9 :=\n  mem_closure_iff.1 ht \u03b5 \u03b5pos", "annotated_tactic": ["obtain \u27e8u : <a>Closeds</a> \u03b1, hu : u \u2208 {t : <a>Closeds</a> \u03b1 | (t : <a>Set</a> \u03b1) \u2286 s}, Dtu : <a>edist</a> t u < \u03b5\u27e9 :=\n      <a>mem_closure_iff</a>.1 ht \u03b5 \u03b5pos", [{"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "TopologicalSpace.Closeds", "def_path": "Mathlib/Topology/Sets/Closeds.lean", "def_pos": [33, 11], "def_end_pos": [33, 18]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [52, 3], "def_end_pos": [52, 8]}, {"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}]], "state_before": "\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5"}, {"tactic": "obtain \u27e8y : \u03b1, hy : y \u2208 u, Dxy : edist x y < \u03b5\u27e9 := exists_edist_lt_of_hausdorffEdist_lt hx Dtu", "annotated_tactic": ["obtain \u27e8y : \u03b1, hy : y \u2208 u, Dxy : <a>edist</a> x y < \u03b5\u27e9 := <a>exists_edist_lt_of_hausdorffEdist_lt</a> hx Dtu", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [52, 3], "def_end_pos": [52, 8]}, {"full_name": "EMetric.exists_edist_lt_of_hausdorffEdist_lt", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [326, 9], "def_end_pos": [326, 45]}]], "state_before": "case intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\ny : \u03b1\nhy : y \u2208 u\nDxy : edist x y < \u03b5\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5"}, {"tactic": "exact \u27e8y, hu hy, Dxy\u27e9", "annotated_tactic": ["exact \u27e8y, hu hy, Dxy\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\ny : \u03b1\nhy : y \u2208 u\nDxy : edist x y < \u03b5\n\u22a2 \u2203 y \u2208 s, edist x y < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.add", "start": [262, 1], "end": [264, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "full_name": "FiniteDimensional.of_locallyCompactSpace", "start": [505, 1], "end": [508, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.iIndepFun.indepFun_prod_mk", "start": [932, 1], "end": [956, 29], "traced_tactics": [{"tactic": "classical\nhave h_right : f k =\n  (fun p : \u2200 j : ({k} : Finset \u03b9), \u03b2 j => p \u27e8k, Finset.mem_singleton_self k\u27e9) \u2218\n  fun a (j : ({k} : Finset \u03b9)) => f j a := rfl\nhave h_meas_right :  Measurable fun p : \u2200 j : ({k} : Finset \u03b9),\n  \u03b2 j => p \u27e8k, Finset.mem_singleton_self k\u27e9 := measurable_pi_apply _\nlet s : Finset \u03b9 := {i, j}\nhave h_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p : \u2200 l : s, \u03b2 l =>\n  (p \u27e8i, Finset.mem_insert_self i _\u27e9,\n  p \u27e8j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)\u27e9)) \u2218 fun a (j : s) => f j a := by\n  ext1 a\n  simp only [Prod.mk.inj_iff]\n  constructor\nhave h_meas_left : Measurable fun p : \u2200 l : s, \u03b2 l =>\n  (p \u27e8i, Finset.mem_insert_self i _\u27e9,\n  p \u27e8j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)\u27e9) :=\n    Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)\nrw [h_left, h_right]\nrefine (hf_Indep.indepFun_finset s {k} ?_ hf_meas).comp h_meas_left h_meas_right\nrw [Finset.disjoint_singleton_right]\nsimp only [s, Finset.mem_insert, Finset.mem_singleton, not_or]\nexact \u27e8hik.symm, hjk.symm\u27e9", "annotated_tactic": ["classical\n  have h_right : f k =\n    (fun p : \u2200 j : ({k} : <a>Finset</a> \u03b9), \u03b2 j => p \u27e8k, <a>Finset.mem_singleton_self</a> k\u27e9) \u2218\n    fun a (j : ({k} : <a>Finset</a> \u03b9)) => f j a := <a>rfl</a>\n  have h_meas_right :  <a>Measurable</a> fun p : \u2200 j : ({k} : <a>Finset</a> \u03b9),\n    \u03b2 j => p \u27e8k, <a>Finset.mem_singleton_self</a> k\u27e9 := <a>measurable_pi_apply</a> _\n  let s : <a>Finset</a> \u03b9 := {i, j}\n  have h_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p : \u2200 l : s, \u03b2 l =>\n    (p \u27e8i, <a>Finset.mem_insert_self</a> i _\u27e9,\n    p \u27e8j, <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_singleton_self</a> _)\u27e9)) \u2218 fun a (j : s) => f j a := by\n    ext1 a\n    simp only [<a>Prod.mk.inj_iff</a>]\n    constructor\n  have h_meas_left : <a>Measurable</a> fun p : \u2200 l : s, \u03b2 l =>\n    (p \u27e8i, <a>Finset.mem_insert_self</a> i _\u27e9,\n    p \u27e8j, <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_singleton_self</a> _)\u27e9) :=\n      <a>Measurable.prod</a> (<a>measurable_pi_apply</a> _) (<a>measurable_pi_apply</a> _)\n  rw [h_left, h_right]\n  refine (hf_Indep.indepFun_finset s {k} ?_ hf_meas).<a>comp</a> h_meas_left h_meas_right\n  rw [<a>Finset.disjoint_singleton_right</a>]\n  simp only [s, <a>Finset.mem_insert</a>, <a>Finset.mem_singleton</a>, <a>not_or</a>]\n  exact \u27e8hik.symm, hjk.symm\u27e9", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [551, 5], "def_end_pos": [551, 15]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 26]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [551, 5], "def_end_pos": [551, 15]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 26]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Measurable.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}, {"full_name": "ProbabilityTheory.kernel.IndepFun.comp", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [803, 9], "def_end_pos": [803, 22]}, {"full_name": "Finset.disjoint_singleton_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1011, 9], "def_end_pos": [1011, 33]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 22]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "no goals"}, {"tactic": "have h_right : f k =\n  (fun p : \u2200 j : ({k} : Finset \u03b9), \u03b2 j => p \u27e8k, Finset.mem_singleton_self k\u27e9) \u2218\n  fun a (j : ({k} : Finset \u03b9)) => f j a := rfl", "annotated_tactic": ["have h_right : f k =\n    (fun p : \u2200 j : ({k} : <a>Finset</a> \u03b9), \u03b2 j => p \u27e8k, <a>Finset.mem_singleton_self</a> k\u27e9) \u2218\n    fun a (j : ({k} : <a>Finset</a> \u03b9)) => f j a := <a>rfl</a>", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc"}, {"tactic": "have h_meas_right :  Measurable fun p : \u2200 j : ({k} : Finset \u03b9),\n  \u03b2 j => p \u27e8k, Finset.mem_singleton_self k\u27e9 := measurable_pi_apply _", "annotated_tactic": ["have h_meas_right :  <a>Measurable</a> fun p : \u2200 j : ({k} : <a>Finset</a> \u03b9),\n    \u03b2 j => p \u27e8k, <a>Finset.mem_singleton_self</a> k\u27e9 := <a>measurable_pi_apply</a> _", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [551, 5], "def_end_pos": [551, 15]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc"}, {"tactic": "let s : Finset \u03b9 := {i, j}", "annotated_tactic": ["let s : <a>Finset</a> \u03b9 := {i, j}", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc"}, {"tactic": "have h_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p : \u2200 l : s, \u03b2 l =>\n  (p \u27e8i, Finset.mem_insert_self i _\u27e9,\n  p \u27e8j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)\u27e9)) \u2218 fun a (j : s) => f j a := by\n  ext1 a\n  simp only [Prod.mk.inj_iff]\n  constructor", "annotated_tactic": ["have h_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p : \u2200 l : s, \u03b2 l =>\n    (p \u27e8i, <a>Finset.mem_insert_self</a> i _\u27e9,\n    p \u27e8j, <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_singleton_self</a> _)\u27e9)) \u2218 fun a (j : s) => f j a := by\n    ext1 a\n    simp only [<a>Prod.mk.inj_iff</a>]\n    constructor", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 26]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc"}, {"tactic": "have h_meas_left : Measurable fun p : \u2200 l : s, \u03b2 l =>\n  (p \u27e8i, Finset.mem_insert_self i _\u27e9,\n  p \u27e8j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)\u27e9) :=\n    Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)", "annotated_tactic": ["have h_meas_left : <a>Measurable</a> fun p : \u2200 l : s, \u03b2 l =>\n    (p \u27e8i, <a>Finset.mem_insert_self</a> i _\u27e9,\n    p \u27e8j, <a>Finset.mem_insert_of_mem</a> (<a>Finset.mem_singleton_self</a> _)\u27e9) :=\n      <a>Measurable.prod</a> (<a>measurable_pi_apply</a> _) (<a>measurable_pi_apply</a> _)", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [551, 5], "def_end_pos": [551, 15]}, {"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 24]}, {"full_name": "Finset.mem_insert_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 26]}, {"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}, {"full_name": "Measurable.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [729, 9], "def_end_pos": [729, 24]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}, {"full_name": "measurable_pi_apply", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc"}, {"tactic": "rw [h_left, h_right]", "annotated_tactic": ["rw [h_left, h_right]", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 IndepFun (fun a => (f i a, f j a)) (f k) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 IndepFun ((fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a) ((fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a) \u03ba \u03bc"}, {"tactic": "refine (hf_Indep.indepFun_finset s {k} ?_ hf_meas).comp h_meas_left h_meas_right", "annotated_tactic": ["refine (hf_Indep.indepFun_finset s {k} ?_ hf_meas).<a>comp</a> h_meas_left h_meas_right", [{"full_name": "ProbabilityTheory.kernel.IndepFun.comp", "def_path": "Mathlib/Probability/Independence/Kernel.lean", "def_pos": [803, 9], "def_end_pos": [803, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 IndepFun ((fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a) ((fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a) \u03ba \u03bc", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 Disjoint s {k}"}, {"tactic": "rw [Finset.disjoint_singleton_right]", "annotated_tactic": ["rw [<a>Finset.disjoint_singleton_right</a>]", [{"full_name": "Finset.disjoint_singleton_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1011, 9], "def_end_pos": [1011, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 Disjoint s {k}", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 k \u2209 s"}, {"tactic": "simp only [s, Finset.mem_insert, Finset.mem_singleton, not_or]", "annotated_tactic": ["simp only [s, <a>Finset.mem_insert</a>, <a>Finset.mem_singleton</a>, <a>not_or</a>]", [{"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 22]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 k \u2209 s", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 \u00ack = i \u2227 \u00ack = j"}, {"tactic": "exact \u27e8hik.symm, hjk.symm\u27e9", "annotated_tactic": ["exact \u27e8hik.symm, hjk.symm\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\nh_left : (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a\nh_meas_left : Measurable fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)\n\u22a2 \u00ack = i \u2227 \u00ack = j", "state_after": "no goals"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\n\u22a2 (fun \u03c9 => (f i \u03c9, f j \u03c9)) = (fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a", "state_after": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\na : \u03a9\n\u22a2 (f i a, f j a) = ((fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a) a"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03a9 : Type u_2\n\u03b9 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\n_m\u03b1 : MeasurableSpace \u03b1\n_m\u03a9 : MeasurableSpace \u03a9\n\u03ba : \u21a5(kernel \u03b1 \u03a9)\n\u03bc : Measure \u03b1\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b2 : \u03b9 \u2192 Type u_8\nm : (i : \u03b9) \u2192 MeasurableSpace (\u03b2 i)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\ninst\u271d : IsMarkovKernel \u03ba\nhf_Indep : iIndepFun m f \u03ba \u03bc\nhf_meas : \u2200 (i : \u03b9), Measurable (f i)\ni j k : \u03b9\nhik : i \u2260 k\nhjk : j \u2260 k\nh_right : f k = (fun p => p \u27e8k, \u22ef\u27e9) \u2218 fun a j => f (\u2191j) a\nh_meas_right : Measurable fun p => p \u27e8k, \u22ef\u27e9\ns : Finset \u03b9 := {i, j}\na : \u03a9\n\u22a2 (f i a, f j a) = ((fun p => (p \u27e8i, \u22ef\u27e9, p \u27e8j, \u22ef\u27e9)) \u2218 fun a j => f (\u2191j) a) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "Units.coe_map'", "start": [419, 1], "end": [420, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Defs.lean", "full_name": "mul_neg", "start": [323, 1], "end": [324, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean", "full_name": "WittVector.RecursionBase.solution_spec", "start": [158, 1], "end": [159, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "full_name": "Ordinal.opow_le_iff_le_log", "start": [331, 1], "end": [336, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.liftp_iff", "start": [101, 1], "end": [114, 26], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2194 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)", "state_after": "case mp\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2192 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)\n\ncase mpr\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)) \u2192 Liftp p x"}, {"tactic": "rintro \u27e8a, f, h\u2080, h\u2081\u27e9", "annotated_tactic": ["rintro \u27e8a, f, h\u2080, h\u2081\u27e9", []], "state_before": "case mpr\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)) \u2192 Liftp p x", "state_after": "case mpr.intro.intro.intro\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 Liftp p x"}, {"tactic": "use abs \u27e8a, fun i => \u27e8f i, h\u2081 i\u27e9\u27e9", "annotated_tactic": ["use <a>abs</a> \u27e8a, fun i => \u27e8f i, h\u2081 i\u27e9\u27e9", [{"full_name": "QPF.abs", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}]], "state_before": "case mpr.intro.intro.intro\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 Liftp p x", "state_after": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 Subtype.val <$> abs \u27e8a, fun i => \u27e8f i, \u22ef\u27e9\u27e9 = x"}, {"tactic": "rw [\u2190 abs_map, h\u2080]", "annotated_tactic": ["rw [\u2190 <a>abs_map</a>, h\u2080]", [{"full_name": "QPF.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "state_before": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 Subtype.val <$> abs \u27e8a, fun i => \u27e8f i, \u22ef\u27e9\u27e9 = x", "state_after": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 abs ((P F).map Subtype.val \u27e8a, fun i => \u27e8f i, \u22ef\u27e9\u27e9) = abs \u27e8a, f\u27e9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh\u2080 : x = abs \u27e8a, f\u27e9\nh\u2081 : \u2200 (i : (P F).B a), p (f i)\n\u22a2 abs ((P F).map Subtype.val \u27e8a, fun i => \u27e8f i, \u22ef\u27e9\u27e9) = abs \u27e8a, f\u27e9", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hy\u27e9", []], "state_before": "case mp\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2192 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)", "state_after": "case mp.intro\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)"}, {"tactic": "cases' h : repr y with a f", "annotated_tactic": ["cases' h : <a>repr</a> y with a f", [{"full_name": "QPF.repr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [54, 3], "def_end_pos": [54, 7]}]], "state_before": "case mp.intro\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)", "state_after": "case mp.intro.mk\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)"}, {"tactic": "use a, fun i => (f i).val", "annotated_tactic": ["use a, fun i => (f i).<a>val</a>", [{"full_name": "Subtype.val", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [587, 3], "def_end_pos": [587, 6]}]], "state_before": "case mp.intro.mk\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : (P F).B a), p (f i)", "state_after": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 x = abs \u27e8a, fun i => \u2191(f i)\u27e9 \u2227 \u2200 (i : (P F).B a), p \u2191(f i)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 x = abs \u27e8a, fun i => \u2191(f i)\u27e9 \u2227 \u2200 (i : (P F).B a), p \u2191(f i)", "state_after": "case h.left\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 x = abs \u27e8a, fun i => \u2191(f i)\u27e9\n\ncase h.right\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 \u2200 (i : (P F).B a), p \u2191(f i)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case h.right\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 \u2200 (i : (P F).B a), p \u2191(f i)", "state_after": "case h.right\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\ni : (P F).B a\n\u22a2 p \u2191(f i)"}, {"tactic": "apply (f i).property", "annotated_tactic": ["apply (f i).<a>property</a>", [{"full_name": "Subtype.property", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [590, 3], "def_end_pos": [590, 11]}]], "state_before": "case h.right\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\ni : (P F).B a\n\u22a2 p \u2191(f i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hy, \u2190 abs_repr y, h, \u2190 abs_map]", "annotated_tactic": ["rw [\u2190 hy, \u2190 <a>abs_repr</a> y, h, \u2190 <a>abs_map</a>]", [{"full_name": "QPF.abs_repr", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [55, 3], "def_end_pos": [55, 11]}, {"full_name": "QPF.abs_map", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [56, 3], "def_end_pos": [56, 10]}]], "state_before": "case h.left\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 x = abs \u27e8a, fun i => \u2191(f i)\u27e9", "state_after": "case h.left\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 abs ((P F).map Subtype.val \u27e8a, f\u27e9) = abs \u27e8a, fun i => \u2191(f i)\u27e9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.left\nF : Type u \u2192 Type u\nq : QPF F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : (P F).B a \u2192 Subtype p\nh : repr y = \u27e8a, f\u27e9\n\u22a2 abs ((P F).map Subtype.val \u27e8a, f\u27e9) = abs \u27e8a, fun i => \u2191(f i)\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "differentiableWithinAt_univ", "start": [631, 1], "end": [633, 79], "traced_tactics": [{"tactic": "simp only [DifferentiableWithinAt, hasFDerivWithinAt_univ, DifferentiableAt]", "annotated_tactic": ["simp only [<a>DifferentiableWithinAt</a>, <a>hasFDerivWithinAt_univ</a>, <a>DifferentiableAt</a>]", [{"full_name": "DifferentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [171, 5], "def_end_pos": [171, 27]}, {"full_name": "hasFDerivWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 31]}, {"full_name": "DifferentiableAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [178, 5], "def_end_pos": [178, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 DifferentiableWithinAt \ud835\udd5c f univ x \u2194 DifferentiableAt \ud835\udd5c f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.add_apply", "start": [438, 1], "end": [439, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.Embedding.coe_injective", "start": [647, 1], "end": [653, 36], "traced_tactics": [{"tactic": "cases f", "annotated_tactic": ["cases f", []], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\nf g : M \u21aa[L] N\nh : \u21d1f = \u21d1g\n\u22a2 f = g", "state_after": "case mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ng : M \u21aa[L] N\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d } = \u21d1g\n\u22a2 { toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d } = g"}, {"tactic": "cases g", "annotated_tactic": ["cases g", []], "state_before": "case mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ng : M \u21aa[L] N\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d } = \u21d1g\n\u22a2 { toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d } = g", "state_after": "case mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d\u00b9.toFun (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh :\n  \u21d1{ toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\n\u22a2 { toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    { toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d\u00b9.toFun (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh :\n  \u21d1{ toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\n\u22a2 { toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    { toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }", "state_after": "case mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d\u00b9.toFun (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh :\n  \u21d1{ toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\n\u22a2 toEmbedding\u271d\u00b9 = toEmbedding\u271d"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d\u00b9.toFun (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh :\n  \u21d1{ toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\n\u22a2 toEmbedding\u271d\u00b9 = toEmbedding\u271d", "state_after": "case mk.mk.e_toEmbedding.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\nP : Type u_1\ninst\u271d\u00b9 : L.Structure P\nQ : Type u_2\ninst\u271d : L.Structure Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d\u00b9.toFun (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n  \u2200 {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M), toEmbedding\u271d.toFun (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : L.Relations n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh :\n  \u21d1{ toEmbedding := toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\nx : M\n\u22a2 toEmbedding\u271d\u00b9 x = toEmbedding\u271d x"}, {"tactic": "exact Function.funext_iff.1 h x", "annotated_tactic": ["exact <a>Function.funext_iff</a>.1 h x", [{"full_name": "Function.funext_iff", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [63, 9], "def_end_pos": [63, 28]}]], 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:= toEmbedding\u271d\u00b9, map_fun' := map_fun'\u271d\u00b9, map_rel' := map_rel'\u271d\u00b9 } =\n    \u21d1{ toEmbedding := toEmbedding\u271d, map_fun' := map_fun'\u271d, map_rel' := map_rel'\u271d }\nx : M\n\u22a2 toEmbedding\u271d\u00b9 x = toEmbedding\u271d x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/WithOne/Defs.lean", "full_name": "WithOne.ne_one_iff_exists", "start": [164, 1], "end": [165, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Monic.lean", "full_name": "Polynomial.Monic.mul_left_ne_zero", "start": [449, 1], "end": [456, 7], "traced_tactics": [{"tactic": "by_cases h : p = 1", "annotated_tactic": ["by_cases h : p = 1", []], "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\n\u22a2 q * p \u2260 0", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : p = 1\n\u22a2 q * p \u2260 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 q * p \u2260 0"}, {"tactic": "rw [Ne, \u2190 degree_eq_bot, hp.degree_mul, WithBot.add_eq_bot, not_or, degree_eq_bot]", "annotated_tactic": ["rw [<a>Ne</a>, \u2190 <a>degree_eq_bot</a>, hp.degree_mul, <a>WithBot.add_eq_bot</a>, <a>not_or</a>, <a>degree_eq_bot</a>]", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}, {"full_name": "WithBot.add_eq_bot", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [634, 9], "def_end_pos": [634, 19]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 q * p \u2260 0", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 \u00acq = 0 \u2227 \u00acp.degree = \u22a5"}, {"tactic": "refine \u27e8hq, ?_\u27e9", "annotated_tactic": ["refine \u27e8hq, ?_\u27e9", []], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 \u00acq = 0 \u2227 \u00acp.degree = \u22a5", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 \u00acp.degree = \u22a5"}, {"tactic": "rw [\u2190 hp.degree_le_zero_iff_eq_one, not_le] at h", "annotated_tactic": ["rw [\u2190 hp.degree_le_zero_iff_eq_one, <a>not_le</a>] at h", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : \u00acp = 1\n\u22a2 \u00acp.degree = \u22a5", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : 0 < p.degree\n\u22a2 \u00acp.degree = \u22a5"}, {"tactic": "refine (lt_trans ?_ h).ne'", "annotated_tactic": ["refine (<a>lt_trans</a> ?_ h).<a>ne'</a>", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : 0 < p.degree\n\u22a2 \u00acp.degree = \u22a5", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np : R[X]\nhp : p.Monic\nq : R[X]\nhq : q \u2260 0\nh : 0 < p.degree\n\u22a2 \u22a5 < 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], 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"Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.toNonUnitalSubring_inj", "start": [195, 1], "end": [198, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order.lean", "full_name": "denseRange_discrete", "start": [293, 1], "end": [294, 56], "traced_tactics": [{"tactic": "rw [DenseRange, dense_discrete, range_iff_surjective]", "annotated_tactic": ["rw [<a>DenseRange</a>, <a>dense_discrete</a>, <a>range_iff_surjective</a>]", [{"full_name": "DenseRange", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [131, 5], "def_end_pos": [131, 15]}, {"full_name": "dense_discrete", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [290, 17], "def_end_pos": [290, 31]}, {"full_name": "Set.range_iff_surjective", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [689, 9], "def_end_pos": [689, 29]}]], "state_before": "\u03b1 : Type u_1\nt t\u2081 t\u2082 : TopologicalSpace \u03b1\ns : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1\n\u22a2 DenseRange f \u2194 Surjective f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "tendsto_nhdsWithin_range", "start": [466, 1], "end": [469, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/StarRingHom.lean", "full_name": "NonUnitalStarRingHom.coe_coe", "start": [107, 11], "end": [109, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Compare.lean", "full_name": "cmp_eq_gt_iff", "start": [236, 1], "end": [237, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Hom/Basic.lean", "full_name": "ContinuousOrderHom.coe_copy", "start": [128, 1], "end": [129, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Pointwise/Stabilizer.lean", "full_name": "MulAction.map_stabilizer_le", "start": [44, 1], "end": [50, 32], "traced_tactics": [{"tactic": "rintro a", "annotated_tactic": ["rintro a", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na : G\nf : G \u2192* H\ns : Set G\n\u22a2 Subgroup.map f (stabilizer G s) \u2264 stabilizer H (\u21d1f '' s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : H\n\u22a2 a \u2208 Subgroup.map f (stabilizer G s) \u2192 a \u2208 stabilizer H (\u21d1f '' s)"}, {"tactic": "simp only [Subgroup.mem_map, mem_stabilizer_iff, exists_prop, forall_exists_index, and_imp]", "annotated_tactic": ["simp only [<a>Subgroup.mem_map</a>, <a>mem_stabilizer_iff</a>, <a>exists_prop</a>, <a>forall_exists_index</a>, <a>and_imp</a>]", [{"full_name": "Subgroup.mem_map", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 16]}, {"full_name": "MulAction.mem_stabilizer_iff", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [718, 9], "def_end_pos": [718, 27]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "forall_exists_index", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [189, 17], "def_end_pos": [189, 36]}, {"full_name": "and_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [115, 17], "def_end_pos": [115, 24]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : H\n\u22a2 a \u2208 Subgroup.map f (stabilizer G s) \u2192 a \u2208 stabilizer H (\u21d1f '' s)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : H\n\u22a2 \u2200 (x : G), x \u2022 s = s \u2192 f x = a \u2192 a \u2022 \u21d1f '' s = \u21d1f '' s"}, {"tactic": "rintro a ha rfl", "annotated_tactic": ["rintro a ha rfl", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : H\n\u22a2 \u2200 (x : G), x \u2022 s = s \u2192 f x = a \u2192 a \u2022 \u21d1f '' s = \u21d1f '' s", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : G\nha : a \u2022 s = s\n\u22a2 f a \u2022 \u21d1f '' s = \u21d1f '' s"}, {"tactic": "rw [\u2190 image_smul_distrib, ha]", "annotated_tactic": ["rw [\u2190 <a>image_smul_distrib</a>, ha]", [{"full_name": "Set.image_smul_distrib", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [772, 9], "def_end_pos": [772, 27]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group H\ninst\u271d : MulAction G \u03b1\na\u271d : G\nf : G \u2192* H\ns : Set G\na : G\nha : a \u2022 s = s\n\u22a2 f a \u2022 \u21d1f '' s = \u21d1f '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Eigenspace/Basic.lean", "full_name": "Module.End.genEigenspace_eq_genEigenspace_finrank_of_le", "start": [300, 1], "end": [303, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Side.lean", "full_name": "AffineSubspace.wSameSide_vadd_left_iff", "start": [272, 1], "end": [281, 37], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\n\u22a2 s.WSameSide (v +\u1d65 x) y \u2194 s.WSameSide x y", "state_after": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\n\u22a2 s.WSameSide (v +\u1d65 x) y \u2192 s.WSameSide x y\n\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\n\u22a2 s.WSameSide x y \u2192 s.WSameSide (v +\u1d65 x) y"}, {"tactic": "rintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9", "annotated_tactic": ["rintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9", []], "state_before": "case mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\n\u22a2 s.WSameSide (v +\u1d65 x) y \u2192 s.WSameSide x y", "state_after": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 s.WSameSide x y"}, {"tactic": "refine\n  \u27e8-v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) hp\u2081, p\u2082, hp\u2082, ?_\u27e9", "annotated_tactic": ["refine\n      \u27e8-v +\u1d65 p\u2081, <a>AffineSubspace.vadd_mem_of_mem_direction</a> (<a>Submodule.neg_mem</a> _ hv) hp\u2081, p\u2082, hp\u2082, ?_\u27e9", [{"full_name": "AffineSubspace.vadd_mem_of_mem_direction", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [249, 9], "def_end_pos": [249, 34]}, {"full_name": "Submodule.neg_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [392, 19], "def_end_pos": [392, 26]}]], "state_before": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 s.WSameSide x y", "state_after": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (x -\u1d65 (-v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)"}, {"tactic": "rwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, \u2190 vadd_vsub_assoc]", "annotated_tactic": ["rwa [<a>vsub_vadd_eq_vsub_sub</a>, <a>sub_neg_eq_add</a>, <a>add_comm</a>, \u2190 <a>vadd_vsub_assoc</a>]", [{"full_name": "vsub_vadd_eq_vsub_sub", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [165, 9], "def_end_pos": [165, 30]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [697, 3], "def_end_pos": [697, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "vadd_vsub_assoc", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [117, 9], "def_end_pos": [117, 24]}]], "state_before": "case mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (x -\u1d65 (-v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)", "state_after": "no goals"}, {"tactic": "rintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9", "annotated_tactic": ["rintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9", []], "state_before": "case mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\n\u22a2 s.WSameSide x y \u2192 s.WSameSide (v +\u1d65 x) y", "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 s.WSameSide (v +\u1d65 x) y"}, {"tactic": "refine \u27e8v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction hv hp\u2081, p\u2082, hp\u2082, ?_\u27e9", "annotated_tactic": ["refine \u27e8v +\u1d65 p\u2081, <a>AffineSubspace.vadd_mem_of_mem_direction</a> hv hp\u2081, p\u2082, hp\u2082, ?_\u27e9", [{"full_name": "AffineSubspace.vadd_mem_of_mem_direction", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [249, 9], "def_end_pos": [249, 34]}]], "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 s.WSameSide (v +\u1d65 x) y", "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (v +\u1d65 x -\u1d65 (v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)"}, {"tactic": "rwa [vadd_vsub_vadd_cancel_left]", "annotated_tactic": ["rwa [<a>vadd_vsub_vadd_cancel_left</a>]", [{"full_name": "vadd_vsub_vadd_cancel_left", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [256, 9], "def_end_pos": [256, 35]}]], "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 s.direction\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (v +\u1d65 x -\u1d65 (v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EqToHom.lean", "full_name": "CategoryTheory.eqToIso_map", "start": [311, 1], "end": [312, 80], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 F.mapIso (eqToIso p) = eqToIso \u22ef", "state_after": "case w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 (F.mapIso (eqToIso p)).hom = (eqToIso \u22ef).hom"}, {"tactic": "cases p", "annotated_tactic": ["cases p", []], "state_before": "case w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 (F.mapIso (eqToIso p)).hom = (eqToIso \u22ef).hom", "state_after": "case w.refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX : C\n\u22a2 (F.mapIso (eqToIso \u22ef)).hom = (eqToIso \u22ef).hom"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case w.refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX : C\n\u22a2 (F.mapIso (eqToIso \u22ef)).hom = (eqToIso \u22ef).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iUnion_congr_Prop", "start": [162, 1], "end": [164, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNRat/BigOperators.lean", "full_name": "NNRat.coe_multiset_sum", "start": [27, 1], "end": [28, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "full_name": "le_iff_compact_le_imp", "start": [359, 1], "end": [363, 58], "traced_tactics": [{"tactic": "rw [\u2190 sSup_compact_le_eq a, \u2190 sSup_compact_le_eq b]", "annotated_tactic": ["rw [\u2190 <a>sSup_compact_le_eq</a> a, \u2190 <a>sSup_compact_le_eq</a> b]", [{"full_name": "sSup_compact_le_eq", "def_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 27]}, {"full_name": "sSup_compact_le_eq", "def_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 27]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na\u271d b\u271d : \u03b1\ns : Set \u03b1\na b : \u03b1\nh : \u2200 (c : \u03b1), CompleteLattice.IsCompactElement c \u2192 c \u2264 a \u2192 c \u2264 b\n\u22a2 a \u2264 b", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na\u271d b\u271d : \u03b1\ns : Set \u03b1\na b : \u03b1\nh : \u2200 (c : \u03b1), CompleteLattice.IsCompactElement c \u2192 c \u2264 a \u2192 c \u2264 b\n\u22a2 sSup {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 a} \u2264 sSup {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 b}"}, {"tactic": "exact sSup_le_sSup fun c hc => \u27e8hc.1, h c hc.1 hc.2\u27e9", "annotated_tactic": ["exact <a>sSup_le_sSup</a> fun c hc => \u27e8hc.1, h c hc.1 hc.2\u27e9", [{"full_name": "sSup_le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [97, 9], "def_end_pos": [97, 21]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na\u271d b\u271d : \u03b1\ns : Set \u03b1\na b : \u03b1\nh : \u2200 (c : \u03b1), CompleteLattice.IsCompactElement c \u2192 c \u2264 a \u2192 c \u2264 b\n\u22a2 sSup {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 a} \u2264 sSup {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.mk_repr", "start": [1112, 1], "end": [1113, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LucasLehmer.lean", "full_name": "LucasLehmer.X.card_units_lt", "start": [404, 8], "end": [407, 15], "traced_tactics": [{"tactic": "have : Fact (1 < (q : \u2115)) := \u27e8w\u27e9", "annotated_tactic": ["have : <a>Fact</a> (1 < (q : \u2115)) := \u27e8w\u27e9", [{"full_name": "Fact", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [114, 7], "def_end_pos": [114, 11]}]], "state_before": "q : \u2115+\nw : 1 < q\n\u22a2 Fintype.card (X q)\u02e3 < \u2191q ^ 2", "state_after": "q : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 Fintype.card (X q)\u02e3 < \u2191q ^ 2"}, {"tactic": "convert card_units_lt (X q)", "annotated_tactic": ["convert <a>card_units_lt</a> (<a>X</a> q)", [{"full_name": "card_units_lt", "def_path": "Mathlib/RingTheory/Fintype.lean", "def_pos": [58, 9], "def_end_pos": [58, 22]}, {"full_name": "LucasLehmer.X", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [226, 5], "def_end_pos": [226, 6]}]], "state_before": "q : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 Fintype.card (X q)\u02e3 < \u2191q ^ 2", "state_after": "case h.e'_4\nq : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 \u2191q ^ 2 = Fintype.card (X q)"}, {"tactic": "rw [card_eq]", "annotated_tactic": ["rw [<a>card_eq</a>]", [{"full_name": "LucasLehmer.X.card_eq", "def_path": "Mathlib/NumberTheory/LucasLehmer.lean", "def_pos": [397, 9], "def_end_pos": [397, 16]}]], "state_before": "case h.e'_4\nq : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 \u2191q ^ 2 = Fintype.card (X q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Alter.lean", "full_name": "Batteries.RBNode.Path.zoom_zoomed\u2082", "start": [137, 1], "end": [145, 33], "traced_tactics": [{"tactic": "revert e", "annotated_tactic": ["revert e", []], "state_before": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne\u271d : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\ne : zoom cut (node c\u271d l\u271d v\u271d r\u271d) path = (t', path')\n\u22a2 Zoomed cut path'", "state_after": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\n\u22a2 zoom cut (node c\u271d l\u271d v\u271d r\u271d) path = (t', path') \u2192 Zoomed cut path'"}, {"tactic": "unfold zoom", "annotated_tactic": ["unfold <a>zoom</a>", [{"full_name": "Batteries.RBNode.zoom", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [468, 19], "def_end_pos": [468, 23]}]], "state_before": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\n\u22a2 zoom cut (node c\u271d l\u271d v\u271d r\u271d) path = (t', path') \u2192 Zoomed cut path'", "state_after": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\n\u22a2 (match cut v\u271d with\n      | Ordering.lt => zoom cut l\u271d (left c\u271d path v\u271d r\u271d)\n      | Ordering.gt => zoom cut r\u271d (right c\u271d l\u271d v\u271d path)\n      | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, path)) =\n      (t', path') \u2192\n    Zoomed cut path'"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\n\u22a2 (match cut v\u271d with\n      | Ordering.lt => zoom cut l\u271d (left c\u271d path v\u271d r\u271d)\n      | Ordering.gt => zoom cut r\u271d (right c\u271d l\u271d v\u271d path)\n      | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, path)) =\n      (t', path') \u2192\n    Zoomed cut path'", "state_after": "case h_1\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\n\u22a2 zoom cut l\u271d (left c\u271d path v\u271d r\u271d) = (t', path') \u2192 Zoomed cut path'\n\ncase h_2\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.gt\n\u22a2 zoom cut r\u271d (right c\u271d l\u271d v\u271d path) = (t', path') \u2192 Zoomed cut path'\n\ncase h_3\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 (node c\u271d l\u271d v\u271d r\u271d, path) = (t', path') \u2192 Zoomed cut path'"}, {"tactic": "next h => exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", "annotated_tactic": ["next h => exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", []], "state_before": "case h_1\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.lt\n\u22a2 zoom cut l\u271d (left c\u271d path v\u271d r\u271d) = (t', path') \u2192 Zoomed cut path'", "state_after": "no goals"}, {"tactic": "exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", "annotated_tactic": ["exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nh : cut v\u271d = Ordering.lt\n\u22a2 zoom cut l\u271d (left c\u271d path v\u271d r\u271d) = (t', path') \u2192 Zoomed cut path'", "state_after": "no goals"}, {"tactic": "next h => exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", "annotated_tactic": ["next h => exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", []], "state_before": "case h_2\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.gt\n\u22a2 zoom cut r\u271d (right c\u271d l\u271d v\u271d path) = (t', path') \u2192 Zoomed cut path'", "state_after": "no goals"}, {"tactic": "exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", "annotated_tactic": ["exact fun e => zoom_zoomed\u2082 e \u27e8h, hp\u27e9", []], "state_before": "\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nh : cut v\u271d = Ordering.gt\n\u22a2 zoom cut r\u271d (right c\u271d l\u271d v\u271d path) = (t', path') \u2192 Zoomed cut path'", "state_after": "no goals"}, {"tactic": "intro e", "annotated_tactic": ["intro e", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 (node c\u271d l\u271d v\u271d r\u271d, path) = (t', path') \u2192 Zoomed cut path'", "state_after": "case h_3\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne\u271d : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\ne : (node c\u271d l\u271d v\u271d r\u271d, path) = (t', path')\n\u22a2 Zoomed cut path'"}, {"tactic": "cases e", "annotated_tactic": ["cases e", []], "state_before": "case h_3\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt'\u271d : RBNode \u03b1\u271d\npath'\u271d : Path \u03b1\u271d\ne\u271d : zoom cut t path = (t'\u271d, path'\u271d)\nhp : Zoomed cut path\npath' : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\ne : (node c\u271d l\u271d v\u271d r\u271d, path) = (t', path')\n\u22a2 Zoomed cut path'", "state_after": "case h_3.refl\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\npath' : Path \u03b1\u271d\ne : zoom cut t path = (t', path')\nhp : Zoomed cut path\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 Zoomed cut path"}, {"tactic": "exact hp", "annotated_tactic": ["exact hp", []], "state_before": "case h_3.refl\n\u03b1\u271d : Type u_1\ncut : \u03b1\u271d \u2192 Ordering\nt : RBNode \u03b1\u271d\npath : Path \u03b1\u271d\nt' : RBNode \u03b1\u271d\npath' : Path \u03b1\u271d\ne : zoom cut t path = (t', path')\nhp : Zoomed cut path\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\u271d\nv\u271d : \u03b1\u271d\nr\u271d : RBNode \u03b1\u271d\nx\u271d : Ordering\nheq\u271d : cut v\u271d = Ordering.eq\n\u22a2 Zoomed cut path", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "full_name": "Submodule.comap_map_eq_of_injective", "start": [349, 1], "end": [350, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/VectorBundle/Basic.lean", "full_name": "Trivialization.linearMapAt_apply", "start": [245, 1], "end": [247, 23], "traced_tactics": [{"tactic": "rw [coe_linearMapAt]", "annotated_tactic": ["rw [<a>coe_linearMapAt</a>]", [{"full_name": "Trivialization.coe_linearMapAt", "def_path": "Mathlib/Topology/VectorBundle/Basic.lean", "def_pos": [235, 9], "def_end_pos": [235, 24]}]], "state_before": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\ny : E b\n\u22a2 (Trivialization.linearMapAt R e b) y = if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).2 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "full_name": "measure_zero_of_dimH_lt", "start": [148, 1], "end": [150, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/CompleteLinearOrder.lean", "full_name": "IsLUB.exists_of_nonempty_of_not_isSuccLimit", "start": [54, 1], "end": [56, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Restrict.lean", "full_name": "Matroid.Basis.exchange", "start": [416, 1], "end": [419, 49], "traced_tactics": [{"tactic": "obtain \u27e8y,hy, h\u27e9 := hIX.restrict_base.exchange hJX.restrict_base he", "annotated_tactic": ["obtain \u27e8y,hy, h\u27e9 := hIX.restrict_base.exchange hJX.restrict_base he", []], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J\u271d X Y B J : Set \u03b1\ne : \u03b1\nhIX : M.Basis I X\nhJX : M.Basis J X\nhe : e \u2208 I \\ J\n\u22a2 \u2203 f \u2208 J \\ I, M.Basis (insert f (I \\ {e})) X", "state_after": "case intro.intro\n\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J\u271d X Y B J : Set \u03b1\ne : \u03b1\nhIX : M.Basis I X\nhJX : M.Basis J X\nhe : e \u2208 I \\ J\ny : \u03b1\nhy : y \u2208 J \\ I\nh : (M \u21be X).Base (insert y (I \\ {e}))\n\u22a2 \u2203 f \u2208 J \\ I, M.Basis (insert f (I \\ {e})) X"}, {"tactic": "exact \u27e8y, hy, by rwa [base_restrict_iff] at h\u27e9", "annotated_tactic": ["exact \u27e8y, hy, by rwa [<a>base_restrict_iff</a>] at h\u27e9", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J\u271d X Y B J : Set \u03b1\ne : \u03b1\nhIX : M.Basis I X\nhJX : M.Basis J X\nhe : e \u2208 I \\ J\ny : \u03b1\nhy : y \u2208 J \\ I\nh : (M \u21be X).Base (insert y (I \\ {e}))\n\u22a2 \u2203 f \u2208 J \\ I, M.Basis (insert f (I \\ {e})) X", "state_after": "no goals"}, {"tactic": "rwa [base_restrict_iff] at h", "annotated_tactic": ["rwa [<a>base_restrict_iff</a>] at h", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J\u271d X Y B J : Set \u03b1\ne : \u03b1\nhIX : M.Basis I X\nhJX : M.Basis J X\nhe : e \u2208 I \\ J\ny : \u03b1\nhy : y \u2208 J \\ I\nh : (M \u21be X).Base (insert y (I \\ {e}))\n\u22a2 M.Basis (insert y (I \\ {e})) X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "full_name": "Matrix.toLinearMap\u209b\u2097\u2082'_toMatrix'", "start": [230, 1], "end": [232, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "Complex.toBasis_orthonormalBasisOneI", "start": [716, 1], "end": [718, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.ofDigits_nil", "start": [196, 1], "end": [196, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Basic.lean", "full_name": "strongLT_of_le_of_strongLT", "start": [956, 1], "end": [957, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Lattice.lean", "full_name": "Multiset.inf_union", "start": [168, 1], "end": [169, 58], "traced_tactics": [{"tactic": "rw [\u2190 inf_dedup, dedup_ext.2, inf_dedup, inf_add]", "annotated_tactic": ["rw [\u2190 <a>inf_dedup</a>, <a>dedup_ext</a>.2, <a>inf_dedup</a>, <a>inf_add</a>]", [{"full_name": "Multiset.inf_dedup", "def_path": "Mathlib/Data/Multiset/Lattice.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Multiset.dedup_ext", "def_path": "Mathlib/Data/Multiset/Dedup.lean", "def_pos": [116, 9], "def_end_pos": [116, 18]}, {"full_name": "Multiset.inf_dedup", "def_path": "Mathlib/Data/Multiset/Lattice.lean", "def_pos": [158, 9], "def_end_pos": [158, 18]}, {"full_name": "Multiset.inf_add", "def_path": "Mathlib/Data/Multiset/Lattice.lean", "def_pos": [137, 9], "def_end_pos": [137, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 (s\u2081 \u222a s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Vector/Snoc.lean", "full_name": "Vector.mapAccumr_snoc", "start": [134, 1], "end": [141, 13], "traced_tactics": [{"tactic": "induction xs", "annotated_tactic": ["induction xs", []], "state_before": "\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\n\u22a2 mapAccumr f (xs.snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f xs q.1;\n    (r.1, r.2.snoc q.2)", "state_after": "case nil\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\n\u22a2 mapAccumr f (nil.snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f nil q.1;\n    (r.1, r.2.snoc q.2)\n\ncase cons\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d :\n  mapAccumr f (w\u271d.snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f w\u271d q.1;\n    (r.1, r.2.snoc q.2)\n\u22a2 mapAccumr f ((x\u271d ::\u1d65 w\u271d).snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f (x\u271d ::\u1d65 w\u271d) q.1;\n    (r.1, r.2.snoc q.2)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\n\u22a2 mapAccumr f (nil.snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f nil q.1;\n    (r.1, r.2.snoc q.2)", "state_after": "no goals"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case cons\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d :\n  mapAccumr f (w\u271d.snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f w\u271d q.1;\n    (r.1, r.2.snoc q.2)\n\u22a2 mapAccumr f ((x\u271d ::\u1d65 w\u271d).snoc x) s =\n    let q := f x s;\n    let r := mapAccumr f (x\u271d ::\u1d65 w\u271d) q.1;\n    (r.1, r.2.snoc q.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "orthogonalProjection_eq_self_iff", "start": [556, 1], "end": [560, 9], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun h => eq_orthogonalProjection_of_mem_of_inner_eq_zero h ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun h => <a>eq_orthogonalProjection_of_mem_of_inner_eq_zero</a> h ?_\u27e9", [{"full_name": "eq_orthogonalProjection_of_mem_of_inner_eq_zero", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [505, 9], "def_end_pos": [505, 56]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 \u2191((orthogonalProjection K) v) = v \u2194 v \u2208 K", "state_after": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191((orthogonalProjection K) v) = v\n\u22a2 v \u2208 K\n\ncase refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : v \u2208 K\n\u22a2 \u2200 w \u2208 K, \u27eav - v, w\u27eb_\ud835\udd5c = 0"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191((orthogonalProjection K) v) = v\n\u22a2 v \u2208 K", "state_after": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191((orthogonalProjection K) v) = v\n\u22a2 \u2191((orthogonalProjection K) v) \u2208 K"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191((orthogonalProjection K) v) = v\n\u22a2 \u2191((orthogonalProjection K) v) \u2208 K", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : v \u2208 K\n\u22a2 \u2200 w \u2208 K, \u27eav - v, w\u27eb_\ud835\udd5c = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "OrthonormalBasis.coe_toBasis", "start": [421, 11], "end": [426, 10], "traced_tactics": [{"tactic": "rw [OrthonormalBasis.toBasis]", "annotated_tactic": ["rw [<a>OrthonormalBasis.toBasis</a>]", [{"full_name": "OrthonormalBasis.toBasis", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [416, 15], "def_end_pos": [416, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u21d1b.toBasis = \u21d1b", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u21d1(Basis.ofEquivFun b.repr.toLinearEquiv) = \u21d1b"}, {"tactic": "ext j", "annotated_tactic": ["ext j", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u21d1(Basis.ofEquivFun b.repr.toLinearEquiv) = \u21d1b", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (Basis.ofEquivFun b.repr.toLinearEquiv) j = b j"}, {"tactic": "classical\n  rw [Basis.coe_ofEquivFun]\n  congr", "annotated_tactic": ["classical\n    rw [<a>Basis.coe_ofEquivFun</a>]\n    congr", [{"full_name": "Basis.coe_ofEquivFun", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [956, 9], "def_end_pos": [956, 29]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (Basis.ofEquivFun b.repr.toLinearEquiv) j = b j", "state_after": "no goals"}, {"tactic": "rw [Basis.coe_ofEquivFun]", "annotated_tactic": ["rw [<a>Basis.coe_ofEquivFun</a>]", [{"full_name": "Basis.coe_ofEquivFun", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [956, 9], "def_end_pos": [956, 29]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (Basis.ofEquivFun b.repr.toLinearEquiv) j = b j", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (fun i => b.repr.symm (update 0 i 1)) j = b j"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : _root_.RCLike \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (fun i => b.repr.symm (update 0 i 1)) j = b j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Preimage.lean", "full_name": "Finset.preimage_union", "start": [63, 1], "end": [67, 33], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns t : Finset \u03b2\nhst : InjOn f (f \u207b\u00b9' \u2191(s \u222a t))\n\u22a2 \u2191((s \u222a t).preimage f hst) = \u2191(s.preimage f \u22ef \u222a t.preimage f \u22ef)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/Dirichlet.lean", "full_name": "ArithmeticFunction.LSeriesSummable_vonMangoldt", "start": [338, 1], "end": [348, 99], "traced_tactics": [{"tactic": "have hf := LSeriesSummable_logMul_of_lt_re\n  (show abscissaOfAbsConv 1 < s.re by rw [abscissaOfAbsConv_one]; exact_mod_cast hs)", "annotated_tactic": ["have hf := <a>LSeriesSummable_logMul_of_lt_re</a>\n    (show <a>abscissaOfAbsConv</a> 1 < s.re by rw [<a>abscissaOfAbsConv_one</a>]; exact_mod_cast hs)", [{"full_name": "LSeriesSummable_logMul_of_lt_re", "def_path": "Mathlib/NumberTheory/LSeries/Deriv.lean", "def_pos": [95, 7], "def_end_pos": [95, 38]}, {"full_name": "LSeries.abscissaOfAbsConv", "def_path": "Mathlib/NumberTheory/LSeries/Convergence.lean", "def_pos": [26, 19], "def_end_pos": [26, 44]}, {"full_name": "LSeries.abscissaOfAbsConv_one", "def_path": "Mathlib/NumberTheory/LSeries/Dirichlet.lean", "def_pos": [231, 7], "def_end_pos": [231, 36]}]], "state_before": "s : \u2102\nhs : 1 < s.re\n\u22a2 LSeriesSummable (fun n => \u2191(\u039b n)) s", "state_after": "s : \u2102\nhs : 1 < s.re\nhf : LSeriesSummable (logMul 1) s\n\u22a2 LSeriesSummable (fun n => \u2191(\u039b n)) s"}, {"tactic": "rw [LSeriesSummable, \u2190 summable_norm_iff] at hf \u22a2", "annotated_tactic": ["rw [<a>LSeriesSummable</a>, \u2190 <a>summable_norm_iff</a>] at hf \u22a2", [{"full_name": "LSeriesSummable", "def_path": "Mathlib/NumberTheory/LSeries/Basic.lean", "def_pos": [131, 5], "def_end_pos": [131, 20]}, {"full_name": "summable_norm_iff", "def_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "def_pos": [685, 9], "def_end_pos": [685, 26]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : LSeriesSummable (logMul 1) s\n\u22a2 LSeriesSummable (fun n => \u2191(\u039b n)) s", "state_after": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\n\u22a2 Summable fun x => \u2016term (fun n => \u2191(\u039b n)) s x\u2016"}, {"tactic": "refine Summable.of_nonneg_of_le (fun _ \u21a6 norm_nonneg _) (fun n \u21a6 norm_term_le s ?_) hf", "annotated_tactic": ["refine <a>Summable.of_nonneg_of_le</a> (fun _ \u21a6 <a>norm_nonneg</a> _) (fun n \u21a6 <a>norm_term_le</a> s ?_) hf", [{"full_name": "Summable.of_nonneg_of_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 33]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "LSeries.norm_term_le", "def_path": "Mathlib/NumberTheory/LSeries/Basic.lean", "def_pos": [92, 7], "def_end_pos": [92, 19]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\n\u22a2 Summable fun x => \u2016term (fun n => \u2191(\u039b n)) s x\u2016", "state_after": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\n\u22a2 \u2016\u2191(\u039b n)\u2016 \u2264 \u2016logMul 1 n\u2016"}, {"tactic": "have h\u039b : \u2016\u2197\u039b n\u2016 \u2264 \u2016Complex.log n\u2016 := by\n  simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg vonMangoldt_nonneg,\n    \u2190 Complex.natCast_log, _root_.abs_of_nonneg <| Real.log_natCast_nonneg n]\n  exact ArithmeticFunction.vonMangoldt_le_log", "annotated_tactic": ["have h\u039b : \u2016\u2197\u039b n\u2016 \u2264 \u2016<a>Complex.log</a> n\u2016 := by\n    simp only [<a>norm_eq_abs</a>, <a>abs_ofReal</a>, <a>_root_.abs_of_nonneg</a> <a>vonMangoldt_nonneg</a>,\n      \u2190 <a>Complex.natCast_log</a>, <a>_root_.abs_of_nonneg</a> <| <a>Real.log_natCast_nonneg</a> n]\n    exact <a>ArithmeticFunction.vonMangoldt_le_log</a>", [{"full_name": "Complex.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": [29, 19], "def_end_pos": [29, 22]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_ofReal", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [75, 9], "def_end_pos": [75, 19]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "ArithmeticFunction.vonMangoldt_nonneg", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [83, 9], "def_end_pos": [83, 27]}, {"full_name": "Complex.natCast_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": [76, 7], "def_end_pos": [76, 18]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Real.log_natCast_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 27]}, {"full_name": "ArithmeticFunction.vonMangoldt_le_log", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [163, 9], "def_end_pos": [163, 27]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\n\u22a2 \u2016\u2191(\u039b n)\u2016 \u2264 \u2016logMul 1 n\u2016", "state_after": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\nh\u039b : \u2016(fun n => \u2191(\u039b n)) n\u2016 \u2264 \u2016Complex.log \u2191n\u2016\n\u22a2 \u2016\u2191(\u039b n)\u2016 \u2264 \u2016logMul 1 n\u2016"}, {"tactic": "exact h\u039b.trans <| by simp only [norm_eq_abs, norm_mul, Pi.one_apply, norm_one, mul_one, le_refl]", "annotated_tactic": ["exact h\u039b.trans <| by simp only [<a>norm_eq_abs</a>, <a>norm_mul</a>, <a>Pi.one_apply</a>, <a>norm_one</a>, <a>mul_one</a>, <a>le_refl</a>]", [{"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 17]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "NormOneClass.norm_one", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [170, 3], "def_end_pos": [170, 11]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\nh\u039b : \u2016(fun n => \u2191(\u039b n)) n\u2016 \u2264 \u2016Complex.log \u2191n\u2016\n\u22a2 \u2016\u2191(\u039b n)\u2016 \u2264 \u2016logMul 1 n\u2016", "state_after": "no goals"}, {"tactic": "rw [abscissaOfAbsConv_one]", "annotated_tactic": ["rw [<a>abscissaOfAbsConv_one</a>]", [{"full_name": "LSeries.abscissaOfAbsConv_one", "def_path": "Mathlib/NumberTheory/LSeries/Dirichlet.lean", "def_pos": [231, 7], "def_end_pos": [231, 36]}]], "state_before": "s : \u2102\nhs : 1 < s.re\n\u22a2 abscissaOfAbsConv 1 < \u2191s.re", "state_after": "s : \u2102\nhs : 1 < s.re\n\u22a2 1 < \u2191s.re"}, {"tactic": "exact_mod_cast hs", "annotated_tactic": ["exact_mod_cast hs", []], "state_before": "s : \u2102\nhs : 1 < s.re\n\u22a2 1 < \u2191s.re", "state_after": "no goals"}, {"tactic": "simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg vonMangoldt_nonneg,\n  \u2190 Complex.natCast_log, _root_.abs_of_nonneg <| Real.log_natCast_nonneg n]", "annotated_tactic": ["simp only [<a>norm_eq_abs</a>, <a>abs_ofReal</a>, <a>_root_.abs_of_nonneg</a> <a>vonMangoldt_nonneg</a>,\n      \u2190 <a>Complex.natCast_log</a>, <a>_root_.abs_of_nonneg</a> <| <a>Real.log_natCast_nonneg</a> n]", [{"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_ofReal", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [75, 9], "def_end_pos": [75, 19]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "ArithmeticFunction.vonMangoldt_nonneg", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [83, 9], "def_end_pos": [83, 27]}, {"full_name": "Complex.natCast_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": [76, 7], "def_end_pos": [76, 18]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Real.log_natCast_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 27]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\n\u22a2 \u2016(fun n => \u2191(\u039b n)) n\u2016 \u2264 \u2016Complex.log \u2191n\u2016", "state_after": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\n\u22a2 \u039b n \u2264 Real.log \u2191n"}, {"tactic": "exact ArithmeticFunction.vonMangoldt_le_log", "annotated_tactic": ["exact <a>ArithmeticFunction.vonMangoldt_le_log</a>", [{"full_name": "ArithmeticFunction.vonMangoldt_le_log", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [163, 9], "def_end_pos": [163, 27]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\n\u22a2 \u039b n \u2264 Real.log \u2191n", "state_after": "no goals"}, {"tactic": "simp only [norm_eq_abs, norm_mul, Pi.one_apply, norm_one, mul_one, le_refl]", "annotated_tactic": ["simp only [<a>norm_eq_abs</a>, <a>norm_mul</a>, <a>Pi.one_apply</a>, <a>norm_one</a>, <a>mul_one</a>, <a>le_refl</a>]", [{"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 17]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "NormOneClass.norm_one", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [170, 3], "def_end_pos": [170, 11]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}]], "state_before": "s : \u2102\nhs : 1 < s.re\nhf : Summable fun x => \u2016term (logMul 1) s x\u2016\nn : \u2115\nh\u039b : \u2016(fun n => \u2191(\u039b n)) n\u2016 \u2264 \u2016Complex.log \u2191n\u2016\n\u22a2 \u2016Complex.log \u2191n\u2016 \u2264 \u2016logMul 1 n\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.coe_compRingHom_apply", "start": [1157, 1], "end": [1158, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "full_name": "GroupFilterBasis.conj", "start": [114, 1], "end": [115, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarAlgHom.mem_equalizer", "start": [463, 1], "end": [465, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderEmbedding.monotone", "start": [649, 11], "end": [650, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Commutator.lean", "full_name": "Subgroup.commutator_comm_le", "start": [121, 1], "end": [123, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.map_id", "start": [171, 1], "end": [172, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.ofDual_apply_coe", "start": [747, 1], "end": [748, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "nilpotent_iff_finite_descending_central_series", "start": [262, 1], "end": [275, 15], "traced_tactics": [{"tactic": "rw [nilpotent_iff_finite_ascending_central_series]", "annotated_tactic": ["rw [<a>nilpotent_iff_finite_ascending_central_series</a>]", [{"full_name": "nilpotent_iff_finite_ascending_central_series", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [219, 9], "def_end_pos": [219, 54]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 Group.IsNilpotent G \u2194 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4) \u2194 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4) \u2194 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "case mp\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4) \u2192 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5\n\ncase mpr\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2192 \u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4"}, {"tactic": "rintro \u27e8n, H, hH, hn\u27e9", "annotated_tactic": ["rintro \u27e8n, H, hH, hn\u27e9", []], "state_before": "case mp\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4) \u2192 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5"}, {"tactic": "refine \u27e8n, fun m => H (n - m), is_decending_rev_series_of_is_ascending G hn hH, ?_\u27e9", "annotated_tactic": ["refine \u27e8n, fun m => H (n - m), <a>is_decending_rev_series_of_is_ascending</a> G hn hH, ?_\u27e9", [{"full_name": "is_decending_rev_series_of_is_ascending", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [230, 9], "def_end_pos": [230, 48]}]], "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 (fun m => H (n - m)) n = \u22a5"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 (fun m => H (n - m)) n = \u22a5", "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 H (n - n) = \u22a5"}, {"tactic": "rw [tsub_self]", "annotated_tactic": ["rw [<a>tsub_self</a>]", [{"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [342, 9], "def_end_pos": [342, 18]}]], "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 H (n - n) = \u22a5", "state_after": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 H 0 = \u22a5"}, {"tactic": "exact hH.1", "annotated_tactic": ["exact hH.1", []], "state_before": "case mp.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = \u22a4\n\u22a2 H 0 = \u22a5", "state_after": "no goals"}, {"tactic": "rintro \u27e8n, H, hH, hn\u27e9", "annotated_tactic": ["rintro \u27e8n, H, hH, hn\u27e9", []], "state_before": "case mpr\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2192 \u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4", "state_after": "case mpr.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = \u22a5\n\u22a2 \u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4"}, {"tactic": "refine \u27e8n, fun m => H (n - m), is_ascending_rev_series_of_is_descending G hn hH, ?_\u27e9", "annotated_tactic": ["refine \u27e8n, fun m => H (n - m), <a>is_ascending_rev_series_of_is_descending</a> G hn hH, ?_\u27e9", [{"full_name": "is_ascending_rev_series_of_is_descending", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [246, 9], "def_end_pos": [246, 49]}]], "state_before": "case mpr.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhH : IsDescendingCentralSeries H\nhn : H n = \u22a5\n\u22a2 \u2203 n H, IsAscendingCentralSeries H \u2227 H n = \u22a4", "state_after": "case mpr.intro.intro.intro\nG : Type 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "mulIndicator_biUnion_finset_eventuallyEq", "start": [89, 1], "end": [94, 46], "traced_tactics": [{"tactic": "rw [iUnion_eq_iUnion_finset s]", "annotated_tactic": ["rw [<a>iUnion_eq_iUnion_finset</a> s]", [{"full_name": "Set.iUnion_eq_iUnion_finset", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun n => (\u22c3 i \u2208 n, s i).mulIndicator f a) =\u1da0[atTop] fun x => (iUnion s).mulIndicator f a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun n => (\u22c3 i \u2208 n, s i).mulIndicator f a) =\u1da0[atTop] fun x => (\u22c3 t, \u22c3 i \u2208 t, s i).mulIndicator f a"}, {"tactic": "apply Monotone.mulIndicator_eventuallyEq_iUnion", "annotated_tactic": ["apply <a>Monotone.mulIndicator_eventuallyEq_iUnion</a>", [{"full_name": "Monotone.mulIndicator_eventuallyEq_iUnion", "def_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "def_pos": [63, 9], "def_end_pos": [63, 50]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun n => (\u22c3 i \u2208 n, s i).mulIndicator f a) =\u1da0[atTop] fun x => (\u22c3 t, \u22c3 i \u2208 t, s i).mulIndicator f a", "state_after": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 Monotone fun i => \u22c3 i_1 \u2208 i, s i_1"}, {"tactic": "exact fun _ _ \u21a6 biUnion_subset_biUnion_left", "annotated_tactic": ["exact fun _ _ \u21a6 <a>biUnion_subset_biUnion_left</a>", [{"full_name": "Set.biUnion_subset_biUnion_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [842, 9], "def_end_pos": [842, 36]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 Monotone fun i => \u22c3 i_1 \u2208 i, s i_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_eq_pi_iff", "start": [219, 1], "end": [221, 68], "traced_tactics": [{"tactic": "rw [angle, \u2190 real_inner_div_norm_mul_norm_eq_neg_one_iff, Real.arccos_eq_pi, LE.le.le_iff_eq]", 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"file_path": "Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean", "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing", "start": [163, 1], "end": [165, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "PSet.not_nonempty_empty", "start": [369, 1], "end": [369, 73], "traced_tactics": [{"tactic": "simp [PSet.Nonempty]", "annotated_tactic": ["simp [<a>PSet.Nonempty</a>]", [{"full_name": "PSet.Nonempty", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [308, 15], "def_end_pos": [308, 23]}]], "state_before": "\u22a2 \u00ac\u2205.Nonempty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "full_name": "Surreal.Multiplication.P2_neg_right", "start": [118, 1], 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goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "StructureGroupoid.restriction_in_maximalAtlas", "start": [1250, 1], "end": [1265, 63], "traced_tactics": [{"tactic": "intro e' he'", "annotated_tactic": ["intro e' he'", []], "state_before": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\n\u22a2 e.subtypeRestr hs \u2208 maximalAtlas (\u21a5s) G", "state_after": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 e' \u2208 G \u2227 e'.symm \u226b\u2095 e.subtypeRestr hs \u2208 G"}, {"tactic": "obtain \u27e8x, this\u27e9 := Opens.chart_eq hs he'", "annotated_tactic": ["obtain \u27e8x, this\u27e9 := <a>Opens.chart_eq</a> hs he'", [{"full_name": "TopologicalSpace.Opens.chart_eq", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [1201, 7], "def_end_pos": [1201, 15]}]], "state_before": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 e' \u2208 G \u2227 e'.symm \u226b\u2095 e.subtypeRestr hs \u2208 G", "state_after": "case intro\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\nx : \u21a5s\nthis : e' = (chartAt H \u2191x).subtypeRestr hs\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 e' \u2208 G \u2227 e'.symm \u226b\u2095 e.subtypeRestr hs \u2208 G"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case intro\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\nx : \u21a5s\nthis : e' = (chartAt H \u2191x).subtypeRestr hs\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 e' \u2208 G \u2227 e'.symm \u226b\u2095 e.subtypeRestr hs \u2208 G", "state_after": "case intro\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\nx : \u21a5s\nthis : e' = (chartAt H \u2191x).subtypeRestr hs\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 (chartAt H \u2191x).subtypeRestr hs \u2208 G \u2227\n    ((chartAt H \u2191x).subtypeRestr hs).symm \u226b\u2095 e.subtypeRestr hs \u2208 G"}, {"tactic": "exact \u27e8G.trans_restricted he (chart_mem_atlas H (x : M)) hs,\n       G.trans_restricted (chart_mem_atlas H (x : M)) he hs\u27e9", "annotated_tactic": ["exact \u27e8G.trans_restricted he (<a>chart_mem_atlas</a> H (x : M)) hs,\n         G.trans_restricted (<a>chart_mem_atlas</a> H (x : M)) he hs\u27e9", [{"full_name": "chart_mem_atlas", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [599, 7], "def_end_pos": [599, 22]}, {"full_name": "chart_mem_atlas", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [599, 7], "def_end_pos": [599, 22]}]], "state_before": "case intro\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ne : PartialHomeomorph M H\nhe : e \u2208 atlas H M\ns : Opens M\nhs : Nonempty \u21a5s\nG : StructureGroupoid H\ninst\u271d\u00b9 : HasGroupoid M G\ninst\u271d : ClosedUnderRestriction G\ne' : PartialHomeomorph (\u21a5s) H\nhe' : e' \u2208 atlas H \u21a5s\nx : \u21a5s\nthis : e' = (chartAt H \u2191x).subtypeRestr hs\n\u22a2 (e.subtypeRestr hs).symm \u226b\u2095 (chartAt H \u2191x).subtypeRestr hs \u2208 G \u2227\n    ((chartAt H \u2191x).subtypeRestr hs).symm \u226b\u2095 e.subtypeRestr hs \u2208 G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsLittleO.right_isBigO_add", "start": [2156, 1], "end": [2158, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "cauchySeq_finset_of_geometric_bound", "start": [439, 1], "end": [442, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "eq_div_of_mul_eq", "start": [370, 1], "end": [370, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "one_lt_of_lt_mul_left", "start": [548, 1], "end": [551, 53], "traced_tactics": [{"tactic": "simpa only [one_mul]", "annotated_tactic": ["simpa only [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : b < a * b\n\u22a2 1 * ?m.28643 < a * ?m.28643", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/IntermediateValue.lean", "full_name": "IsPreconnected.intermediate_value_Iic", "start": [161, 1], "end": [165, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.image_fst_graphOn", "start": [826, 1], "end": [828, 30], "traced_tactics": [{"tactic": "simp [graphOn, image_image]", "annotated_tactic": ["simp [<a>graphOn</a>, <a>image_image</a>]", [{"full_name": "Set.graphOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [295, 5], "def_end_pos": [295, 12]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [269, 9], "def_end_pos": [269, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\np : Set \u03b3\nf\u271d f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2\ng g\u2081 g\u2082 : \u03b2 \u2192 \u03b3\nf' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1\ng' : \u03b3 \u2192 \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 Prod.fst '' graphOn f s = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "full_name": "GromovHausdorff.HD_below_aux2", "start": [307, 1], "end": [310, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Union.lean", "full_name": "Finset.disjiUnion_cons", "start": [62, 1], "end": [69, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "EMetric.mem_nhds_iff", "start": [668, 1], "end": [669, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Pointwise.lean", "full_name": "Submodule.neg_top", "start": [139, 1], "end": [140, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzOnWith.extend_pi", "start": [420, 1], "end": [431, 22], "traced_tactics": [{"tactic": "have : \u2200 i, \u2203 g : \u03b1 \u2192 \u211d, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s := fun i => by\n  have : LipschitzOnWith K (fun x : \u03b1 => f x i) s :=\n    LipschitzOnWith.of_dist_le_mul fun x hx y hy =>\n      (dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy)\n  exact this.extend_real", "annotated_tactic": ["have : \u2200 i, \u2203 g : \u03b1 \u2192 \u211d, <a>LipschitzWith</a> K g \u2227 <a>EqOn</a> (fun x => f x i) g s := fun i => by\n    have : <a>LipschitzOnWith</a> K (fun x : \u03b1 => f x i) s :=\n      <a>LipschitzOnWith.of_dist_le_mul</a> fun x hx y hy =>\n        (<a>dist_le_pi_dist</a> _ _ i).<a>trans</a> (hf.dist_le_mul x hx y hy)\n    exact this.extend_real", [{"full_name": "LipschitzWith", "def_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "def_pos": [52, 5], "def_end_pos": [52, 18]}, {"full_name": "Set.EqOn", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [263, 5], "def_end_pos": [263, 9]}, {"full_name": "LipschitzOnWith", "def_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "def_pos": [58, 5], "def_end_pos": [58, 20]}, {"full_name": "LipschitzOnWith.of_dist_le_mul", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [58, 37], "def_end_pos": [58, 67]}, {"full_name": "dist_le_pi_dist", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "def_pos": [377, 7], "def_end_pos": [377, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : \u2200 (i : \u03b9), \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s"}, {"tactic": "choose g hg using this", "annotated_tactic": ["choose g hg using this", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : \u2200 (i : \u03b9), \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s"}, {"tactic": "refine \u27e8fun x i => g i x, LipschitzWith.of_dist_le_mul fun x y => ?_, fun x hx \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8fun x i => g i x, <a>LipschitzWith.of_dist_le_mul</a> fun x y => ?_, fun x hx \u21a6 ?_\u27e9", [{"full_name": "LipschitzWith.of_dist_le_mul", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [47, 35], "def_end_pos": [47, 63]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s", "state_after": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx y : \u03b1\n\u22a2 (dist (fun i => g i x) fun i => g i y) \u2264 \u2191K * dist x y\n\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = (fun x i => g i x) x"}, {"tactic": "have : LipschitzOnWith K (fun x : \u03b1 => f x i) s :=\n  LipschitzOnWith.of_dist_le_mul fun x hx y hy =>\n    (dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy)", "annotated_tactic": ["have : <a>LipschitzOnWith</a> K (fun x : \u03b1 => f x i) s :=\n      <a>LipschitzOnWith.of_dist_le_mul</a> fun x hx y hy =>\n        (<a>dist_le_pi_dist</a> _ _ i).<a>trans</a> (hf.dist_le_mul x hx y hy)", [{"full_name": "LipschitzOnWith", "def_path": "Mathlib/Topology/EMetricSpace/Lipschitz.lean", "def_pos": [58, 5], "def_end_pos": [58, 20]}, {"full_name": "LipschitzOnWith.of_dist_le_mul", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [58, 37], "def_end_pos": [58, 67]}, {"full_name": "dist_le_pi_dist", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "def_pos": [377, 7], "def_end_pos": [377, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ni : \u03b9\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ni : \u03b9\nthis : LipschitzOnWith K (fun x => f x i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s"}, {"tactic": "exact this.extend_real", "annotated_tactic": ["exact this.extend_real", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ni : \u03b9\nthis : LipschitzOnWith K (fun x => f x i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s", "state_after": "no goals"}, {"tactic": "exact (dist_pi_le_iff (mul_nonneg K.2 dist_nonneg)).2 fun i => (hg i).1.dist_le_mul x y", "annotated_tactic": ["exact (<a>dist_pi_le_iff</a> (<a>mul_nonneg</a> K.2 <a>dist_nonneg</a>)).2 fun i => (hg i).1.<a>dist_le_mul</a> x y", [{"full_name": "dist_pi_le_iff", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "def_pos": [334, 7], "def_end_pos": [334, 21]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "dist_nonneg", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [259, 9], "def_end_pos": [259, 20]}, {"full_name": "LipschitzWith.dist_le_mul", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [47, 8], "def_end_pos": [47, 33]}]], "state_before": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx y : \u03b1\n\u22a2 (dist (fun i => g i x) fun i => g i y) \u2264 \u2191K * dist x y", "state_after": "no goals"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "case refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = (fun x i => g i x) x", "state_after": "case refine_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\ni : \u03b9\n\u22a2 f x i = (fun x i => g i x) x i"}, {"tactic": "exact (hg i).2 hx", "annotated_tactic": ["exact (hg i).2 hx", []], "state_before": "case refine_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\ni : \u03b9\n\u22a2 f x i = (fun x i => g i x) x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.mem_sym_iff", "start": [190, 1], "end": [206, 77], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 0\n\u22a2 m \u2208 s.sym 0 \u2194 \u2200 a \u2208 m, a \u2208 s\n\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\n\u22a2 m \u2208 s.sym (n + 1) \u2194 \u2200 a \u2208 m, a \u2208 s"}, {"tactic": "refine mem_sup.trans \u27e8?_, fun h \u21a6 ?_\u27e9", "annotated_tactic": ["refine mem_sup.trans \u27e8?_, fun h \u21a6 ?_\u27e9", []], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\n\u22a2 m \u2208 s.sym (n + 1) \u2194 \u2200 a \u2208 m, a \u2208 s", "state_after": "case succ.refine_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\n\u22a2 (\u2203 v \u2208 s, m \u2208 image (Sym.cons v) (s.sym n)) \u2192 \u2200 a \u2208 m, a \u2208 s\n\ncase succ.refine_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\nh : \u2200 a \u2208 m, a \u2208 s\n\u22a2 \u2203 v \u2208 s, m \u2208 image (Sym.cons v) (s.sym n)"}, {"tactic": "refine mem_singleton.trans \u27e8?_, fun _ \u21a6 Sym.eq_nil_of_card_zero _\u27e9", "annotated_tactic": ["refine mem_singleton.trans \u27e8?_, fun _ \u21a6 <a>Sym.eq_nil_of_card_zero</a> _\u27e9", [{"full_name": "Sym.eq_nil_of_card_zero", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [276, 9], "def_end_pos": [276, 28]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 0\n\u22a2 m \u2208 s.sym 0 \u2194 \u2200 a \u2208 m, a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 0\n\u22a2 m = \u2205 \u2192 \u2200 a \u2208 m, a \u2208 s"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 0\n\u22a2 m = \u2205 \u2192 \u2200 a \u2208 m, a \u2208 s", "state_after": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 \u2200 a \u2208 \u2205, a \u2208 s"}, {"tactic": "exact fun a ha \u21a6 (Finset.not_mem_empty _ ha).elim", "annotated_tactic": ["exact fun a ha \u21a6 (<a>Finset.not_mem_empty</a> _ ha).<a>elim</a>", [{"full_name": "Finset.not_mem_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [550, 9], "def_end_pos": [550, 22]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 \u2200 a \u2208 \u2205, a \u2208 s", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, ha, he\u27e9 b hb", "annotated_tactic": ["rintro \u27e8a, ha, he\u27e9 b hb", []], "state_before": "case succ.refine_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\n\u22a2 (\u2203 v \u2208 s, m \u2208 image (Sym.cons v) (s.sym n)) \u2192 \u2200 a \u2208 m, a \u2208 s", "state_after": "case succ.refine_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\na : \u03b1\nha : a \u2208 s\nhe : m \u2208 image (Sym.cons a) (s.sym n)\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "rw [mem_image] at he", "annotated_tactic": ["rw [<a>mem_image</a>] at he", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}]], "state_before": "case succ.refine_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\na : \u03b1\nha : a \u2208 s\nhe : m \u2208 image (Sym.cons a) (s.sym n)\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\na : \u03b1\nha : a \u2208 s\nhe : \u2203 a_1 \u2208 s.sym n, a ::\u209b a_1 = m\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "obtain \u27e8m, he, rfl\u27e9 := he", "annotated_tactic": ["obtain \u27e8m, he, rfl\u27e9 := he", []], "state_before": "case succ.refine_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\na : \u03b1\nha : a \u2208 s\nhe : \u2203 a_1 \u2208 s.sym n, a ::\u209b a_1 = m\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b \u2208 a ::\u209b m\n\u22a2 b \u2208 s"}, {"tactic": "rw [Sym.mem_cons] at hb", "annotated_tactic": ["rw [<a>Sym.mem_cons</a>] at hb", [{"full_name": "Sym.mem_cons", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 17]}]], "state_before": "case succ.refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b \u2208 a ::\u209b m\n\u22a2 b \u2208 s", "state_after": "case succ.refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b = a \u2228 b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "obtain rfl | hb := hb", "annotated_tactic": ["obtain rfl | hb := hb", []], "state_before": "case succ.refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b = a \u2228 b \u2208 m\n\u22a2 b \u2208 s", "state_after": "case succ.refine_1.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nha : b \u2208 s\n\u22a2 b \u2208 s\n\ncase succ.refine_1.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b \u2208 m\n\u22a2 b \u2208 s"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case succ.refine_1.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nha : b \u2208 s\n\u22a2 b \u2208 s", "state_after": "no goals"}, {"tactic": "exact ih.1 he _ hb", "annotated_tactic": ["exact ih.1 he _ hb", []], "state_before": "case succ.refine_1.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 s.sym n\nhb : b \u2208 m\n\u22a2 b \u2208 s", "state_after": "no goals"}, {"tactic": "obtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ", "annotated_tactic": ["obtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ", []], "state_before": "case succ.refine_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\nm : Sym \u03b1 (n + 1)\nh : \u2200 a \u2208 m, a \u2208 s\n\u22a2 \u2203 v \u2208 s, m \u2208 image (Sym.cons v) (s.sym n)", "state_after": "case succ.refine_2.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nm : Sym \u03b1 n\nh : \u2200 a_1 \u2208 a ::\u209b m, a_1 \u2208 s\n\u22a2 \u2203 v \u2208 s, a ::\u209b m \u2208 image (Sym.cons v) (s.sym n)"}, {"tactic": "exact\n  \u27e8a, h _ <| Sym.mem_cons_self _ _,\n    mem_image_of_mem _ <| ih.2 fun b hb \u21a6 h _ <| Sym.mem_cons_of_mem hb\u27e9", "annotated_tactic": ["exact\n      \u27e8a, h _ <| <a>Sym.mem_cons_self</a> _ _,\n        <a>mem_image_of_mem</a> _ <| ih.2 fun b hb \u21a6 h _ <| <a>Sym.mem_cons_of_mem</a> hb\u27e9", [{"full_name": "Sym.mem_cons_self", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 22]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}, {"full_name": "Sym.mem_cons_of_mem", "def_path": "Mathlib/Data/Sym/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 24]}]], "state_before": "case succ.refine_2.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nn\u271d n : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s\na : \u03b1\nm : Sym \u03b1 n\nh : \u2200 a_1 \u2208 a ::\u209b m, a_1 \u2208 s\n\u22a2 \u2203 v \u2208 s, a ::\u209b m \u2208 image (Sym.cons v) (s.sym n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Orientation.lean", "full_name": "OrthonormalBasis.orthonormal_adjustToOrientation", "start": [103, 1], "end": [105, 65], "traced_tactics": [{"tactic": "apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg", "annotated_tactic": ["apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Orthonormal \u211d \u21d1(e.toBasis.adjustToOrientation x)", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 \u2200 (i : \u03b9), (e.toBasis.adjustToOrientation x) i = e i \u2228 (e.toBasis.adjustToOrientation x) i = -e i"}, {"tactic": "simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x", "annotated_tactic": ["simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 \u2200 (i : \u03b9), (e.toBasis.adjustToOrientation x) i = e i \u2228 (e.toBasis.adjustToOrientation x) i = -e i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "full_name": "HurwitzZeta.completedHurwitzZetaOdd_one_sub", "start": [380, 1], "end": [385, 34], "traced_tactics": [{"tactic": "rw [completedHurwitzZetaOdd, completedSinZeta,\n  (by { push_cast; ring } : (1 - s + 1) / 2 = \u2191(3 / 2 : \u211d) - (s + 1) / 2),\n  \u2190 hurwitzOddFEPair_k, (hurwitzOddFEPair a).functional_equation ((s + 1) / 2),\n  hurwitzOddFEPair_\u03b5, one_smul]", "annotated_tactic": ["rw [<a>completedHurwitzZetaOdd</a>, <a>completedSinZeta</a>,\n    (by { push_cast; ring } : (1 - s + 1) / 2 = \u2191(3 / 2 : \u211d) - (s + 1) / 2),\n    \u2190 <a>hurwitzOddFEPair_k</a>, (<a>hurwitzOddFEPair</a> a).<a>functional_equation</a> ((s + 1) / 2),\n    <a>hurwitzOddFEPair_\u03b5</a>, <a>one_smul</a>]", [{"full_name": "HurwitzZeta.completedHurwitzZetaOdd", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "def_pos": [342, 5], "def_end_pos": [342, 28]}, {"full_name": "HurwitzZeta.completedSinZeta", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "def_pos": [354, 5], "def_end_pos": [354, 21]}, {"full_name": "HurwitzZeta.hurwitzOddFEPair_k", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "def_pos": [303, 3], "def_end_pos": [303, 8]}, {"full_name": "HurwitzZeta.hurwitzOddFEPair", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "def_pos": [304, 5], "def_end_pos": [304, 21]}, {"full_name": "StrongFEPair.functional_equation", "def_path": "Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean", "def_pos": [214, 9], "def_end_pos": [214, 28]}, {"full_name": "HurwitzZeta.hurwitzOddFEPair_\u03b5", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean", "def_pos": [303, 3], "def_end_pos": [303, 8]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "a : UnitAddCircle\ns : \u2102\n\u22a2 completedHurwitzZetaOdd a (1 - s) = completedSinZeta a s", "state_after": "no goals"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "a : UnitAddCircle\ns : \u2102\n\u22a2 (1 - s + 1) / 2 = \u2191(3 / 2) - (s + 1) / 2", "state_after": "a : UnitAddCircle\ns : \u2102\n\u22a2 (1 - s + 1) / 2 = 3 / 2 - (s + 1) / 2"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "a : UnitAddCircle\ns : \u2102\n\u22a2 (1 - s + 1) / 2 = 3 / 2 - (s + 1) / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Prod.lean", "full_name": "deriv_pi", "start": [105, 1], "end": [107, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.cons_diff_of_mem", "start": [1007, 1], "end": [1008, 76], "traced_tactics": [{"tactic": "rw [cons_diff, if_pos h]", "annotated_tactic": ["rw [<a>cons_diff</a>, <a>if_pos</a> h]", [{"full_name": "List.cons_diff", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [996, 9], "def_end_pos": [996, 18]}, {"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : LawfulBEq \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nh : a \u2208 l\u2082\nl\u2081 : List \u03b1\n\u22a2 (a :: l\u2081).diff l\u2082 = l\u2081.diff (l\u2082.erase a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Covering.lean", "full_name": "IsCoveringMap.continuous", "start": [162, 11], "end": [163, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.ModEq.mul", "start": [201, 11], "end": [202, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "Substring.Valid.next", "start": [968, 1], "end": [972, 56], "traced_tactics": [{"tactic": "let \u27e8l, m, r, h\u27e9 := h.validFor", "annotated_tactic": ["let \u27e8l, m, r, h\u27e9 := h.validFor", []], "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081 ++ c :: m\u2082\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081 ++ c :: m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }"}, {"tactic": "simp only [h.toString] at e", "annotated_tactic": ["simp only [h.toString] at e", []], "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\ne : x\u271d.toString.data = m\u2081 ++ c :: m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081 ++ c :: m\u2082\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }"}, {"tactic": "subst e", "annotated_tactic": ["subst e", []], "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081 ++ c :: m\u2082\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl r : List Char\nh : ValidFor l (m\u2081 ++ c :: m\u2082) r x\u271d\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }"}, {"tactic": "simp [h.next]", "annotated_tactic": ["simp [h.next]", []], "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : x\u271d.Valid\nl r : List Char\nh : ValidFor l (m\u2081 ++ c :: m\u2082) r x\u271d\n\u22a2 x\u271d.next { byteIdx := utf8Len m\u2081 } = { byteIdx := utf8Len m\u2081 + c.utf8Size }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dimension/Localization.lean", "full_name": "exists_set_linearIndependent_of_isDomain", "start": [85, 1], "end": [93, 96], "traced_tactics": [{"tactic": "obtain \u27e8w, hw\u27e9 :=\n  IsLocalizedModule.linearIndependent_lift R\u2070 (LocalizedModule.mkLinearMap R\u2070 M) le_rfl\n    (Module.Free.chooseBasis (FractionRing R) (LocalizedModule R\u2070 M)).linearIndependent", "annotated_tactic": ["obtain \u27e8w, hw\u27e9 :=\n    <a>IsLocalizedModule.linearIndependent_lift</a> R\u2070 (<a>LocalizedModule.mkLinearMap</a> R\u2070 M) <a>le_rfl</a>\n      (<a>Module.Free.chooseBasis</a> (<a>FractionRing</a> R) (<a>LocalizedModule</a> R\u2070 M)).<a>linearIndependent</a>", [{"full_name": "IsLocalizedModule.linearIndependent_lift", "def_path": "Mathlib/LinearAlgebra/Dimension/Localization.lean", "def_pos": [33, 7], "def_end_pos": [33, 47]}, {"full_name": "LocalizedModule.mkLinearMap", "def_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "def_pos": [490, 5], "def_end_pos": [490, 16]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "Module.Free.chooseBasis", "def_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean", "def_pos": [85, 19], "def_end_pos": [85, 30]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}, {"full_name": "LocalizedModule", "def_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "def_pos": [82, 5], "def_end_pos": [82, 27]}, {"full_name": "Basis.linearIndependent", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [555, 19], "def_end_pos": [555, 36]}]], "state_before": "R : Type u\nS : Type u'\nM : Type v\nN : Type v'\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Module S N\ninst\u271d\u00b3 : IsScalarTower R S N\np : Submonoid R\ninst\u271d\u00b2 : IsLocalization p S\nf : M \u2192\u2097[R] N\ninst\u271d\u00b9 : IsLocalizedModule p f\nhp : p \u2264 R\u2070\ninst\u271d : IsDomain R\n\u22a2 \u2203 s, #\u2191s = Module.rank R M \u2227 LinearIndependent (\u03b9 := \u2191s) R Subtype.val", "state_after": "case intro\nR : Type u\nS : Type u'\nM : Type v\nN : Type v'\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Module S N\ninst\u271d\u00b3 : IsScalarTower R S N\np : Submonoid R\ninst\u271d\u00b2 : IsLocalization p S\nf : M \u2192\u2097[R] N\ninst\u271d\u00b9 : IsLocalizedModule p f\nhp : p \u2264 R\u2070\ninst\u271d : IsDomain R\nw : Module.Free.ChooseBasisIndex (FractionRing R) (LocalizedModule R\u2070 M) \u2192 M\nhw : LinearIndependent R w\n\u22a2 \u2203 s, #\u2191s = Module.rank R M \u2227 LinearIndependent (\u03b9 := \u2191s) R Subtype.val"}, {"tactic": "refine \u27e8Set.range w, ?_, 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"commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup'_eq_sup", "start": [1184, 1], "end": [1185, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Closed.lean", "full_name": "CategoryTheory.topologyOfClosureOperator_self", "start": [294, 1], "end": [297, 50], "traced_tactics": [{"tactic": "ext X S", "annotated_tactic": ["ext X S", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nJ\u2081 J\u2082 : GrothendieckTopology C\n\u22a2 topologyOfClosureOperator J\u2081.closureOperator \u22ef = J\u2081", "state_after": "case h.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nJ\u2081 J\u2082 : GrothendieckTopology C\nX : C\nS : Sieve X\n\u22a2 S \u2208 (topologyOfClosureOperator J\u2081.closureOperator \u22ef).sieves X \u2194 S \u2208 J\u2081.sieves X"}, {"tactic": "apply 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: \u2115\nhn : p.Coprime n\ng : G\n\u22a2 (hG.powEquiv hn).symm g = g ^ (orderOf g).gcdB n", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d : Group G\nhG : IsPGroup p G\nn : \u2115\nhn : p.Coprime n\ng : G\n\u22a2 (hG.powEquiv hn).symm g = g ^ (Nat.card \u21a5(Subgroup.zpowers g)).gcdB n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d : Group G\nhG : IsPGroup p G\nn : \u2115\nhn : p.Coprime n\ng : G\n\u22a2 (hG.powEquiv hn).symm g = g ^ (Nat.card \u21a5(Subgroup.zpowers g)).gcdB n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.cos_nat_mul_pi_sub", "start": [438, 1], "end": [439, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable", "start": [795, 1], "end": [869, 57], "traced_tactics": [{"tactic": "let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' \u03bc", "annotated_tactic": ["let A := <a>MeasureTheory.Measure.finiteSpanningSetsInOpen'</a> \u03bc", [{"full_name": "MeasureTheory.Measure.finiteSpanningSetsInOpen'", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [1567, 1], "def_end_pos": [1604, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)"}, {"tactic": "rcases h'f.isSeparable_range with \u27e8t, t_count, ht\u27e9", "annotated_tactic": ["rcases h'f.isSeparable_range with \u27e8t, t_count, ht\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)"}, {"tactic": "filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x", "annotated_tactic": ["filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)"}, {"tactic": "have M :\n  \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b y in a, \u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a)\n    (v.filterAt x) (\ud835\udcdd \u2016f x - c\u2016\u208a) := by\n  intro c hc\n  obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x\n  specialize hx n c hc\n  simp only [xn, indicator_of_mem] at hx\n  apply hx.congr' _\n  filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n    v.eventually_filterAt_measurableSet x] with a ha h'a\n  congr 1\n  apply setLIntegral_congr_fun h'a\n  filter_upwards with y hy using (by simp only [ha hy, indicator_of_mem])", "annotated_tactic": ["have M :\n    \u2200 c \u2208 t, <a>Tendsto</a> (fun a => (\u222b\u207b y in a, \u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a)\n      (v.filterAt x) (\ud835\udcdd \u2016f x - c\u2016\u208a) := by\n    intro c hc\n    obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using <a>mem_univ</a> x\n    specialize hx n c hc\n    simp only [xn, <a>indicator_of_mem</a>] at hx\n    apply hx.congr' _\n    filter_upwards [v.eventually_filterAt_subset_of_nhds (<a>IsOpen.mem_nhds</a> (A.set_mem n) xn),\n      v.eventually_filterAt_measurableSet x] with a ha h'a\n    congr 1\n    apply <a>setLIntegral_congr_fun</a> h'a\n    filter_upwards with y hy using (by simp only [ha hy, <a>indicator_of_mem</a>])", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "MeasureTheory.setLIntegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [343, 9], "def_end_pos": [343, 31]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)"}, {"tactic": "apply ENNReal.tendsto_nhds_zero.2 fun \u03b5 \u03b5pos => ?_", "annotated_tactic": ["apply <a>ENNReal.tendsto_nhds_zero</a>.2 fun \u03b5 \u03b5pos => ?_", [{"full_name": "ENNReal.tendsto_nhds_zero", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [293, 19], "def_end_pos": [293, 36]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in v.filterAt x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc x_1 \u2264 \u03b5"}, {"tactic": "obtain \u27e8c, ct, xc\u27e9 : \u2203 c \u2208 t, (\u2016f x - c\u2016\u208a : \u211d\u22650\u221e) < \u03b5 / 2 := by\n  simp_rw [\u2190 edist_eq_coe_nnnorm_sub]\n  have : f x \u2208 closure t := ht (mem_range_self _)\n  exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos))", "annotated_tactic": ["obtain \u27e8c, ct, xc\u27e9 : \u2203 c \u2208 t, (\u2016f x - c\u2016\u208a : \u211d\u22650\u221e) < \u03b5 / 2 := by\n    simp_rw [\u2190 <a>edist_eq_coe_nnnorm_sub</a>]\n    have : f x \u2208 <a>closure</a> t := ht (<a>mem_range_self</a> _)\n    exact <a>EMetric.mem_closure_iff</a>.1 this (\u03b5 / 2) (<a>ENNReal.half_pos</a> (<a>ne_of_gt</a> \u03b5pos))", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [870, 3], "def_end_pos": [870, 14]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}, {"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [492, 19], "def_end_pos": [492, 27]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in v.filterAt x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc x_1 \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in v.filterAt x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc x_1 \u2264 \u03b5"}, {"tactic": "filter_upwards [(tendsto_order.1 (M c ct)).2 (\u03b5 / 2) xc, h'x, v.eventually_measure_lt_top x] with\n  a ha h'a h''a", "annotated_tactic": ["filter_upwards [(<a>tendsto_order</a>.1 (M c ct)).2 (\u03b5 / 2) xc, h'x, v.eventually_measure_lt_top x] with\n    a ha h'a h''a", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in v.filterAt x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc x_1 \u2264 \u03b5", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a \u2264 \u03b5"}, {"tactic": "apply ENNReal.div_le_of_le_mul", "annotated_tactic": ["apply <a>ENNReal.div_le_of_le_mul</a>", [{"full_name": "ENNReal.div_le_of_le_mul", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [332, 9], "def_end_pos": [332, 25]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u03bc a \u2264 \u03b5", "state_after": "case h.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc \u2264 \u03b5 * \u03bc a"}, {"tactic": "simp_rw [ae_all_iff]", "annotated_tactic": ["simp_rw [<a>ae_all_iff</a>]", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [102, 9], "def_end_pos": [102, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200 (i : \u2115),\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set i).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt a)\n          (\ud835\udcdd \u2191\u2016f a - (A.set i).indicator (fun x => c) a\u2016\u208a)"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200 (i : \u2115),\n    \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set i).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt a)\n          (\ud835\udcdd \u2191\u2016f a - (A.set i).indicator (fun x => c) a\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nx : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt a)\n        (\ud835\udcdd \u2191\u2016f a - (A.set x).indicator (fun x => c) a\u2016\u208a)"}, {"tactic": "rw [ae_ball_iff t_count]", "annotated_tactic": ["rw [<a>ae_ball_iff</a> t_count]", [{"full_name": "MeasureTheory.ae_ball_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [115, 9], "def_end_pos": [115, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nx : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt a)\n        (\ud835\udcdd \u2191\u2016f a - (A.set x).indicator (fun x => c) a\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nx : \u2115\n\u22a2 \u2200 i \u2208 t,\n    \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => i) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x_1)\n        (\ud835\udcdd \u2191\u2016f x_1 - (A.set x).indicator (fun x => i) x_1\u2016\u208a)"}, {"tactic": "revert x", "annotated_tactic": ["revert x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nx : \u2115\n\u22a2 \u2200 i \u2208 t,\n    \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bc,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => i) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x_1)\n        (\ud835\udcdd \u2191\u2016f x_1 - (A.set x).indicator (fun x => i) x_1\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200 (x : \u2115),\n    \u2200 i \u2208 t,\n      \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bc,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => i) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x_1)\n          (\ud835\udcdd \u2191\u2016f x_1 - (A.set x).indicator (fun x => i) x_1\u2016\u208a)"}, {"tactic": "intro n c _", "annotated_tactic": ["intro n c _", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\n\u22a2 \u2200 (x : \u2115),\n    \u2200 i \u2208 t,\n      \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bc,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set x).indicator (fun x => i) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x_1)\n          (\ud835\udcdd \u2191\u2016f x_1 - (A.set x).indicator (fun x => i) x_1\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n      (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)"}, {"tactic": "apply ae_tendsto_lintegral_div'", "annotated_tactic": ["apply <a>ae_tendsto_lintegral_div'</a>", [{"full_name": "VitaliFamily.ae_tendsto_lintegral_div'", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [769, 9], "def_end_pos": [769, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n      (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Measurable fun y => \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a\n\ncase h'f\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "refine (h'f.sub ?_).ennnorm", "annotated_tactic": ["refine (h'f.sub ?_).<a>ennnorm</a>", [{"full_name": "MeasureTheory.StronglyMeasurable.ennnorm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [895, 19], "def_end_pos": [895, 26]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Measurable fun y => \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a", "state_after": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 StronglyMeasurable ((A.set n).indicator fun x => c)"}, {"tactic": "exact stronglyMeasurable_const.indicator (IsOpen.measurableSet (A.set_mem n))", "annotated_tactic": ["exact stronglyMeasurable_const.indicator (<a>IsOpen.measurableSet</a> (A.set_mem n))", [{"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [261, 9], "def_end_pos": [261, 29]}]], "state_before": "case hf\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 StronglyMeasurable ((A.set n).indicator fun x => c)", "state_after": "no goals"}, {"tactic": "apply ne_of_lt", "annotated_tactic": ["apply <a>ne_of_lt</a>", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case h'f\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc \u2260 \u22a4", "state_after": "case h'f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "calc\n  (\u222b\u207b y, \u2191\u2016f y - (A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc) \u2264\n      \u222b\u207b y, \u2016f y\u2016\u208a + \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc := by\n    apply lintegral_mono\n    intro x\n    dsimp\n    rw [\u2190 ENNReal.coe_add]\n    exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)\n  _ = (\u222b\u207b y, \u2016f y\u2016\u208a \u2202\u03bc) + \u222b\u207b y, \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc :=\n    (lintegral_add_left h'f.ennnorm _)\n  _ < \u221e + \u221e :=\n    haveI I : Integrable ((A.set n).indicator fun _ : \u03b1 => c) \u03bc := by\n      simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n        integrableOn_const, A.finite n, or_true_iff]\n    ENNReal.add_lt_add hf.2 I.2", "annotated_tactic": ["calc\n        (\u222b\u207b y, \u2191\u2016f y - (A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc) \u2264\n            \u222b\u207b y, \u2016f y\u2016\u208a + \u2016(A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc := by\n          apply <a>lintegral_mono</a>\n          intro x\n          dsimp\n          rw [\u2190 <a>ENNReal.coe_add</a>]\n          exact <a>ENNReal.coe_le_coe</a>.2 (<a>nnnorm_sub_le</a> _ _)\n        _ = (\u222b\u207b y, \u2016f y\u2016\u208a \u2202\u03bc) + \u222b\u207b y, \u2016(A.set n).<a>indicator</a> (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc :=\n          (<a>lintegral_add_left</a> h'f.ennnorm _)\n        _ < \u221e + \u221e :=\n          haveI I : <a>Integrable</a> ((A.set n).<a>indicator</a> fun _ : \u03b1 => c) \u03bc := by\n            simp only [<a>integrable_indicator_iff</a> (<a>IsOpen.measurableSet</a> (A.set_mem n)),\n              <a>integrableOn_const</a>, A.finite n, <a>or_true_iff</a>]\n          <a>ENNReal.add_lt_add</a> hf.2 I.2", [{"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}, {"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [100, 9], "def_end_pos": [100, 23]}, {"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [410, 26], "def_end_pos": [410, 33]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [371, 28], "def_end_pos": [371, 38]}, {"full_name": "nnnorm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [823, 3], "def_end_pos": [823, 14]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}, {"full_name": "MeasureTheory.lintegral_add_left", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [586, 9], "def_end_pos": [586, 27]}, {"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [438, 5], "def_end_pos": [438, 15]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}, {"full_name": "MeasureTheory.integrable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 33]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [261, 9], "def_end_pos": [261, 29]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [112, 9], "def_end_pos": [112, 27]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 20]}, {"full_name": "ENNReal.add_lt_add", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [300, 19], "def_end_pos": [300, 29]}]], "state_before": "case h'f.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "apply lintegral_mono", "annotated_tactic": ["apply <a>lintegral_mono</a>", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [100, 9], "def_end_pos": [100, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc \u2264\n    \u222b\u207b (y : \u03b1), \u2191\u2016f y\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 (fun a => \u2191\u2016f a - (A.set n).indicator (fun x => c) a\u2016\u208a) \u2264 fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) a\u2016\u208a"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 (fun a => \u2191\u2016f a - (A.set n).indicator (fun x => c) a\u2016\u208a) \u2264 fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) a\u2016\u208a", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 (fun a => \u2191\u2016f a - (A.set n).indicator (fun x => c) a\u2016\u208a) x \u2264\n    (fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) a\u2016\u208a) x"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 (fun a => \u2191\u2016f a - (A.set n).indicator (fun x => c) a\u2016\u208a) x \u2264\n    (fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) a\u2016\u208a) x", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a \u2264 \u2191\u2016f x\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) x\u2016\u208a"}, {"tactic": "rw [\u2190 ENNReal.coe_add]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.coe_add</a>]", [{"full_name": "ENNReal.coe_add", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [410, 26], "def_end_pos": [410, 33]}]], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a \u2264 \u2191\u2016f x\u2016\u208a + \u2191\u2016(A.set n).indicator (fun x => c) x\u2016\u208a", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a \u2264 \u2191(\u2016f x\u2016\u208a + \u2016(A.set n).indicator (fun x => c) x\u2016\u208a)"}, {"tactic": "exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)", "annotated_tactic": ["exact <a>ENNReal.coe_le_coe</a>.2 (<a>nnnorm_sub_le</a> _ _)", [{"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [371, 28], "def_end_pos": [371, 38]}, {"full_name": "nnnorm_sub_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [823, 3], "def_end_pos": [823, 14]}]], "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\nx : \u03b1\n\u22a2 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a \u2264 \u2191(\u2016f x\u2016\u208a + \u2016(A.set n).indicator (fun x => c) x\u2016\u208a)", "state_after": "no goals"}, {"tactic": "simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n  integrableOn_const, A.finite n, or_true_iff]", "annotated_tactic": ["simp only [<a>integrable_indicator_iff</a> (<a>IsOpen.measurableSet</a> (A.set_mem n)),\n              <a>integrableOn_const</a>, A.finite n, <a>or_true_iff</a>]", [{"full_name": "MeasureTheory.integrable_indicator_iff", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [268, 9], "def_end_pos": [268, 33]}, {"full_name": "IsOpen.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [261, 9], "def_end_pos": [261, 29]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [112, 9], "def_end_pos": [112, 27]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nn : \u2115\nc : E\nhi\u271d : c \u2208 t\n\u22a2 Integrable ((A.set n).indicator fun x => c) \u03bc", "state_after": "no goals"}, {"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\n\u22a2 \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x", "annotated_tactic": ["obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using <a>mem_univ</a> x", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "specialize hx n c hc", "annotated_tactic": ["specialize hx n c hc", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "simp only [xn, indicator_of_mem] at hx", "annotated_tactic": ["simp only [xn, <a>indicator_of_mem</a>] at hx", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)"}, {"tactic": "apply hx.congr' _", "annotated_tactic": ["apply hx.congr' _", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) =\u1da0[v.filterAt x] fun a =>\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a"}, {"tactic": "filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n  v.eventually_filterAt_measurableSet x] with a ha h'a", "annotated_tactic": ["filter_upwards [v.eventually_filterAt_subset_of_nhds (<a>IsOpen.mem_nhds</a> (A.set_mem n) xn),\n      v.eventually_filterAt_measurableSet x] with a ha h'a", [{"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u22a2 (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) =\u1da0[v.filterAt x] fun a =>\n    (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a = (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a = (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc = \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc"}, {"tactic": "apply setLIntegral_congr_fun h'a", "annotated_tactic": ["apply <a>setLIntegral_congr_fun</a> h'a", [{"full_name": "MeasureTheory.setLIntegral_congr_fun", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [343, 9], "def_end_pos": [343, 31]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc = \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 a \u2192 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a = \u2191\u2016f x - c\u2016\u208a"}, {"tactic": "filter_upwards with y hy using (by simp only [ha hy, indicator_of_mem])", "annotated_tactic": ["filter_upwards with y hy using (by simp only [ha hy, <a>indicator_of_mem</a>])", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 a \u2192 \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a = \u2191\u2016f x - c\u2016\u208a", "state_after": "no goals"}, {"tactic": "simpa [\u2190 A.spanning] using mem_univ x", "annotated_tactic": ["simpa [\u2190 A.spanning] using <a>mem_univ</a> x", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\n\u22a2 \u2203 n, x \u2208 A.set n", "state_after": "no goals"}, {"tactic": "simp only [ha hy, indicator_of_mem]", "annotated_tactic": ["simp only [ha hy, <a>indicator_of_mem</a>]", [{"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nc : E\nhc : c \u2208 t\nn : \u2115\nxn : x \u2208 A.set n\nhx :\n  Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n    (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\na : Set \u03b1\nha : a \u2286 A.set n\nh'a : MeasurableSet a\ny : \u03b1\nhy : y \u2208 a\n\u22a2 \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 edist_eq_coe_nnnorm_sub]", "annotated_tactic": ["simp_rw [\u2190 <a>edist_eq_coe_nnnorm_sub</a>]", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [870, 3], "def_end_pos": [870, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c \u2208 t, \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c \u2208 t, edist (f x) c < \u03b5 / 2"}, {"tactic": "have : f x \u2208 closure t := ht (mem_range_self _)", "annotated_tactic": ["have : f x \u2208 <a>closure</a> t := ht (<a>mem_range_self</a> _)", [{"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 c \u2208 t, edist (f x) c < \u03b5 / 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : f x \u2208 closure t\n\u22a2 \u2203 c \u2208 t, edist (f x) c < \u03b5 / 2"}, {"tactic": "exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos))", "annotated_tactic": ["exact <a>EMetric.mem_closure_iff</a>.1 this (\u03b5 / 2) (<a>ENNReal.half_pos</a> (<a>ne_of_gt</a> \u03b5pos))", [{"full_name": "EMetric.mem_closure_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 24]}, {"full_name": "ENNReal.half_pos", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [492, 19], "def_end_pos": [492, 27]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : f x \u2208 closure t\n\u22a2 \u2203 c \u2208 t, edist (f x) c < \u03b5 / 2", "state_after": "no goals"}, {"tactic": "apply lintegral_mono fun x => ?_", "annotated_tactic": ["apply <a>lintegral_mono</a> fun x => ?_", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [100, 9], "def_end_pos": [100, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a + \u2191\u2016f x - c\u2016\u208a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx\u271d : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x\u271d)\n        (\ud835\udcdd \u2191\u2016f x\u271d - (A.set n).indicator (fun x => c) x\u271d\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x\u271d, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x\u271d - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016f x - f x\u271d\u2016\u208a \u2264 \u2191\u2016f x - c\u2016\u208a + \u2191\u2016f x\u271d - c\u2016\u208a"}, {"tactic": "simpa only [\u2190 edist_eq_coe_nnnorm_sub] using edist_triangle_right _ _ _", "annotated_tactic": ["simpa only [\u2190 <a>edist_eq_coe_nnnorm_sub</a>] using <a>edist_triangle_right</a> _ _ _", [{"full_name": "edist_eq_coe_nnnorm_sub", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [870, 3], "def_end_pos": [870, 14]}, {"full_name": "edist_triangle_right", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [114, 9], "def_end_pos": [114, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx\u271d : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x\u271d)\n        (\ud835\udcdd \u2191\u2016f x\u271d - (A.set n).indicator (fun x => c) x\u271d\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x\u271d, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x\u271d - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016f x - f x\u271d\u2016\u208a \u2264 \u2191\u2016f x - c\u2016\u208a + \u2191\u2016f x\u271d - c\u2016\u208a", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc + \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a + \u03b5 / 2 * \u03bc a", "state_after": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a\n\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a"}, {"tactic": "rw [ENNReal.div_lt_iff (Or.inl h'a.ne') (Or.inl h''a.ne)] at ha", "annotated_tactic": ["rw [<a>ENNReal.div_lt_iff</a> (<a>Or.inl</a> h'a.ne') (<a>Or.inl</a> h''a.ne)] at ha", [{"full_name": "ENNReal.div_lt_iff", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [355, 19], "def_end_pos": [355, 29]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a", "state_after": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc < \u03b5 / 2 * \u03bc a\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a"}, {"tactic": "exact ha.le", "annotated_tactic": ["exact ha.le", []], "state_before": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc < \u03b5 / 2 * \u03bc a\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter]", "annotated_tactic": ["simp only [<a>lintegral_const</a>, <a>Measure.restrict_apply</a>, <a>MeasurableSet.univ</a>, <a>univ_inter</a>]", [{"full_name": "MeasureTheory.lintegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [137, 9], "def_end_pos": [137, 24]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [103, 19], "def_end_pos": [103, 37]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [985, 9], "def_end_pos": [985, 19]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u03bc a", "state_after": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u2191\u2016f x - c\u2016\u208a * \u03bc a \u2264 \u03b5 / 2 * \u03bc a"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u2191\u2016f x - c\u2016\u208a * \u03bc a \u2264 \u03b5 / 2 * \u03bc a", "state_after": "no goals"}, {"tactic": "rw [\u2190 add_mul, ENNReal.add_halves]", "annotated_tactic": ["rw [\u2190 <a>add_mul</a>, <a>ENNReal.add_halves</a>]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "ENNReal.add_halves", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [477, 19], "def_end_pos": [477, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nh'f : StronglyMeasurable f\nA : \u03bc.FiniteSpanningSetsIn {K | IsOpen K} := \u03bc.finiteSpanningSetsInOpen'\nt : Set E\nt_count : t.Countable\nht : range f \u2286 closure t\nmain :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2200 (n : \u2115),\n      \u2200 c \u2208 t,\n        Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n          (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nx : \u03b1\nhx :\n  \u2200 (n : \u2115),\n    \u2200 c \u2208 t,\n      Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - (A.set n).indicator (fun x => c) y\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x)\n        (\ud835\udcdd \u2191\u2016f x - (A.set n).indicator (fun x => c) x\u2016\u208a)\nh'x : \u2200\u1da0 (a : Set \u03b1) in v.filterAt x, 0 < \u03bc a\nM : \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a) (v.filterAt x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a)\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nc : E\nct : c \u2208 t\nxc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2\na : Set \u03b1\nha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a < \u03b5 / 2\nh'a : 0 < \u03bc a\nh''a : \u03bc a < \u22a4\n\u22a2 \u03b5 / 2 * \u03bc a + \u03b5 / 2 * \u03bc a = \u03b5 * \u03bc a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_univ", "start": [91, 1], "end": [91, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "Units.mul_right_eq_zero", "start": [47, 1], "end": [48, 89], "traced_tactics": [{"tactic": "simpa using mul_eq_zero_of_right (\u2191u\u207b\u00b9) h", "annotated_tactic": ["simpa using <a>mul_eq_zero_of_right</a> (\u2191u\u207b\u00b9) h", [{"full_name": "mul_eq_zero_of_right", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [274, 9], "def_end_pos": [274, 29]}]], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MonoidWithZero M\u2080\nu : M\u2080\u02e3\na : M\u2080\nh : \u2191u * a = 0\n\u22a2 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.finset_sup_le_sum", "start": [410, 1], "end": [415, 15], "traced_tactics": [{"tactic": "classical\nrefine Finset.sup_le_iff.mpr ?_\nintro i hi\nrw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]\nexact bot_le", "annotated_tactic": ["classical\n  refine Finset.sup_le_iff.mpr ?_\n  intro i hi\n  rw [<a>Finset.sum_eq_sum_diff_singleton_add</a> hi, <a>le_add_iff_nonneg_left</a>]\n  exact <a>bot_le</a>", [{"full_name": "Finset.sum_eq_sum_diff_singleton_add", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1865, 3], "def_end_pos": [1865, 14]}, {"full_name": "le_add_iff_nonneg_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [465, 30], "def_end_pos": [465, 52]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\n\u22a2 s.sup p \u2264 \u2211 i \u2208 s, p i", "state_after": "no goals"}, {"tactic": "refine Finset.sup_le_iff.mpr ?_", "annotated_tactic": ["refine Finset.sup_le_iff.mpr ?_", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\n\u22a2 s.sup p \u2264 \u2211 i \u2208 s, p i", "state_after": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\n\u22a2 \u2200 b \u2208 s, p b \u2264 \u2211 i \u2208 s, p i"}, {"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\n\u22a2 \u2200 b \u2208 s, p b \u2264 \u2211 i \u2208 s, p i", "state_after": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 p i \u2264 \u2211 i \u2208 s, p i"}, {"tactic": "rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]", "annotated_tactic": ["rw [<a>Finset.sum_eq_sum_diff_singleton_add</a> hi, <a>le_add_iff_nonneg_left</a>]", [{"full_name": "Finset.sum_eq_sum_diff_singleton_add", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1865, 3], "def_end_pos": [1865, 14]}, {"full_name": "le_add_iff_nonneg_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [465, 30], "def_end_pos": [465, 52]}]], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 p i \u2264 \u2211 i \u2208 s, p i", "state_after": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 \u2211 x \u2208 s \\ {i}, p x"}, {"tactic": "exact bot_le", "annotated_tactic": ["exact <a>bot_le</a>", [{"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : AddCommGroup E\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup E\u2083\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c E\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083\ninst\u271d\u2074 : Module \ud835\udd5c F\ninst\u271d\u00b3 : Module \ud835\udd5c G\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\np : \u03b9 \u2192 Seminorm \ud835\udd5c E\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 \u2211 x \u2208 s \\ {i}, p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/CPolynomial.lean", "full_name": "hasFiniteFPowerSeriesOnBall_const", "start": [141, 1], "end": [143, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Separable.lean", "full_name": "Polynomial.Separable.injective_of_prod_X_sub_C", "start": [253, 1], "end": [255, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean", "full_name": "SimpleGraph.ComponentCompl.exists_eq_mk", "start": [120, 11], "end": [122, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isPredLimitRecOn_limit", "start": [410, 1], "end": [412, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Trace.lean", "full_name": "LinearMap.trace_tensorProduct", "start": [238, 1], "end": [247, 42], "traced_tactics": [{"tactic": "apply\n  (compl\u2081\u2082_inj (show Surjective (dualTensorHom R M M) from (dualTensorHomEquiv R M M).surjective)\n      (show Surjective (dualTensorHom R N N) from (dualTensorHomEquiv R N N).surjective)).1", "annotated_tactic": ["apply\n    (<a>compl\u2081\u2082_inj</a> (show <a>Surjective</a> (<a>dualTensorHom</a> R M M) from (<a>dualTensorHomEquiv</a> R M M).<a>surjective</a>)\n        (show <a>Surjective</a> (<a>dualTensorHom</a> R N N) from (<a>dualTensorHomEquiv</a> R N N).<a>surjective</a>)).1", [{"full_name": "LinearMap.compl\u2081\u2082_inj", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [368, 9], "def_end_pos": [368, 20]}, {"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}, {"full_name": "dualTensorHom", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [61, 5], "def_end_pos": [61, 18]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearEquiv.surjective", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [560, 19], "def_end_pos": [560, 29]}, {"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}, {"full_name": "dualTensorHom", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [61, 5], "def_end_pos": [61, 18]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearEquiv.surjective", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [560, 19], "def_end_pos": [560, 29]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\n\u22a2 (mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N)) = (lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\n\u22a2 ((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N) =\n    ((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N)"}, {"tactic": "ext f m g n", "annotated_tactic": ["ext f m g n", []], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\n\u22a2 ((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N) =\n    ((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N)", "state_after": "case a.h.h.a.h.h\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : Module.Dual R M\nm : M\ng : Module.Dual R N\nn : N\n\u22a2 ((AlgebraTensorModule.curry\n          (((AlgebraTensorModule.curry\n                (((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))).compl\u2081\u2082 (dualTensorHom R M M)\n                  (dualTensorHom R N N)))\n              f)\n            m))\n        g)\n      n =\n    ((AlgebraTensorModule.curry\n          (((AlgebraTensorModule.curry\n                (((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N)))\n              f)\n            m))\n        g)\n      n"}, {"tactic": "simp only [AlgebraTensorModule.curry_apply, toFun_eq_coe, TensorProduct.curry_apply,\n  coe_restrictScalars, compl\u2081\u2082_apply, compr\u2082_apply, mapBilinear_apply,\n  trace_eq_contract_apply, contractLeft_apply, lsmul_apply, Algebra.id.smul_eq_mul,\n  map_dualTensorHom, dualDistrib_apply]", "annotated_tactic": ["simp only [<a>AlgebraTensorModule.curry_apply</a>, <a>toFun_eq_coe</a>, <a>TensorProduct.curry_apply</a>,\n    <a>coe_restrictScalars</a>, <a>compl\u2081\u2082_apply</a>, <a>compr\u2082_apply</a>, <a>mapBilinear_apply</a>,\n    <a>trace_eq_contract_apply</a>, <a>contractLeft_apply</a>, <a>lsmul_apply</a>, <a>Algebra.id.smul_eq_mul</a>,\n    <a>map_dualTensorHom</a>, <a>dualDistrib_apply</a>]", [{"full_name": "TensorProduct.AlgebraTensorModule.curry_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Tower.lean", "def_pos": [82, 3], "def_end_pos": [82, 8]}, {"full_name": "LinearMap.toFun_eq_coe", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 21]}, {"full_name": "TensorProduct.curry_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [661, 9], "def_end_pos": [661, 20]}, {"full_name": "LinearMap.coe_restrictScalars", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [471, 9], "def_end_pos": [471, 28]}, {"full_name": "LinearMap.compl\u2081\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [358, 9], "def_end_pos": [358, 22]}, {"full_name": "LinearMap.compr\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [390, 9], "def_end_pos": [390, 21]}, {"full_name": "TensorProduct.mapBilinear_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [987, 9], "def_end_pos": [987, 26]}, {"full_name": "LinearMap.trace_eq_contract_apply", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [172, 9], "def_end_pos": [172, 32]}, {"full_name": "contractLeft_apply", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}, {"full_name": "LinearMap.lsmul_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [413, 9], "def_end_pos": [413, 20]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "map_dualTensorHom", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [113, 9], "def_end_pos": [113, 26]}, {"full_name": "TensorProduct.dualDistrib_apply", "def_path": "Mathlib/LinearAlgebra/Dual.lean", "def_pos": [1805, 9], "def_end_pos": [1805, 26]}]], "state_before": "case a.h.h.a.h.h\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : Module.Free R N\ninst\u271d\u00b2 : Module.Finite R N\ninst\u271d\u00b9 : Module.Free R P\ninst\u271d : Module.Finite R P\nf : Module.Dual R M\nm : M\ng : Module.Dual R N\nn : N\n\u22a2 ((AlgebraTensorModule.curry\n          (((AlgebraTensorModule.curry\n                (((mapBilinear R M N M N).compr\u2082 (trace R (M \u2297[R] N))).compl\u2081\u2082 (dualTensorHom R M M)\n                  (dualTensorHom R N N)))\n              f)\n            m))\n        g)\n      n =\n    ((AlgebraTensorModule.curry\n          (((AlgebraTensorModule.curry\n                (((lsmul R R).compl\u2081\u2082 (trace R M) (trace R N)).compl\u2081\u2082 (dualTensorHom R M M) (dualTensorHom R N N)))\n              f)\n            m))\n        g)\n      n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Ring.lean", "full_name": "Finset.sum_pow'", "start": [174, 1], "end": [176, 77], "traced_tactics": [{"tactic": "convert @prod_univ_sum (Fin n) _ _ _ _ _ (fun _i \u21a6 s) fun _i d \u21a6 f d", "annotated_tactic": ["convert @<a>prod_univ_sum</a> (<a>Fin</a> n) _ _ _ _ _ (fun _i \u21a6 s) fun _i d \u21a6 f d", [{"full_name": "Finset.prod_univ_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [165, 7], "def_end_pos": [165, 20]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b9\ni : \u03b9\na : \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2211 a \u2208 s, f a) ^ n = \u2211 p \u2208 piFinset fun _i => s, \u220f i : Fin n, f (p i)", "state_after": "case h.e'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b9\ni : \u03b9\na : \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2211 a \u2208 s, f a) ^ n = \u220f i : Fin n, \u2211 j \u2208 s, f j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03ba : \u03b9 \u2192 Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b9\ni : \u03b9\na : \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2211 a \u2208 s, f a) ^ n = \u220f i : Fin n, \u2211 j \u2208 s, f j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/CC/Lemmas.lean", "full_name": "Mathlib.Tactic.CC.eq_true_of_and_eq_true_right", "start": [102, 1], "end": [103, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordnode.lean", "full_name": "Ordnode.size_nil", "start": [139, 9], "end": [140, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/IdentDistrib.lean", "full_name": "ProbabilityTheory.IdentDistrib.sq", "start": [262, 11], "end": [264, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_set_union", "start": [364, 1], "end": [365, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.surj_on_of_inj_on_of_card_le", "start": [475, 1], "end": [486, 41], "traced_tactics": [{"tactic": "classical\nhave h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).card = s.card := by\n  rw [\u2190 @card_attach _ s, card_image_of_injective]\n  intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h\n  exact Subtype.eq <| hinj _ _ _ _ h\nobtain rfl : image (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n  eq_of_subset_of_card_le (image_subset_iff.2 $ by simpa) (by simp [hst, h])\nsimp only [mem_image, mem_attach, true_and, Subtype.exists, forall_exists_index]\nexact fun b a ha hb \u21a6 \u27e8a, ha, hb.symm\u27e9", "annotated_tactic": ["classical\n  have h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).<a>card</a> = s.card := by\n    rw [\u2190 @<a>card_attach</a> _ s, <a>card_image_of_injective</a>]\n    intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h\n    exact <a>Subtype.eq</a> <| hinj _ _ _ _ h\n  obtain rfl : <a>image</a> (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n    <a>eq_of_subset_of_card_le</a> (<a>image_subset_iff</a>.2 $ by simpa) (by simp [hst, h])\n  simp only [<a>mem_image</a>, <a>mem_attach</a>, <a>true_and</a>, <a>Subtype.exists</a>, <a>forall_exists_index</a>]\n  exact fun b a ha hb \u21a6 \u27e8a, ha, hb.symm\u27e9", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [42, 5], "def_end_pos": [42, 9]}, {"full_name": "Finset.card_attach", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 32]}, {"full_name": "Subtype.eq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1110, 19], "def_end_pos": [1110, 21]}, {"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [340, 5], "def_end_pos": [340, 10]}, {"full_name": "Finset.eq_of_subset_of_card_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Finset.image_subset_iff", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [480, 9], "def_end_pos": [480, 25]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Finset.mem_attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2501, 9], "def_end_pos": [2501, 19]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [63, 19], "def_end_pos": [63, 27]}, {"full_name": "forall_exists_index", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [189, 17], "def_end_pos": [189, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\n\u22a2 \u2200 b \u2208 t, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha", "state_after": "no goals"}, {"tactic": "have h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).card = s.card := by\n  rw [\u2190 @card_attach _ s, card_image_of_injective]\n  intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h\n  exact Subtype.eq <| hinj _ _ _ _ h", "annotated_tactic": ["have h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).<a>card</a> = s.card := by\n    rw [\u2190 @<a>card_attach</a> _ s, <a>card_image_of_injective</a>]\n    intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h\n    exact <a>Subtype.eq</a> <| hinj _ _ _ _ h", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [42, 5], "def_end_pos": [42, 9]}, {"full_name": "Finset.card_attach", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 32]}, {"full_name": "Subtype.eq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1110, 19], "def_end_pos": [1110, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\n\u22a2 \u2200 b \u2208 t, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\n\u22a2 \u2200 b \u2208 t, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha"}, {"tactic": "obtain rfl : image (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n  eq_of_subset_of_card_le (image_subset_iff.2 $ by simpa) (by simp [hst, h])", "annotated_tactic": ["obtain rfl : <a>image</a> (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n    <a>eq_of_subset_of_card_le</a> (<a>image_subset_iff</a>.2 $ by simpa) (by simp [hst, h])", [{"full_name": "Finset.image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [340, 5], "def_end_pos": [340, 10]}, {"full_name": "Finset.eq_of_subset_of_card_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Finset.image_subset_iff", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [480, 9], "def_end_pos": [480, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\n\u22a2 \u2200 b \u2208 t, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 image (fun a => f \u2191a \u22ef) s.attach\nhst : (image (fun a => f \u2191a \u22ef) s.attach).card \u2264 s.card\n\u22a2 \u2200 b \u2208 image (fun a => f \u2191a \u22ef) s.attach, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha"}, {"tactic": "simp only [mem_image, mem_attach, true_and, Subtype.exists, forall_exists_index]", "annotated_tactic": ["simp only [<a>mem_image</a>, <a>mem_attach</a>, <a>true_and</a>, <a>Subtype.exists</a>, <a>forall_exists_index</a>]", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Finset.mem_attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2501, 9], "def_end_pos": [2501, 19]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [63, 19], "def_end_pos": [63, 27]}, {"full_name": "forall_exists_index", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [189, 17], "def_end_pos": [189, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 image (fun a => f \u2191a \u22ef) s.attach\nhst : (image (fun a => f \u2191a \u22ef) s.attach).card \u2264 s.card\n\u22a2 \u2200 b \u2208 image (fun a => f \u2191a \u22ef) s.attach, \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 image (fun a => f \u2191a \u22ef) s.attach\nhst : (image (fun a => f \u2191a \u22ef) s.attach).card \u2264 s.card\n\u22a2 \u2200 (b : \u03b2) (x : \u03b1) (x_1 : x \u2208 s), f x \u22ef = b \u2192 \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha"}, {"tactic": "exact fun b a ha hb \u21a6 \u27e8a, ha, hb.symm\u27e9", "annotated_tactic": ["exact fun b a ha hb \u21a6 \u27e8a, ha, hb.symm\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 image (fun a => f \u2191a \u22ef) s.attach\nhst : (image (fun a => f \u2191a \u22ef) s.attach).card \u2264 s.card\n\u22a2 \u2200 (b : \u03b2) (x : \u03b1) (x_1 : x \u2208 s), f x \u22ef = b \u2192 \u2203 a, \u2203 (ha : a \u2208 s), b = f a ha", "state_after": "no goals"}, {"tactic": "rw [\u2190 @card_attach _ s, card_image_of_injective]", "annotated_tactic": ["rw [\u2190 @<a>card_attach</a> _ s, <a>card_image_of_injective</a>]", [{"full_name": "Finset.card_attach", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [286, 9], "def_end_pos": [286, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\n\u22a2 (image (fun a => f \u2191a \u22ef) s.attach).card = s.card", "state_after": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\n\u22a2 Injective fun a => f \u2191a \u22ef"}, {"tactic": "intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h", "annotated_tactic": ["intro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h", []], "state_before": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\n\u22a2 Injective fun a => f \u2191a \u22ef", "state_after": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nval\u271d\u00b9 : \u03b1\nproperty\u271d\u00b9 : val\u271d\u00b9 \u2208 s\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 s\nh : (fun a => f \u2191a \u22ef) \u27e8val\u271d\u00b9, property\u271d\u00b9\u27e9 = (fun a => f \u2191a \u22ef) \u27e8val\u271d, property\u271d\u27e9\n\u22a2 \u27e8val\u271d\u00b9, property\u271d\u00b9\u27e9 = \u27e8val\u271d, property\u271d\u27e9"}, {"tactic": "exact Subtype.eq <| hinj _ _ _ _ h", "annotated_tactic": ["exact <a>Subtype.eq</a> <| hinj _ _ _ _ h", [{"full_name": "Subtype.eq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1110, 19], "def_end_pos": [1110, 21]}]], "state_before": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nval\u271d\u00b9 : \u03b1\nproperty\u271d\u00b9 : val\u271d\u00b9 \u2208 s\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 s\nh : (fun a => f \u2191a \u22ef) \u27e8val\u271d\u00b9, property\u271d\u00b9\u27e9 = (fun a => f \u2191a \u22ef) \u27e8val\u271d, property\u271d\u27e9\n\u22a2 \u27e8val\u271d\u00b9, property\u271d\u00b9\u27e9 = \u27e8val\u271d, property\u271d\u27e9", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\n\u22a2 \u2200 x \u2208 s.attach, f \u2191x \u22ef \u2208 t", "state_after": "no goals"}, {"tactic": "simp [hst, h]", "annotated_tactic": ["simp [hst, h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : t.card \u2264 s.card\nh : (image (fun a => f \u2191a \u22ef) s.attach).card = s.card\n\u22a2 t.card \u2264 (image (fun a => f \u2191a \u22ef) s.attach).card", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Properties.lean", "full_name": "LinearMap.BilinForm.IsAlt.neg", "start": [168, 11], "end": [169, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.comap_fst_neBot", "start": [2591, 1], "end": [2593, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/NeLocus.lean", "full_name": "Finsupp.neLocus_self_sub_right", "start": [169, 1], "end": [170, 59], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, neLocus_self_add_right, support_neg]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>neLocus_self_add_right</a>, <a>support_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "Finsupp.neLocus_self_add_right", "def_path": "Mathlib/Data/Finsupp/NeLocus.lean", "def_pos": [159, 9], "def_end_pos": [159, 31]}, {"full_name": "Finsupp.support_neg", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [1359, 9], "def_end_pos": [1359, 20]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 f.neLocus (f - g) = g.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.einfsep_anti", "start": [136, 1], "end": [137, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.finite_mem_finset", "start": [1007, 1], "end": [1008, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "full_name": "OrderRingHom.coe_orderAddMonoidHom_id", "start": [267, 1], "end": [268, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_val", "start": [2581, 1], "end": [2582, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "full_name": "UpperHalfPlane.center_zero", "start": [158, 1], "end": [159, 57], "traced_tactics": [{"tactic": "rw [center_im, Real.cosh_zero, mul_one]", "annotated_tactic": ["rw [<a>center_im</a>, <a>Real.cosh_zero</a>, <a>mul_one</a>]", [{"full_name": "UpperHalfPlane.center_im", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean", "def_pos": [153, 9], "def_end_pos": [153, 18]}, {"full_name": "Real.cosh_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "z\u271d w : \u210d\nr R : \u211d\nz : \u210d\n\u22a2 (z.center 0).im = z.im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.prod_ext_iff", "start": [1660, 1], "end": [1663, 6], "traced_tactics": [{"tactic": "simp only [\u2190 coe_inj, LinearMap.prod_ext_iff]", "annotated_tactic": ["simp only [\u2190 <a>coe_inj</a>, <a>LinearMap.prod_ext_iff</a>]", [{"full_name": "ContinuousLinearMap.coe_inj", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [448, 9], "def_end_pos": [448, 16]}, {"full_name": "LinearMap.prod_ext_iff", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [296, 9], "def_end_pos": [296, 21]}]], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nS : Type u_4\nS\u2083 : Type u_5\ninst\u271d\u00b2\u2079 : Semiring R\ninst\u271d\u00b2\u2078 : Semiring R\u2082\ninst\u271d\u00b2\u2077 : Semiring R\u2083\ninst\u271d\u00b2\u2076 : Semiring S\ninst\u271d\u00b2\u2075 : Semiring S\u2083\nM : Type u_6\ninst\u271d\u00b2\u2074 : TopologicalSpace M\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nM\u2082 : Type u_7\ninst\u271d\u00b2\u00b9 : TopologicalSpace M\u2082\ninst\u271d\u00b2\u2070 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2079 : Module R\u2082 M\u2082\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace M\u2083\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2076 : Module R\u2083 M\u2083\nN\u2082 : Type u_9\ninst\u271d\u00b9\u2075 : TopologicalSpace N\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u00b3 : Module R N\u2082\nN\u2083 : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace N\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2083\ninst\u271d\u00b9\u2070 : Module R N\u2083\ninst\u271d\u2079 : Module S\u2083 M\u2083\ninst\u271d\u2078 : SMulCommClass R\u2083 S\u2083 M\u2083\ninst\u271d\u2077 : ContinuousConstSMul S\u2083 M\u2083\ninst\u271d\u2076 : Module S N\u2082\ninst\u271d\u2075 : ContinuousConstSMul S N\u2082\ninst\u271d\u2074 : SMulCommClass R S N\u2082\ninst\u271d\u00b3 : Module S N\u2083\ninst\u271d\u00b2 : SMulCommClass R S N\u2083\ninst\u271d\u00b9 : ContinuousConstSMul S N\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\nc : S\nh : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf\u271d g\u271d : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nx y z : M\nf g : M \u00d7 N\u2082 \u2192L[R] N\u2083\n\u22a2 f = g \u2194 f.comp (inl R M N\u2082) = g.comp (inl R M N\u2082) \u2227 f.comp (inr R M N\u2082) = g.comp (inr R M N\u2082)", "state_after": "R : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nS : Type u_4\nS\u2083 : Type u_5\ninst\u271d\u00b2\u2079 : Semiring R\ninst\u271d\u00b2\u2078 : Semiring R\u2082\ninst\u271d\u00b2\u2077 : Semiring R\u2083\ninst\u271d\u00b2\u2076 : Semiring S\ninst\u271d\u00b2\u2075 : Semiring S\u2083\nM : Type u_6\ninst\u271d\u00b2\u2074 : TopologicalSpace M\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nM\u2082 : Type u_7\ninst\u271d\u00b2\u00b9 : TopologicalSpace M\u2082\ninst\u271d\u00b2\u2070 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2079 : Module R\u2082 M\u2082\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace M\u2083\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2076 : Module R\u2083 M\u2083\nN\u2082 : Type u_9\ninst\u271d\u00b9\u2075 : TopologicalSpace N\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u00b3 : Module R N\u2082\nN\u2083 : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace N\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2083\ninst\u271d\u00b9\u2070 : Module R N\u2083\ninst\u271d\u2079 : Module S\u2083 M\u2083\ninst\u271d\u2078 : SMulCommClass R\u2083 S\u2083 M\u2083\ninst\u271d\u2077 : ContinuousConstSMul S\u2083 M\u2083\ninst\u271d\u2076 : Module S N\u2082\ninst\u271d\u2075 : ContinuousConstSMul S N\u2082\ninst\u271d\u2074 : SMulCommClass R S N\u2082\ninst\u271d\u00b3 : Module S N\u2083\ninst\u271d\u00b2 : SMulCommClass R S N\u2083\ninst\u271d\u00b9 : ContinuousConstSMul S N\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\nc : S\nh : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf\u271d g\u271d : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nx y z : M\nf g : M \u00d7 N\u2082 \u2192L[R] N\u2083\n\u22a2 \u2191f \u2218\u2097 LinearMap.inl R M N\u2082 = \u2191g \u2218\u2097 LinearMap.inl R M N\u2082 \u2227 \u2191f \u2218\u2097 LinearMap.inr R M N\u2082 = \u2191g \u2218\u2097 LinearMap.inr R M N\u2082 \u2194\n    \u2191(f.comp (inl R M N\u2082)) = \u2191(g.comp (inl R M N\u2082)) \u2227 \u2191(f.comp (inr R M N\u2082)) = \u2191(g.comp (inr R M N\u2082))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nS : Type u_4\nS\u2083 : Type u_5\ninst\u271d\u00b2\u2079 : Semiring R\ninst\u271d\u00b2\u2078 : Semiring R\u2082\ninst\u271d\u00b2\u2077 : Semiring R\u2083\ninst\u271d\u00b2\u2076 : Semiring S\ninst\u271d\u00b2\u2075 : Semiring S\u2083\nM : Type u_6\ninst\u271d\u00b2\u2074 : TopologicalSpace M\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nM\u2082 : Type u_7\ninst\u271d\u00b2\u00b9 : TopologicalSpace M\u2082\ninst\u271d\u00b2\u2070 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2079 : Module R\u2082 M\u2082\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace M\u2083\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2076 : Module R\u2083 M\u2083\nN\u2082 : Type u_9\ninst\u271d\u00b9\u2075 : TopologicalSpace N\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u00b3 : Module R N\u2082\nN\u2083 : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace N\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2083\ninst\u271d\u00b9\u2070 : Module R N\u2083\ninst\u271d\u2079 : Module S\u2083 M\u2083\ninst\u271d\u2078 : SMulCommClass R\u2083 S\u2083 M\u2083\ninst\u271d\u2077 : ContinuousConstSMul S\u2083 M\u2083\ninst\u271d\u2076 : Module S N\u2082\ninst\u271d\u2075 : ContinuousConstSMul S N\u2082\ninst\u271d\u2074 : SMulCommClass R S N\u2082\ninst\u271d\u00b3 : Module S N\u2083\ninst\u271d\u00b2 : SMulCommClass R S N\u2083\ninst\u271d\u00b9 : ContinuousConstSMul S N\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\nc : S\nh : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf\u271d g\u271d : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nx y z : M\nf g : M \u00d7 N\u2082 \u2192L[R] N\u2083\n\u22a2 \u2191f \u2218\u2097 LinearMap.inl R M N\u2082 = \u2191g \u2218\u2097 LinearMap.inl R M N\u2082 \u2227 \u2191f \u2218\u2097 LinearMap.inr R M N\u2082 = \u2191g \u2218\u2097 LinearMap.inr R M N\u2082 \u2194\n    \u2191(f.comp (inl R M N\u2082)) = \u2191(g.comp (inl R M N\u2082)) \u2227 \u2191(f.comp (inr R M N\u2082)) = \u2191(g.comp (inr R M N\u2082))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventually_map", "start": [1920, 1], "end": [1921, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "orthogonalProjection_inner_eq_zero", "start": [490, 1], "end": [492, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.eval\u2082_neg", "start": [1309, 1], "end": [1310, 69], "traced_tactics": [{"tactic": "rw [eq_neg_iff_add_eq_zero, \u2190 eval\u2082_add, add_left_neg, eval\u2082_zero]", "annotated_tactic": ["rw [<a>eq_neg_iff_add_eq_zero</a>, \u2190 <a>eval\u2082_add</a>, <a>add_left_neg</a>, <a>eval\u2082_zero</a>]", [{"full_name": "eq_neg_iff_add_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [931, 3], 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{"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsRelPrime.neg_neg", "start": [491, 11], "end": [491, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "nilpotent_of_mulEquiv", "start": [569, 1], "end": [571, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.invOn_id", "start": [1307, 1], "end": [1307, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/AddTorsor.lean", "full_name": "eq_of_vsub_eq_zero", "start": [129, 1], "end": 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "full_name": "VitaliFamily.FineSubfamilyOn.covering_disjoint_subtype", "start": [144, 1], "end": [145, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "full_name": "Projectivization.submodule_injective", "start": [151, 1], "end": [156, 46], "traced_tactics": [{"tactic": "induction' u using ind with u hu", "annotated_tactic": ["induction' u using <a>ind</a> with u hu", [{"full_name": "Projectivization.ind", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 12]}]], "state_before": "K : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu v : \u2119 K V\nh : u.submodule = v.submodule\n\u22a2 u = v", "state_after": "case h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\nu : V\nhu : u \u2260 0\nh : (mk K u hu).submodule = v.submodule\n\u22a2 mk K u hu = v"}, {"tactic": "induction' v using ind with v hv", "annotated_tactic": ["induction' v using <a>ind</a> with v hv", [{"full_name": "Projectivization.ind", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 12]}]], "state_before": "case h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\nu : V\nhu : u \u2260 0\nh : (mk K u hu).submodule = v.submodule\n\u22a2 mk K u hu = v", "state_after": "case h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\nh : (mk K u hu).submodule = (mk K v hv).submodule\n\u22a2 mk K u hu = mk K v hv"}, {"tactic": "rw [submodule_mk, submodule_mk, Submodule.span_singleton_eq_span_singleton] at h", "annotated_tactic": ["rw [<a>submodule_mk</a>, <a>submodule_mk</a>, <a>Submodule.span_singleton_eq_span_singleton</a>] at h", [{"full_name": "Projectivization.submodule_mk", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 21]}, {"full_name": "Projectivization.submodule_mk", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 21]}, {"full_name": "Submodule.span_singleton_eq_span_singleton", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [628, 9], "def_end_pos": [628, 41]}]], "state_before": "case h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\nh : (mk K u hu).submodule = (mk K v hv).submodule\n\u22a2 mk K u hu = mk K v hv", "state_after": "case h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\nh : \u2203 z, z \u2022 u = v\n\u22a2 mk K u hu = mk K v hv"}, {"tactic": "exact ((mk_eq_mk_iff K v u hv hu).2 h).symm", "annotated_tactic": ["exact ((<a>mk_eq_mk_iff</a> K v u hv hu).2 h).<a>symm</a>", [{"full_name": "Projectivization.mk_eq_mk_iff", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 21]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\nh : \u2203 z, z \u2022 u = v\n\u22a2 mk K u hu = mk K v hv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "Rel.mul_edgeDensity_le_edgeDensity", "start": [187, 1], "end": [193, 30], "traced_tactics": [{"tactic": "have hst : (s\u2082.card : \u211a) * t\u2082.card \u2260 0 := by simp [hs\u2082.ne_empty, ht\u2082.ne_empty]", "annotated_tactic": ["have hst : (s\u2082.card : \u211a) * t\u2082.card \u2260 0 := by simp [hs\u2082.ne_empty, ht\u2082.ne_empty]", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\n\u22a2 \u2191s\u2082.card / \u2191s\u2081.card * (\u2191t\u2082.card / \u2191t\u2081.card) * edgeDensity r s\u2082 t\u2082 \u2264 edgeDensity r s\u2081 t\u2081", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 \u2191s\u2082.card / \u2191s\u2081.card * (\u2191t\u2082.card / \u2191t\u2081.card) * edgeDensity r s\u2082 t\u2082 \u2264 edgeDensity r s\u2081 t\u2081"}, {"tactic": "rw [edgeDensity, edgeDensity, div_mul_div_comm, mul_comm, div_mul_div_cancel _ hst]", "annotated_tactic": ["rw [<a>edgeDensity</a>, <a>edgeDensity</a>, <a>div_mul_div_comm</a>, <a>mul_comm</a>, <a>div_mul_div_cancel</a> _ hst]", [{"full_name": "Rel.edgeDensity", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}, {"full_name": "Rel.edgeDensity", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}, {"full_name": "div_mul_div_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 25]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "div_mul_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [396, 7], "def_end_pos": [396, 25]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 \u2191s\u2082.card / \u2191s\u2081.card * (\u2191t\u2082.card / \u2191t\u2081.card) * edgeDensity r s\u2082 t\u2082 \u2264 edgeDensity r s\u2081 t\u2081", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 \u2191(interedges r s\u2082 t\u2082).card / (\u2191s\u2081.card * \u2191t\u2081.card) \u2264 \u2191(interedges r s\u2081 t\u2081).card / (\u2191s\u2081.card * \u2191t\u2081.card)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 \u2191(interedges r s\u2082 t\u2082).card / (\u2191s\u2081.card * \u2191t\u2081.card) \u2264 \u2191(interedges r s\u2081 t\u2081).card / (\u2191s\u2081.card * \u2191t\u2081.card)", "state_after": "case hab.h.a\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 interedges r s\u2082 t\u2082 \u2286 interedges r s\u2081 t\u2081"}, {"tactic": "exact interedges_mono hs ht", "annotated_tactic": ["exact <a>interedges_mono</a> hs ht", [{"full_name": "Rel.interedges_mono", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [70, 9], "def_end_pos": [70, 24]}]], "state_before": "case hab.h.a\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\nhst : \u2191s\u2082.card * \u2191t\u2082.card \u2260 0\n\u22a2 interedges r s\u2082 t\u2082 \u2286 interedges r s\u2081 t\u2081", "state_after": "no goals"}, {"tactic": "simp [hs\u2082.ne_empty, ht\u2082.ne_empty]", "annotated_tactic": ["simp [hs\u2082.ne_empty, ht\u2082.ne_empty]", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\nhs : s\u2082 \u2286 s\u2081\nht : t\u2082 \u2286 t\u2081\nhs\u2082 : s\u2082.Nonempty\nht\u2082 : t\u2082.Nonempty\n\u22a2 \u2191s\u2082.card * \u2191t\u2082.card \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "full_name": "left_eq_midpoint_iff", "start": [174, 1], "end": [175, 37], "traced_tactics": [{"tactic": "rw [eq_comm, midpoint_eq_left_iff]", "annotated_tactic": ["rw [<a>eq_comm</a>, <a>midpoint_eq_left_iff</a>]", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "midpoint_eq_left_iff", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "def_pos": [169, 9], "def_end_pos": [169, 29]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 x = midpoint R x y \u2194 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.not_injective_iff", "start": [109, 1], "end": [110, 49], "traced_tactics": [{"tactic": "simp only [Injective, not_forall, exists_prop]", "annotated_tactic": ["simp only [<a>Injective</a>, <a>not_forall</a>, <a>exists_prop</a>]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}, {"full_name": "Classical.not_forall", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [137, 21], "def_end_pos": [137, 31]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u00acInjective f \u2194 \u2203 a b, f a = f b \u2227 a \u2260 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monovary.lean", "full_name": "AntivaryOn.mul_right\u2080", "start": [211, 1], "end": [213, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup'_union", "start": [869, 1], "end": [872, 60], "traced_tactics": [{"tactic": "simp [or_imp, forall_and]", "annotated_tactic": ["simp [<a>or_imp</a>, <a>forall_and</a>]", [{"full_name": "or_imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 15]}, {"full_name": "forall_and", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [256, 9], "def_end_pos": [256, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ns : Finset \u03b2\nH : s.Nonempty\nf\u271d : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ns\u2081 s\u2082 : Finset \u03b2\nh\u2081 : s\u2081.Nonempty\nh\u2082 : s\u2082.Nonempty\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (s\u2081 \u222a s\u2082).sup' \u22ef f \u2264 a \u2194 s\u2081.sup' h\u2081 f \u2294 s\u2082.sup' h\u2082 f \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean", "full_name": "Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow", "start": [100, 1], "end": [110, 59], "traced_tactics": [{"tactic": "rcases m.eq_zero_or_pos with (rfl | hm)", "annotated_tactic": ["rcases m.eq_zero_or_pos with (rfl | hm)", []], "state_before": "p : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn m : \u2115\nhmn : m \u2264 n\nh : Irreducible (cyclotomic (p ^ n) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "case inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 \u2264 n\n\u22a2 Irreducible (cyclotomic (p ^ 0) R)\n\ncase inr\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn m : \u2115\nhmn : m \u2264 n\nh : Irreducible (cyclotomic (p ^ n) R)\nhm : m > 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)"}, {"tactic": "obtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 := <a>Nat.exists_eq_add_of_le</a> hmn", [{"full_name": "Nat.exists_eq_add_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [192, 19], "def_end_pos": [192, 38]}]], "state_before": "case inr\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn m : \u2115\nhmn : m \u2264 n\nh : Irreducible (cyclotomic (p ^ n) R)\nhm : m > 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "case inr.intro\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhmn : m \u2264 m + k\nh : Irreducible (cyclotomic (p ^ (m + k)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)"}, {"tactic": "induction' k with k hk", "annotated_tactic": ["induction' k with k hk", []], "state_before": "case inr.intro\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhmn : m \u2264 m + k\nh : Irreducible (cyclotomic (p ^ (m + k)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "case inr.intro.zero\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nhmn : m \u2264 m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n\ncase inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible (cyclotomic (p ^ (m + (k + 1))) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)"}, {"tactic": "have : m + k \u2260 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne'", "annotated_tactic": ["have : m + k \u2260 0 := (<a>add_pos_of_pos_of_nonneg</a> hm k.zero_le).<a>ne'</a>", [{"full_name": "add_pos_of_pos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1099, 24], "def_end_pos": [1099, 48]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible (cyclotomic (p ^ (m + (k + 1))) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "case inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible (cyclotomic (p ^ (m + (k + 1))) R)\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)"}, {"tactic": "rw [Nat.add_succ, pow_succ, \u2190 cyclotomic_expand_eq_cyclotomic hp <| dvd_pow_self p this] at h", "annotated_tactic": ["rw [<a>Nat.add_succ</a>, <a>pow_succ</a>, \u2190 <a>cyclotomic_expand_eq_cyclotomic</a> hp <| <a>dvd_pow_self</a> p this] at h", [{"full_name": "Nat.add_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [148, 9], "def_end_pos": [148, 17]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "Polynomial.cyclotomic_expand_eq_cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean", "def_pos": [78, 9], "def_end_pos": [78, 40]}, {"full_name": "dvd_pow_self", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [168, 7], "def_end_pos": [168, 19]}]], "state_before": "case inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible (cyclotomic (p ^ (m + (k + 1))) R)\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "case inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible ((expand R p) (cyclotomic (p ^ (m + k)) R))\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)"}, {"tactic": "exact hk (by omega) (of_irreducible_expand hp.ne_zero h)", "annotated_tactic": ["exact hk (by omega) (<a>of_irreducible_expand</a> hp.ne_zero h)", [{"full_name": "Polynomial.of_irreducible_expand", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [317, 9], "def_end_pos": [317, 30]}]], "state_before": "case inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible ((expand R p) (cyclotomic (p ^ (m + k)) R))\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "no goals"}, {"tactic": "simpa using irreducible_X_sub_C (1 : R)", "annotated_tactic": ["simpa using <a>irreducible_X_sub_C</a> (1 : R)", [{"full_name": "Polynomial.irreducible_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/RingDivision.lean", "def_pos": [694, 9], "def_end_pos": [694, 28]}]], "state_before": "case inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 \u2264 n\n\u22a2 Irreducible (cyclotomic (p ^ 0) R)", "state_after": "no goals"}, {"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "case inr.intro.zero\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nhmn : m \u2264 m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "p : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + (k + 1)\nh : Irreducible ((expand R p) (cyclotomic (p ^ (m + k)) R))\nthis : m + k \u2260 0\n\u22a2 m \u2264 m + k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sign.lean", "full_name": "SignType.nonneg_iff_ne_neg_one", "start": [165, 1], "end": [165, 87], "traced_tactics": [{"tactic": "cases a <;> decide", "annotated_tactic": ["cases a <;> decide", []], "state_before": "a : SignType\n\u22a2 0 \u2264 a \u2194 a \u2260 -1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Coxeter/Length.lean", "full_name": "CoxeterSystem.length_eq_one_iff", "start": [152, 1], "end": [159, 29], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "B : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n\u22a2 cs.length w = 1 \u2194 \u2203 i, w = cs.simple i", "state_after": "case mp\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n\u22a2 cs.length w = 1 \u2192 \u2203 i, w = cs.simple i\n\ncase mpr\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n\u22a2 (\u2203 i, w = cs.simple i) \u2192 cs.length w = 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n\u22a2 cs.length w = 1 \u2192 \u2203 i, w = cs.simple i", "state_after": "case mp\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\nh : cs.length w = 1\n\u22a2 \u2203 i, w = cs.simple i"}, {"tactic": "rcases cs.exists_reduced_word w with \u27e8\u03c9, h\u03c9, rfl\u27e9", "annotated_tactic": ["rcases cs.exists_reduced_word w with \u27e8\u03c9, h\u03c9, rfl\u27e9", []], "state_before": "case mp\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\nh : cs.length w = 1\n\u22a2 \u2203 i, w = cs.simple i", "state_after": "case mp.intro.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\n\u03c9 : List B\nh : cs.length (cs.wordProd \u03c9) = 1\nh\u03c9 : \u03c9.length = cs.length (cs.wordProd \u03c9)\n\u22a2 \u2203 i, cs.wordProd \u03c9 = cs.simple i"}, {"tactic": "rcases List.length_eq_one.mp (h\u03c9.trans h) with \u27e8i, rfl\u27e9", "annotated_tactic": ["rcases List.length_eq_one.mp (h\u03c9.trans h) with \u27e8i, rfl\u27e9", []], "state_before": "case mp.intro.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\n\u03c9 : List B\nh : cs.length (cs.wordProd \u03c9) = 1\nh\u03c9 : \u03c9.length = cs.length (cs.wordProd \u03c9)\n\u22a2 \u2203 i, cs.wordProd \u03c9 = cs.simple i", "state_after": "case mp.intro.intro.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nh : cs.length (cs.wordProd [i]) = 1\nh\u03c9 : [i].length = cs.length (cs.wordProd [i])\n\u22a2 \u2203 i_1, cs.wordProd [i] = cs.simple i_1"}, {"tactic": "exact \u27e8i, cs.wordProd_singleton i\u27e9", "annotated_tactic": ["exact \u27e8i, cs.wordProd_singleton i\u27e9", []], "state_before": "case mp.intro.intro.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nh : cs.length (cs.wordProd [i]) = 1\nh\u03c9 : [i].length = cs.length (cs.wordProd [i])\n\u22a2 \u2203 i_1, cs.wordProd [i] = cs.simple i_1", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, rfl\u27e9", "annotated_tactic": ["rintro \u27e8i, rfl\u27e9", []], "state_before": "case mpr\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n\u22a2 (\u2203 i, w = cs.simple i) \u2192 cs.length w = 1", "state_after": "case mpr.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n\u22a2 cs.length (cs.simple i) = 1"}, {"tactic": "exact cs.length_simple i", "annotated_tactic": ["exact cs.length_simple i", []], "state_before": "case mpr.intro\nB : Type u_1\nW : Type u_2\ninst\u271d : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n\u22a2 cs.length (cs.simple i) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.mem_lp_const_smul", "start": [619, 1], "end": [620, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Coloring.lean", "full_name": "SimpleGraph.Coloring.mem_colorClasses", "start": [107, 1], "end": [108, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean", "full_name": "Topology.WithLowerSet.toLowerSet_ofLowerSet", "start": [131, 1], "end": [131, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "Monovary.sum_mul_comp_perm_lt_sum_mul_iff", "start": [487, 1], "end": [489, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Basic.lean", "full_name": "LieModuleEquiv.symm_trans_self", "start": [1128, 1], "end": [1129, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/HahnSeries/Basic.lean", "full_name": "HahnSeries.single_coeff", "start": [188, 1], "end": [189, 32], "traced_tactics": [{"tactic": "split_ifs with h <;> simp [h]", "annotated_tactic": ["split_ifs with h <;> simp [h]", []], "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na b : \u0393\nr : R\n\u22a2 ((single a) r).coeff b = if b = a then r else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.measure_inter_lt_top_of_left_ne_top", "start": [282, 1], "end": [283, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Properties.lean", "full_name": "LinearMap.BilinForm.isAlt_zero", "start": [180, 1], "end": [180, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "full_name": "Filter.EventuallyEq.mfderivWithin_eq", "start": [613, 1], "end": [619, 51], "traced_tactics": [{"tactic": "by_cases h : MDifferentiableWithinAt I I' f s x", "annotated_tactic": ["by_cases h : <a>MDifferentiableWithinAt</a> I I' f s x", [{"full_name": "MDifferentiableWithinAt", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Defs.lean", "def_pos": [199, 5], "def_end_pos": [199, 28]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x", "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x\n\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x"}, {"tactic": "exact (h.hasMFDerivWithinAt.congr_of_eventuallyEq hL hx).mfderivWithin hs", "annotated_tactic": ["exact (h.hasMFDerivWithinAt.congr_of_eventuallyEq hL hx).<a>mfderivWithin</a> hs", [{"full_name": "HasMFDerivWithinAt.mfderivWithin", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "def_pos": [267, 9], "def_end_pos": [267, 41]}]], "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x", "state_after": "no goals"}, {"tactic": "unfold mfderivWithin", "annotated_tactic": ["unfold <a>mfderivWithin</a>", [{"full_name": "mfderivWithin", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Defs.lean", "def_pos": [319, 5], "def_end_pos": [319, 18]}]], "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x", "state_after": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 (if MDifferentiableWithinAt I I' f\u2081 s x then\n      fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f\u2081) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n    else 0) =\n    if MDifferentiableWithinAt I I' f s x then\n      fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n    else 0"}, {"tactic": "rw [if_neg h, if_neg]", "annotated_tactic": ["rw [<a>if_neg</a> h, <a>if_neg</a>]", [{"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}]], "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 (if MDifferentiableWithinAt I I' f\u2081 s x then\n      fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f\u2081) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n    else 0) =\n    if MDifferentiableWithinAt I I' f s x then\n      fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (\u2191(extChartAt I x).symm \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n    else 0", "state_after": "case neg.hnc\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 \u00acMDifferentiableWithinAt I I' f\u2081 s x"}, {"tactic": "rwa [\u2190 hL.mdifferentiableWithinAt_iff I I' hx]", "annotated_tactic": ["rwa [\u2190 hL.mdifferentiableWithinAt_iff I I' hx]", []], "state_before": "case neg.hnc\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set 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ContinuousMultilinearMap R M\u2081 M\u2082\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u2081 i), f.toMultilinearMap x = f'.toMultilinearMap x) \u2194 \u2200 (x : (i : \u03b9) \u2192 M\u2081 i), f x = f' x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_range", "start": [1345, 1], "end": [1346, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean", "full_name": "SimpleGraph.Subgraph.subgraphOfAdj_connected", "start": [73, 1], "end": [78, 46], "traced_tactics": [{"tactic": "refine \u27e8\u27e8?_\u27e9\u27e9", "annotated_tactic": ["refine \u27e8\u27e8?_\u27e9\u27e9", []], "state_before": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\n\u22a2 (G.subgraphOfAdj hvw).Connected", "state_after": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\n\u22a2 (G.subgraphOfAdj hvw).coe.Preconnected"}, {"tactic": "rintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "annotated_tactic": ["rintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", []], "state_before": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\n\u22a2 (G.subgraphOfAdj hvw).coe.Preconnected", "state_after": "case mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\na : V\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nb : V\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u27e9 \u27e8b, hb\u27e9"}, {"tactic": "simp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb", "annotated_tactic": ["simp only [<a>subgraphOfAdj_verts</a>, <a>Set.mem_insert_iff</a>, <a>Set.mem_singleton_iff</a>] at ha hb", [{"full_name": "SimpleGraph.subgraphOfAdj_verts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "def_pos": [84, 3], "def_end_pos": [84, 8]}, {"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1102, 9], "def_end_pos": [1102, 23]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\na : V\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nb : V\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "state_after": "case mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\na : V\nha\u271d : a \u2208 (G.subgraphOfAdj hvw).verts\nb : V\nhb\u271d : b \u2208 (G.subgraphOfAdj hvw).verts\nha : a = v \u2228 a = w\nhb : b = v \u2228 b = w\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u271d\u27e9 \u27e8b, hb\u271d\u27e9"}, {"tactic": "obtain rfl | rfl := ha <;> obtain rfl | rfl := hb <;>\n  first | rfl | (apply Adj.reachable; simp)", "annotated_tactic": ["obtain rfl | rfl := ha <;> obtain rfl | rfl := hb <;>\n    first | rfl | (apply <a>Adj.reachable</a>; simp)", [{"full_name": "SimpleGraph.Adj.reachable", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [2017, 19], "def_end_pos": [2017, 32]}]], "state_before": "case mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : G.Adj v w\na : V\nha\u271d : a \u2208 (G.subgraphOfAdj hvw).verts\nb : V\nhb\u271d : b \u2208 (G.subgraphOfAdj hvw).verts\nha : a = v \u2228 a = w\nhb : b = v \u2228 b = w\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u271d\u27e9 \u27e8b, hb\u271d\u27e9", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.inr.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv b : V\nhvw : G.Adj v b\nha hb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8b, ha\u27e9 \u27e8b, hb\u27e9", "state_after": "no goals"}, {"tactic": "(apply Adj.reachable; simp)", "annotated_tactic": ["(apply <a>Adj.reachable</a>; simp)", [{"full_name": "SimpleGraph.Adj.reachable", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [2017, 19], "def_end_pos": [2017, 32]}]], "state_before": "case mk.mk.inr.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : G.Adj b a\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "state_after": "no goals"}, {"tactic": "apply Adj.reachable", "annotated_tactic": ["apply <a>Adj.reachable</a>", [{"full_name": "SimpleGraph.Adj.reachable", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [2017, 19], "def_end_pos": [2017, 32]}]], "state_before": "case mk.mk.inr.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : G.Adj b a\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Reachable \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "state_after": "case mk.mk.inr.inl.h\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : G.Adj b a\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Adj \u27e8a, ha\u27e9 \u27e8b, hb\u27e9"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.mk.inr.inl.h\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : G.Adj b a\nha : a \u2208 (G.subgraphOfAdj hvw).verts\nhb : b \u2208 (G.subgraphOfAdj hvw).verts\n\u22a2 (G.subgraphOfAdj hvw).coe.Adj \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Distributions/Exponential.lean", "full_name": "ProbabilityTheory.stronglyMeasurable_exponentialPDFReal", "start": [68, 1], "end": [70, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean", "full_name": "CliffordAlgebra.\u03b9_sq_scalar", "start": [117, 1], "end": [119, 6], "traced_tactics": [{"tactic": "erw [\u2190 AlgHom.map_mul, RingQuot.mkAlgHom_rel R (Rel.of m), AlgHom.commutes]", "annotated_tactic": ["erw [\u2190 <a>AlgHom.map_mul</a>, <a>RingQuot.mkAlgHom_rel</a> R (<a>Rel.of</a> m), <a>AlgHom.commutes</a>]", [{"full_name": "AlgHom.map_mul", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [246, 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"annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nn : \u2115\nm : M\n\u22a2 (algebraMap R (RingQuot (Rel Q))) (Q m) = (algebraMap R (CliffordAlgebra Q)) (Q m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Tower.lean", "full_name": "AlgHom.coe_restrictScalars", "start": [216, 1], "end": [216, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Basic.lean", "full_name": "Real.mk_neg", "start": [323, 1], "end": [323, 84], "traced_tactics": [{"tactic": "simp [mk, \u2190 ofCauchy_neg]", "annotated_tactic": ["simp [<a>mk</a>, \u2190 <a>ofCauchy_neg</a>]", [{"full_name": "Real.mk", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [284, 5], "def_end_pos": [284, 7]}, {"full_name": "Real.ofCauchy_neg", "def_path": "Mathlib/Data/Real/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 21]}]], "state_before": "x y : \u211d\nf : CauSeq \u211a abs\n\u22a2 mk (-f) = -mk f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Pairwise.lean", "full_name": "Pairwise.eq", "start": [41, 11], "end": [42, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.coe_comp", "start": [704, 1], "end": [706, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.eval\u2082_mul_noncomm", "start": [176, 1], "end": [181, 63], "traced_tactics": [{"tactic": "rcases p with \u27e8p\u27e9", "annotated_tactic": ["rcases p with \u27e8p\u27e9", []], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nhf : \u2200 (k : \u2115), Commute (f (q.coeff k)) x\n\u22a2 eval\u2082 f x (p * q) = eval\u2082 f x p * eval\u2082 f x q", "state_after": "case ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nq r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nhf : \u2200 (k : \u2115), Commute (f (q.coeff k)) x\np : R[\u2115]\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * q) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x q"}, {"tactic": "rcases q with \u27e8q\u27e9", "annotated_tactic": ["rcases q with \u27e8q\u27e9", []], "state_before": "case ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nq r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nhf : \u2200 (k : \u2115), Commute (f (q.coeff k)) x\np : R[\u2115]\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * q) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x q", "state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f ({ toFinsupp := q }.coeff k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }"}, {"tactic": "simp only [coeff] at hf", "annotated_tactic": ["simp only [<a>coeff</a>] at hf", [{"full_name": "Polynomial.coeff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [663, 5], "def_end_pos": [663, 10]}]], "state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f ({ toFinsupp := q }.coeff k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }", "state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f (q k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }"}, {"tactic": "simp only [\u2190 ofFinsupp_mul, eval\u2082_ofFinsupp]", "annotated_tactic": ["simp only [\u2190 <a>ofFinsupp_mul</a>, <a>eval\u2082_ofFinsupp</a>]", [{"full_name": "Polynomial.ofFinsupp_mul", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 22]}, {"full_name": "Polynomial.eval\u2082_ofFinsupp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [170, 9], "def_end_pos": [170, 24]}]], "state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f (q k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }", "state_after": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f (q k)) x\n\u22a2 (liftNC \u2191f \u21d1((powersHom S) x)) (p * q) = (liftNC \u2191f \u21d1((powersHom S) x)) p * (liftNC \u2191f \u21d1((powersHom S) x)) q"}, {"tactic": "exact liftNC_mul _ _ p q fun {k n} _hn => (hf k).pow_right n", "annotated_tactic": ["exact <a>liftNC_mul</a> _ _ p q fun {k n} _hn => (hf k).<a>pow_right</a> n", [{"full_name": "AddMonoidAlgebra.liftNC_mul", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1361, 9], "def_end_pos": [1361, 19]}, {"full_name": "Commute.pow_right", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [176, 9], "def_end_pos": [176, 18]}]], "state_before": "case ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : R[\u2115]\nhf : \u2200 (k : \u2115), Commute (f (q k)) x\n\u22a2 (liftNC \u2191f \u21d1((powersHom S) x)) (p * q) = (liftNC \u2191f \u21d1((powersHom S) x)) p * (liftNC \u2191f \u21d1((powersHom S) x)) q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.get_succ_iterate", "start": [292, 1], "end": [293, 91], "traced_tactics": [{"tactic": "rw [get_succ, tail_iterate]", "annotated_tactic": ["rw [<a>get_succ</a>, <a>tail_iterate</a>]", [{"full_name": "Stream'.get_succ", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [79, 9], "def_end_pos": [79, 17]}, {"full_name": "Stream'.tail_iterate", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [274, 9], "def_end_pos": [274, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 (iterate f a).get n.succ = (iterate f (f a)).get n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Dioph.lean", "full_name": "Dioph.le_dioph", "start": [614, 1], "end": [616, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "full_name": "Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'", "start": [254, 1], "end": [257, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.continuous_setToFun_of_dominated", "start": [1802, 1], "end": [1809, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.swap_inv", "start": [521, 1], "end": [522, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartitionOfUnity.lean", "full_name": "BumpCovering.le_one", "start": [344, 1], "end": [345, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_ball", "start": [108, 1], "end": [109, 72], "traced_tactics": [{"tactic": "rw [ball_eq_Ioo, volume_Ioo, \u2190 sub_add, add_sub_cancel_left, two_mul]", "annotated_tactic": ["rw [<a>ball_eq_Ioo</a>, <a>volume_Ioo</a>, \u2190 <a>sub_add</a>, <a>add_sub_cancel_left</a>, <a>two_mul</a>]", [{"full_name": "Real.ball_eq_Ioo", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1364, 9], "def_end_pos": [1364, 25]}, {"full_name": "Real.volume_Ioo", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "sub_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [775, 3], "def_end_pos": [775, 14]}, {"full_name": "add_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1325, 3], "def_end_pos": [1325, 14]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [179, 9], "def_end_pos": [179, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na r : \u211d\n\u22a2 volume (Metric.ball a r) = ofReal (2 * r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean", "full_name": "ProbabilityTheory.IsCondKernelCDF.setLIntegral", "start": [446, 1], "end": [452, 27], "traced_tactics": [{"tactic": "rw [\u2190 ofReal_integral_eq_lintegral_ofReal (hf.integrable a x).restrict\n  (ae_of_all _ (fun _ \u21a6 hf.nonneg _ _)), hf.setIntegral a hs x, ENNReal.ofReal_toReal]", "annotated_tactic": ["rw [\u2190 <a>ofReal_integral_eq_lintegral_ofReal</a> (hf.integrable a x).<a>restrict</a>\n    (<a>ae_of_all</a> _ (fun _ \u21a6 hf.nonneg _ _)), hf.setIntegral a hs x, <a>ENNReal.ofReal_toReal</a>]", [{"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1204, 9], "def_end_pos": [1204, 44]}, {"full_name": "MeasureTheory.Integrable.restrict", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1585, 7], "def_end_pos": [1585, 26]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 (\u03b2 \u00d7 \u211d))\n\u03bd : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\ninst\u271d : IsFiniteKernel \u03ba\nf : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\nhf : IsCondKernelCDF f \u03ba \u03bd\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u211d\n\u22a2 \u222b\u207b (b : \u03b2) in s, ENNReal.ofReal (\u2191(f (a, b)) x) \u2202\u03bd a = (\u03ba a) (s \u00d7\u02e2 Iic x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 (\u03b2 \u00d7 \u211d))\n\u03bd : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\ninst\u271d : IsFiniteKernel \u03ba\nf : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\nhf : IsCondKernelCDF f \u03ba \u03bd\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u211d\n\u22a2 (\u03ba a) (s \u00d7\u02e2 Iic x) \u2260 \u22a4"}, {"tactic": "exact measure_ne_top _ _", "annotated_tactic": ["exact <a>measure_ne_top</a> _ _", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 (\u03b2 \u00d7 \u211d))\n\u03bd : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\ninst\u271d : IsFiniteKernel \u03ba\nf : \u03b1 \u00d7 \u03b2 \u2192 StieltjesFunction\nhf : IsCondKernelCDF f \u03ba \u03bd\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u211d\n\u22a2 (\u03ba a) (s \u00d7\u02e2 Iic x) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "full_name": "Affine.Simplex.reindex_reindex_symm", "start": [907, 1], "end": [908, 101], "traced_tactics": [{"tactic": "rw [\u2190 reindex_trans, Equiv.self_trans_symm, reindex_refl]", "annotated_tactic": ["rw [\u2190 <a>reindex_trans</a>, <a>Equiv.self_trans_symm</a>, <a>reindex_refl</a>]", [{"full_name": "Affine.Simplex.reindex_trans", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [899, 9], "def_end_pos": [899, 22]}, {"full_name": "Equiv.self_trans_symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [368, 17], "def_end_pos": [368, 32]}, {"full_name": "Affine.Simplex.reindex_refl", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [893, 9], "def_end_pos": [893, 21]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nm n : \u2115\ns : Simplex k P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 (s.reindex e).reindex e.symm = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean", "full_name": "AlgebraicGeometry.diagonal_targetAffineLocally_eq_targetAffineLocally", "start": [140, 1], "end": [144, 52], "traced_tactics": [{"tactic": "ext _ _ f", "annotated_tactic": ["ext _ _ f", []], "state_before": "P : AffineTargetMorphismProperty\nhP : P.IsLocal\n\u22a2 (targetAffineLocally P).diagonal = targetAffineLocally P.diagonal", "state_after": "case h\nP : AffineTargetMorphismProperty\nhP : P.IsLocal\nX\u271d Y\u271d : Scheme\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (targetAffineLocally P).diagonal f \u2194 targetAffineLocally P.diagonal f"}, {"tactic": "exact ((hP.diagonal_affine_openCover_TFAE f).out 0 1).trans\n  ((hP.diagonal.affine_openCover_TFAE f).out 1 0)", "annotated_tactic": ["exact ((hP.diagonal_affine_openCover_TFAE f).<a>out</a> 0 1).<a>trans</a>\n    ((hP.diagonal.affine_openCover_TFAE f).<a>out</a> 1 0)", [{"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "Iff.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [803, 9], "def_end_pos": [803, 18]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}]], "state_before": "case h\nP : AffineTargetMorphismProperty\nhP : P.IsLocal\nX\u271d Y\u271d : Scheme\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (targetAffineLocally P).diagonal f \u2194 targetAffineLocally P.diagonal f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.frequently_iff", "start": [369, 1], "end": [371, 66], "traced_tactics": [{"tactic": "simp only [Filter.Frequently, hl.eventually_iff]", "annotated_tactic": ["simp only [<a>Filter.Frequently</a>, hl.eventually_iff]", [{"full_name": "Filter.Frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1279, 15], "def_end_pos": [1279, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2203\u1da0 (x : \u03b1) in l, q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u00ac\u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 \u00acq x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u00ac\u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 \u00acq x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : l.HasBasis p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x \u2208 s i, q x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.injOn_of_subsingleton", "start": [697, 1], "end": [698, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Nullstellensatz.lean", "full_name": "MvPolynomial.vanishingIdeal_pointToPoint", "start": [149, 1], "end": [159, 74], "traced_tactics": [{"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "k : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 PrimeSpectrum.vanishingIdeal (pointToPoint '' V)\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 (vanishingIdeal {x}).IsPrime", "state_after": "no goals"}, {"tactic": "exact \u27e8x, \u27e8hx, rfl\u27e9\u27e9", "annotated_tactic": ["exact \u27e8x, \u27e8hx, <a>rfl</a>\u27e9\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "k : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 PrimeSpectrum.vanishingIdeal (pointToPoint '' V)\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 { asIdeal := vanishingIdeal {x}, isPrime := \u22ef } \u2208 pointToPoint '' V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Permutation.lean", "full_name": "List.permutationsAux_cons", "start": [223, 1], "end": [227, 49], "traced_tactics": [{"tactic": "rw [permutationsAux, permutationsAux.rec]", "annotated_tactic": ["rw [<a>permutationsAux</a>, <a>permutationsAux.rec</a>]", [{"full_name": "List.permutationsAux", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [232, 5], "def_end_pos": [232, 20]}, {"full_name": "List.permutationsAux.rec", "def_path": "Mathlib/Data/List/Defs.lean", "def_pos": [221, 5], "def_end_pos": [221, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts is : List \u03b1\n\u22a2 (t :: ts).permutationsAux is =\n    foldr (fun y r => (permutationsAux2 t ts r y id).2) (ts.permutationsAux (t :: is)) is.permutations", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts is : List \u03b1\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).2)\n      (permutationsAux.rec (fun x => [])\n        (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) ts (t :: is))\n      (is ::\n        permutationsAux.rec (fun x => [])\n          (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) is []) =\n    foldr (fun y r => (permutationsAux2 t ts r y id).2)\n      (permutationsAux.rec (fun x => [])\n        (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) ts (t :: is))\n      is.permutations"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts is : List \u03b1\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).2)\n      (permutationsAux.rec (fun x => [])\n        (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) ts (t :: is))\n      (is ::\n        permutationsAux.rec (fun x => [])\n          (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) is []) =\n    foldr (fun y r => (permutationsAux2 t ts r y id).2)\n      (permutationsAux.rec (fun x => [])\n        (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).2) IH1 (is :: IH2)) ts (t :: is))\n      is.permutations", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "full_name": "Nat.factorization_eq_zero_of_not_dvd", "start": [143, 1], "end": [144, 38], "traced_tactics": [{"tactic": "simp [factorization_eq_zero_iff, h]", "annotated_tactic": ["simp [<a>factorization_eq_zero_iff</a>, h]", [{"full_name": "Nat.factorization_eq_zero_iff", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 34]}]], "state_before": "a b m n\u271d p\u271d n p : \u2115\nh : \u00acp \u2223 n\n\u22a2 n.factorization p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean", "full_name": "differentiableWithinAt_pow", "start": [68, 1], "end": [70, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/CategoryTheory/Elementwise.lean", "full_name": "Tactic.Elementwise.forall_congr_forget_Type", "start": [45, 1], "end": [46, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "full_name": "Field.lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic", "start": [974, 1], "end": [979, 76], "traced_tactics": [{"tactic": "have h := lift_rank_mul_lift_sepDegree_of_isSeparable F (separableClosure F E) K", "annotated_tactic": ["have h := <a>lift_rank_mul_lift_sepDegree_of_isSeparable</a> F (<a>separableClosure</a> F E) K", [{"full_name": "Field.lift_rank_mul_lift_sepDegree_of_isSeparable", "def_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "def_pos": [947, 7], "def_end_pos": [947, 50]}, {"full_name": "separableClosure", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [78, 5], "def_end_pos": [78, 21]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : Algebra.IsAlgebraic F E\n\u22a2 Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)", "state_after": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : Algebra.IsAlgebraic F E\nh :\n  Cardinal.lift.{w, v} (Module.rank F \u21a5(separableClosure F E)) *\n      Cardinal.lift.{v, w} (sepDegree (\u21a5(separableClosure F E)) K) =\n    Cardinal.lift.{v, w} (sepDegree F K)\n\u22a2 Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)"}, {"tactic": "haveI := separableClosure.isPurelyInseparable F E", "annotated_tactic": ["haveI := <a>separableClosure.isPurelyInseparable</a> F E", [{"full_name": "separableClosure.isPurelyInseparable", "def_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "def_pos": [571, 9], "def_end_pos": [571, 45]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : Algebra.IsAlgebraic F E\nh :\n  Cardinal.lift.{w, v} (Module.rank F \u21a5(separableClosure F E)) *\n      Cardinal.lift.{v, w} (sepDegree (\u21a5(separableClosure F E)) K) =\n    Cardinal.lift.{v, w} (sepDegree F K)\n\u22a2 Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)", "state_after": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : Algebra.IsAlgebraic F E\nh :\n  Cardinal.lift.{w, v} (Module.rank F \u21a5(separableClosure F E)) *\n      Cardinal.lift.{v, w} (sepDegree (\u21a5(separableClosure F E)) K) =\n    Cardinal.lift.{v, w} (sepDegree F K)\nthis : IsPurelyInseparable (\u21a5(separableClosure F E)) E\n\u22a2 Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)"}, {"tactic": "rwa [sepDegree_eq_of_isPurelyInseparable (separableClosure F E) E K] at h", "annotated_tactic": ["rwa [<a>sepDegree_eq_of_isPurelyInseparable</a> (<a>separableClosure</a> F E) E K] at h", [{"full_name": "Field.sepDegree_eq_of_isPurelyInseparable", "def_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "def_pos": [963, 7], "def_end_pos": [963, 42]}, {"full_name": "separableClosure", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [78, 5], "def_end_pos": [78, 21]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : Algebra.IsAlgebraic F E\nh :\n  Cardinal.lift.{w, v} (Module.rank F \u21a5(separableClosure F E)) *\n      Cardinal.lift.{v, w} (sepDegree (\u21a5(separableClosure F E)) K) =\n    Cardinal.lift.{v, w} (sepDegree F K)\nthis : IsPurelyInseparable (\u21a5(separableClosure F E)) E\n\u22a2 Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "full_name": "HasLineDerivAt.le_of_lipschitzOn", "start": [400, 1], "end": [405, 79], "traced_tactics": [{"tactic": "refine hf.le_of_lip' C.coe_nonneg ?_", "annotated_tactic": ["refine hf.le_of_lip' C.coe_nonneg ?_", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 : E \u2192 F\nf'\u271d : F\ns\u271d t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nf : E \u2192 F\nf' : F\nx\u2080 : E\nhf : HasLineDerivAt \ud835\udd5c f f' x\u2080 v\ns : Set E\nhs : s \u2208 \ud835\udcdd x\u2080\nC : \u211d\u22650\nhlip : LipschitzOnWith C f s\n\u22a2 \u2016f'\u2016 \u2264 \u2191C * \u2016v\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 : E \u2192 F\nf'\u271d : F\ns\u271d t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nf : E \u2192 F\nf' : F\nx\u2080 : E\nhf : HasLineDerivAt \ud835\udd5c f f' x\u2080 v\ns : Set E\nhs : s \u2208 \ud835\udcdd x\u2080\nC : \u211d\u22650\nhlip : LipschitzOnWith C f s\n\u22a2 \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 \u2191C * \u2016x - x\u2080\u2016"}, {"tactic": 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"Ring.closure_mono", "start": [201, 1], "end": [202, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor/Prime.lean", "full_name": "FloorRing.exists_prime_mul_pow_div_factorial_lt_one", "start": [36, 1], "end": [40, 45], "traced_tactics": [{"tactic": "simp_rw [div_lt_one (\u03b1 := K) (Nat.cast_pos.mpr (Nat.factorial_pos _))]", "annotated_tactic": ["simp_rw [<a>div_lt_one</a> (\u03b1 := K) (Nat.cast_pos.mpr (<a>Nat.factorial_pos</a> _))]", [{"full_name": "div_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [372, 9], "def_end_pos": [372, 19]}, {"full_name": "Nat.factorial_pos", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [64, 9], "def_end_pos": [64, 22]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\n\u22a2 \u2203 p > n, Nat.Prime p 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"def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Fold.lean", "def_pos": [14, 3], "def_end_pos": [14, 7]}]], "state_before": "\u03b1 : Sort u_1\nf : \u03b1 \u2192 Fin 0 \u2192 \u03b1\nx : \u03b1\n\u22a2 foldl 0 f x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Algebraic.lean", "full_name": "AlgEquiv.isAlgebraic_iff", "start": [173, 1], "end": [175, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.sin_add_sin", "start": [602, 1], "end": [603, 33], "traced_tactics": [{"tactic": "simpa using sin_sub_sin x (-y)", "annotated_tactic": ["simpa using <a>sin_sub_sin</a> x (-y)", [{"full_name": "Complex.sin_sub_sin", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [584, 9], "def_end_pos": [584, 20]}]], "state_before": "x y : \u2102\n\u22a2 sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Order.lean", "full_name": "Complex.monotone_ofReal", "start": [121, 1], "end": [123, 47], "traced_tactics": [{"tactic": "intro x y hxy", "annotated_tactic": ["intro x y hxy", []], "state_before": "\u22a2 Monotone ofReal'", "state_after": "x y : \u211d\nhxy : x \u2264 y\n\u22a2 \u2191x \u2264 \u2191y"}, {"tactic": "simp only [ofReal_eq_coe, real_le_real, hxy]", "annotated_tactic": ["simp only [<a>ofReal_eq_coe</a>, <a>real_le_real</a>, hxy]", [{"full_name": "Complex.ofReal_eq_coe", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [766, 9], "def_end_pos": [766, 22]}, {"full_name": "Complex.real_le_real", "def_path": "Mathlib/Data/Complex/Order.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}]], "state_before": "x y : \u211d\nhxy : x \u2264 y\n\u22a2 \u2191x \u2264 \u2191y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "full_name": "LinearMap.orthogonal_span_singleton_eq_to_lin_ker", "start": [399, 1], "end": [407, 29], "traced_tactics": [{"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx : V\n\u22a2 (Submodule.span K {x}).orthogonalBilin B = ker (B x)", "state_after": "case h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 y \u2208 (Submodule.span K {x}).orthogonalBilin B \u2194 y \u2208 ker (B x)"}, {"tactic": "simp_rw [Submodule.mem_orthogonalBilin_iff, LinearMap.mem_ker, Submodule.mem_span_singleton]", "annotated_tactic": ["simp_rw [<a>Submodule.mem_orthogonalBilin_iff</a>, <a>LinearMap.mem_ker</a>, <a>Submodule.mem_span_singleton</a>]", [{"full_name": "Submodule.mem_orthogonalBilin_iff", "def_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "def_pos": [362, 9], "def_end_pos": [362, 32]}, {"full_name": "LinearMap.mem_ker", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [65, 9], "def_end_pos": [65, 16]}, {"full_name": "Submodule.mem_span_singleton", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [486, 9], "def_end_pos": [486, 27]}]], "state_before": "case h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 y \u2208 (Submodule.span K {x}).orthogonalBilin B \u2194 y \u2208 ker (B x)", "state_after": "case h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y) \u2194 (B x) y = 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y) \u2194 (B x) y = 0", "state_after": "case h.mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y) \u2192 (B x) y = 0\n\ncase h.mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (B x) y = 0 \u2192 \u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y"}, {"tactic": "exact fun h \u21a6 h x \u27e81, one_smul _ _\u27e9", "annotated_tactic": ["exact fun h \u21a6 h x \u27e81, <a>one_smul</a> _ _\u27e9", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case h.mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y) \u2192 (B x) y = 0", "state_after": "no goals"}, {"tactic": "rintro h _ \u27e8z, rfl\u27e9", "annotated_tactic": ["rintro h _ \u27e8z, rfl\u27e9", []], "state_before": "case h.mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\n\u22a2 (B x) y = 0 \u2192 \u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 B.IsOrtho n y", "state_after": "case h.mpr.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\nh : (B x) y = 0\nz : K\n\u22a2 B.IsOrtho (z \u2022 x) y"}, {"tactic": "rw [isOrtho_def, map_smul\u209b\u2097\u2082, smul_eq_zero]", "annotated_tactic": ["rw [<a>isOrtho_def</a>, <a>map_smul\u209b\u2097\u2082</a>, <a>smul_eq_zero</a>]", [{"full_name": "LinearMap.isOrtho_def", "def_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "LinearMap.map_smul\u209b\u2097\u2082", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [159, 9], "def_end_pos": [159, 20]}, {"full_name": "smul_eq_zero", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}]], "state_before": "case h.mpr.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\nh : (B x) y = 0\nz : K\n\u22a2 B.IsOrtho (z \u2022 x) y", "state_after": "case h.mpr.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\nh : (B x) y = 0\nz : K\n\u22a2 (RingHom.id K) z = 0 \u2228 (B x) y = 0"}, {"tactic": "exact Or.intro_right _ h", "annotated_tactic": ["exact <a>Or.intro_right</a> _ h", [{"full_name": "Or.intro_right", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [543, 9], "def_end_pos": [543, 23]}]], "state_before": "case h.mpr.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\nM\u2097\u2081 : Type u_9\nM\u2097\u2081' : Type u_10\nM\u2097\u2082 : Type u_11\nM\u2097\u2082' : Type u_12\nK : Type u_13\nK\u2081 : Type u_14\nK\u2082 : Type u_15\nV : Type u_16\nV\u2081 : Type u_17\nV\u2082 : Type u_18\nn : Type u_19\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\ninst\u271d\u2074 : Field K\u2081\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K\u2081 V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nJ : K \u2192+* K\nJ\u2081 J\u2081' : K\u2081 \u2192+* K\nB : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082\nx y : V\nh : (B x) y = 0\nz : K\n\u22a2 (RingHom.id K) z = 0 \u2228 (B x) y = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIocMod_periodic", "start": [757, 1], "end": [758, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.ge_of_eq", "start": [127, 1], "end": [127, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableAt.sum", "start": [380, 1], "end": [382, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPowerSeries/Inverse.lean", "full_name": "MvPowerSeries.inv_eq_zero", "start": [207, 1], "end": [213, 63], "traced_tactics": [{"tactic": "simpa using congr_arg (constantCoeff \u03c3 k) h", "annotated_tactic": ["simpa using <a>congr_arg</a> (<a>constantCoeff</a> \u03c3 k) h", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "MvPowerSeries.constantCoeff", "def_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "def_pos": [450, 5], "def_end_pos": [450, 18]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\nk : Type u_3\ninst\u271d : Field k\n\u03c6 : MvPowerSeries \u03c3 k\nh : \u03c6\u207b\u00b9 = 0\n\u22a2 (constantCoeff \u03c3 k) \u03c6 = 0", "state_after": "no goals"}, {"tactic": "classical\nrw [coeff_inv]\nsplit_ifs <;>\n  simp only [h, map_zero, zero_mul, inv_zero, neg_zero]", "annotated_tactic": ["classical\n      rw [<a>coeff_inv</a>]\n      split_ifs <;>\n        simp only [h, <a>map_zero</a>, <a>zero_mul</a>, <a>inv_zero</a>, <a>neg_zero</a>]", [{"full_name": "MvPowerSeries.coeff_inv", "def_path": "Mathlib/RingTheory/MvPowerSeries/Inverse.lean", "def_pos": [191, 9], "def_end_pos": [191, 18]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "GroupWithZero.inv_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [212, 3], "def_end_pos": [212, 11]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\nk : Type u_3\ninst\u271d : Field k\n\u03c6 : MvPowerSeries \u03c3 k\nh : (constantCoeff \u03c3 k) \u03c6 = 0\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 (coeff k n) \u03c6\u207b\u00b9 = (coeff k n) 0", "state_after": "no goals"}, {"tactic": "rw [coeff_inv]", "annotated_tactic": ["rw [<a>coeff_inv</a>]", [{"full_name": "MvPowerSeries.coeff_inv", "def_path": "Mathlib/RingTheory/MvPowerSeries/Inverse.lean", "def_pos": [191, 9], "def_end_pos": [191, 18]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\nk : Type u_3\ninst\u271d : Field k\n\u03c6 : MvPowerSeries \u03c3 k\nh : (constantCoeff \u03c3 k) \u03c6 = 0\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 (coeff k n) \u03c6\u207b\u00b9 = (coeff k n) 0", "state_after": "\u03c3 : Type u_1\nR : Type u_2\nk : Type u_3\ninst\u271d : Field k\n\u03c6 : MvPowerSeries \u03c3 k\nh : (constantCoeff \u03c3 k) \u03c6 = 0\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if n = 0 then ((constantCoeff \u03c3 k) \u03c6)\u207b\u00b9\n    else\n      -((constantCoeff \u03c3 k) \u03c6)\u207b\u00b9 * \u2211 x \u2208 antidiagonal n, if x.2 < n then (coeff k x.1) \u03c6 * (coeff k x.2) \u03c6\u207b\u00b9 else 0) =\n    (coeff k n) 0"}, {"tactic": "split_ifs <;>\n  simp only [h, map_zero, zero_mul, inv_zero, neg_zero]", "annotated_tactic": ["split_ifs <;>\n        simp only [h, <a>map_zero</a>, <a>zero_mul</a>, <a>inv_zero</a>, <a>neg_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "GroupWithZero.inv_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [212, 3], "def_end_pos": [212, 11]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\nk : Type u_3\ninst\u271d : Field k\n\u03c6 : MvPowerSeries \u03c3 k\nh : (constantCoeff \u03c3 k) \u03c6 = 0\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if n = 0 then ((constantCoeff \u03c3 k) \u03c6)\u207b\u00b9\n    else\n      -((constantCoeff \u03c3 k) \u03c6)\u207b\u00b9 * \u2211 x \u2208 antidiagonal n, if x.2 < n then (coeff k x.1) \u03c6 * (coeff k x.2) \u03c6\u207b\u00b9 else 0) =\n    (coeff k n) 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "nnnorm_one'", "start": [792, 1], "end": [793, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_C_mul_X_pow_le", "start": [311, 1], "end": [313, 27], "traced_tactics": [{"tactic": "rw [C_mul_X_pow_eq_monomial]", "annotated_tactic": ["rw [<a>C_mul_X_pow_eq_monomial</a>]", [{"full_name": "Polynomial.C_mul_X_pow_eq_monomial", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [770, 9], "def_end_pos": [770, 32]}]], "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 (C a * X ^ n).degree \u2264 \u2191n", "state_after": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 ((monomial n) a).degree \u2264 \u2191n"}, {"tactic": "apply degree_monomial_le", "annotated_tactic": ["apply <a>degree_monomial_le</a>", [{"full_name": "Polynomial.degree_monomial_le", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [305, 9], "def_end_pos": [305, 27]}]], "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 ((monomial n) a).degree \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.bliminf_eq_liminf", "start": [511, 1], "end": [513, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.le_one_toAddSubmonoid", "start": [87, 1], "end": [89, 44], "traced_tactics": [{"tactic": "rintro x \u27e8n, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8n, rfl\u27e9", []], "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 1 \u2264 toAddSubmonoid 1", "state_after": "case intro\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n\u271d : A\nn : \u2115\n\u22a2 (Nat.castAddMonoidHom A) n \u2208 toAddSubmonoid 1"}, {"tactic": "exact \u27e8n, map_natCast (algebraMap R A) n\u27e9", "annotated_tactic": ["exact \u27e8n, <a>map_natCast</a> (<a>algebraMap</a> R A) n\u27e9", [{"full_name": "map_natCast", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}]], "state_before": "case intro\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n\u271d : A\nn : \u2115\n\u22a2 (Nat.castAddMonoidHom A) n \u2208 toAddSubmonoid 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Div.lean", "full_name": "Polynomial.natDegree_modByMonic_le", "start": [233, 1], "end": [235, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.part_surjOn", "start": [512, 1], "end": [515, 88], "traced_tactics": [{"tactic": "obtain \u27e8x, hx\u27e9 := P.nonempty_of_mem_parts hp", "annotated_tactic": ["obtain \u27e8x, hx\u27e9 := P.nonempty_of_mem_parts hp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t u : Finset \u03b1\nP : Finpartition s\na : \u03b1\np : Finset \u03b1\nhp : p \u2208 \u2191P.parts\n\u22a2 p \u2208 P.part '' \u2191s", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t u : Finset \u03b1\nP : Finpartition s\na : \u03b1\np : Finset \u03b1\nhp : p \u2208 \u2191P.parts\nx : \u03b1\nhx : x \u2208 p\n\u22a2 p \u2208 P.part '' \u2191s"}, {"tactic": "have hx' := mem_of_subset (P.le hp) hx", "annotated_tactic": ["have hx' := <a>mem_of_subset</a> (P.le hp) hx", [{"full_name": "Finset.mem_of_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t u : Finset \u03b1\nP : Finpartition s\na : \u03b1\np : Finset \u03b1\nhp : p \u2208 \u2191P.parts\nx : \u03b1\nhx : x \u2208 p\n\u22a2 p \u2208 P.part '' \u2191s", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t u : Finset \u03b1\nP : Finpartition s\na : \u03b1\np : Finset \u03b1\nhp : p \u2208 \u2191P.parts\nx : \u03b1\nhx : x \u2208 p\nhx' : x \u2208 s\n\u22a2 p \u2208 P.part '' \u2191s"}, {"tactic": "use x, hx', (P.existsUnique_mem hx').unique \u27e8P.part_mem hx', P.mem_part hx'\u27e9 \u27e8hp, hx\u27e9", "annotated_tactic": ["use x, hx', (P.existsUnique_mem hx').<a>unique</a> \u27e8P.part_mem hx', P.mem_part hx'\u27e9 \u27e8hp, hx\u27e9", [{"full_name": "ExistsUnique.unique", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [263, 9], "def_end_pos": [263, 28]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t u : Finset \u03b1\nP : Finpartition s\na : \u03b1\np : Finset \u03b1\nhp : p \u2208 \u2191P.parts\nx : \u03b1\nhx : x \u2208 p\nhx' : x \u2208 s\n\u22a2 p \u2208 P.part '' \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.Nonempty.one_mem_div", "start": [1248, 1], "end": [1250, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trNat_default", "start": [1186, 1], "end": [1187, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/FullSubcategory.lean", "full_name": "CategoryTheory.FullSubcategory.id_def", "start": [124, 1], "end": [124, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "full_name": "UniformFun.uniformSpace_eq_inf_precomp_of_cover", "start": [547, 1], "end": [563, 20], "traced_tactics": [{"tactic": "ext : 1", "annotated_tactic": ["ext : 1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 uniformSpace \u03b1 \u03b2 =\n    UniformSpace.comap (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) (uniformSpace \u03b4\u2081 \u03b2) \u2293\n      UniformSpace.comap (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) (uniformSpace \u03b4\u2082 \u03b2)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) = \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)"}, {"tactic": "refine le_antisymm (le_inf ?_ ?_) ?_", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>le_inf</a> ?_ ?_) ?_", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) = \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)", "state_after": "case h.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)\n\ncase h.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)\n\ncase h.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)"}, {"tactic": "exact tendsto_iff_comap.mp UniformFun.precomp_uniformContinuous", "annotated_tactic": ["exact tendsto_iff_comap.mp <a>UniformFun.precomp_uniformContinuous</a>", [{"full_name": "UniformFun.precomp_uniformContinuous", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [448, 19], "def_end_pos": [448, 44]}]], "state_before": "case h.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)", "state_after": "no goals"}, {"tactic": "exact tendsto_iff_comap.mp UniformFun.precomp_uniformContinuous", "annotated_tactic": ["exact tendsto_iff_comap.mp <a>UniformFun.precomp_uniformContinuous</a>", [{"full_name": "UniformFun.precomp_uniformContinuous", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [448, 19], "def_end_pos": [448, 44]}]], "state_before": "case h.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)", "state_after": "no goals"}, {"tactic": "refine\n  (UniformFun.hasBasis_uniformity \u03b4\u2081 \u03b2 |>.comap _).inf\n  (UniformFun.hasBasis_uniformity \u03b4\u2082 \u03b2 |>.comap _)\n    |>.le_basis_iff (UniformFun.hasBasis_uniformity \u03b1 \u03b2) |>.mpr fun U hU \u21a6\n    \u27e8\u27e8U, U\u27e9, \u27e8hU, hU\u27e9, fun \u27e8f, g\u27e9 hfg x \u21a6 ?_\u27e9", "annotated_tactic": ["refine\n      (<a>UniformFun.hasBasis_uniformity</a> \u03b4\u2081 \u03b2 |>.<a>comap</a> _).<a>inf</a>\n      (<a>UniformFun.hasBasis_uniformity</a> \u03b4\u2082 \u03b2 |>.<a>comap</a> _)\n        |>.<a>le_basis_iff</a> (<a>UniformFun.hasBasis_uniformity</a> \u03b1 \u03b2) |>.<a>mpr</a> fun U hU \u21a6\n        \u27e8\u27e8U, U\u27e9, \u27e8hU, hU\u27e9, fun \u27e8f, g\u27e9 hfg x \u21a6 ?_\u27e9", [{"full_name": "UniformFun.hasBasis_uniformity", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [310, 19], "def_end_pos": [310, 38]}, {"full_name": "Filter.HasBasis.comap", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [802, 9], "def_end_pos": [802, 23]}, {"full_name": "Filter.HasBasis.inf", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [499, 9], "def_end_pos": [499, 21]}, {"full_name": "UniformFun.hasBasis_uniformity", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [310, 19], "def_end_pos": [310, 38]}, {"full_name": "Filter.HasBasis.comap", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [802, 9], "def_end_pos": [802, 23]}, {"full_name": "Filter.HasBasis.le_basis_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [471, 9], "def_end_pos": [471, 30]}, {"full_name": "UniformFun.hasBasis_uniformity", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [310, 19], "def_end_pos": [310, 38]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case h.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\n\u22a2 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2) \u2264 \ud835\udce4 (\u03b1 \u2192\u1d64 \u03b2)", "state_after": "case h.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx\u271d\u00b9 : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nx : \u03b1\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\n\u22a2 (toFun (f, g).1 x, toFun (f, g).2 x) \u2208 U"}, {"tactic": "rcases h_cover.ge <| mem_univ x with (\u27e8y, rfl\u27e9|\u27e8y, rfl\u27e9)", "annotated_tactic": ["rcases h_cover.ge <| <a>mem_univ</a> x with (\u27e8y, rfl\u27e9|\u27e8y, rfl\u27e9)", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "case h.refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx\u271d\u00b9 : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nx : \u03b1\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\n\u22a2 (toFun (f, g).1 x, toFun (f, g).2 x) \u2208 U", "state_after": "case h.refine_3.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\ny : \u03b4\u2081\n\u22a2 (toFun (f, g).1 (\u03c6\u2081 y), toFun (f, g).2 (\u03c6\u2081 y)) \u2208 U\n\ncase h.refine_3.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\ny : \u03b4\u2082\n\u22a2 (toFun (f, g).1 (\u03c6\u2082 y), toFun (f, g).2 (\u03c6\u2082 y)) \u2208 U"}, {"tactic": "exact hfg.1 y", "annotated_tactic": ["exact hfg.1 y", []], "state_before": "case h.refine_3.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\ny : \u03b4\u2081\n\u22a2 (toFun (f, g).1 (\u03c6\u2081 y), toFun (f, g).2 (\u03c6\u2081 y)) \u2208 U", "state_after": "no goals"}, {"tactic": "exact hfg.2 y", "annotated_tactic": ["exact hfg.2 y", []], "state_before": "case h.refine_3.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b4 : \u03b9 \u2192 Type u_5\ninst\u271d : (i : \u03b9) \u2192 UniformSpace (\u03b4 i)\n\u03b4\u2081 : Type u_6\n\u03b4\u2082 : Type u_7\n\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1\n\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1\nh_cover : range \u03c6\u2081 \u222a range \u03c6\u2082 = univ\nU : Set (\u03b2 \u00d7 \u03b2)\nhU : U \u2208 \ud835\udce4 \u03b2\nx\u271d : (\u03b1 \u2192\u1d64 \u03b2) \u00d7 (\u03b1 \u2192\u1d64 \u03b2)\nf g : \u03b1 \u2192\u1d64 \u03b2\nhfg :\n  (f, g) \u2208\n    (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2081) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2081 \u03b2 (U, U).1 \u2229\n      (fun p => ((\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.1, (\u21d1ofFun \u2218 (fun x => x \u2218 \u03c6\u2082) \u2218 \u21d1toFun) p.2)) \u207b\u00b9'\n        UniformFun.gen \u03b4\u2082 \u03b2 (U, U).2\ny : \u03b4\u2082\n\u22a2 (toFun (f, g).1 (\u03c6\u2082 y), toFun (f, g).2 (\u03c6\u2082 y)) \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Category/Cat.lean", "full_name": "CategoryTheory.Cat.comp_map", "start": [104, 1], "end": [106, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelSeries.lean", "full_name": "RelSeries.map_apply", "start": [304, 1], "end": [305, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "ContinuousOn.inner", "start": [2289, 1], "end": [2290, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.pow_dvd_card_of_pow_dvd_card", "start": [691, 1], "end": [695, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "HasDerivAt.continuousOn", "start": [763, 11], "end": [764, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsExtrFilter.filter_inf", "start": [277, 1], "end": [278, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero", "start": [225, 1], "end": [228, 62], "traced_tactics": [{"tactic": "rw [\u2190 neg_eq_zero, \u2190 inner_neg_right] at h", "annotated_tactic": ["rw [\u2190 <a>neg_eq_zero</a>, \u2190 <a>inner_neg_right</a>] at h", [{"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}, {"full_name": "inner_neg_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 24]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 angle x (x - y) = Real.arccos (\u2016x\u2016 / \u2016x - y\u2016)", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, -y\u27eb_\u211d = 0\n\u22a2 angle x (x - y) = Real.arccos (\u2016x\u2016 / \u2016x - y\u2016)"}, {"tactic": "rw [sub_eq_add_neg, angle_add_eq_arccos_of_inner_eq_zero h]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>angle_add_eq_arccos_of_inner_eq_zero</a> h]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "def_pos": [69, 9], "def_end_pos": [69, 45]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, -y\u27eb_\u211d = 0\n\u22a2 angle x (x - y) = Real.arccos (\u2016x\u2016 / \u2016x - y\u2016)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/ZPow.lean", "full_name": "differentiableWithinAt_zpow", "start": [76, 1], "end": [78, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "full_name": "MonoidHom.eq_of_eqOn_topM", "start": [628, 1], "end": [629, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.Subperm.cons_right", "start": [236, 1], "end": [237, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Regular/Basic.lean", "full_name": "IsLeftRegular.all", "start": [363, 1], "end": [364, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "full_name": "CategoryTheory.Abelian.exact_iff_exact_coimage_\u03c0", "start": [243, 1], "end": [245, 27], "traced_tactics": [{"tactic": "conv_lhs => rw [\u2190 Abelian.coimage.fac g]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Abelian.coimage.fac</a> g]", [{"full_name": "CategoryTheory.Abelian.coimage.fac", "def_path": "Mathlib/CategoryTheory/Abelian/Images.lean", "def_pos": [87, 19], "def_end_pos": [87, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f g \u2194 Exact f (coimage.\u03c0 g)", "state_after": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f (coimage.\u03c0 g \u226b Abelian.factorThruCoimage g) \u2194 Exact f (coimage.\u03c0 g)"}, {"tactic": "rw [exact_comp_mono_iff]", "annotated_tactic": ["rw [<a>exact_comp_mono_iff</a>]", [{"full_name": "CategoryTheory.exact_comp_mono_iff", "def_path": "Mathlib/Algebra/Homology/Exact.lean", "def_pos": [211, 9], "def_end_pos": [211, 28]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f (coimage.\u03c0 g \u226b Abelian.factorThruCoimage g) \u2194 Exact f (coimage.\u03c0 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Map.lean", "full_name": "Matroid.comap_base_iff", "start": [216, 1], "end": [218, 48], "traced_tactics": [{"tactic": "rw [\u2190 basis_ground_iff, comap_basis_iff]", "annotated_tactic": ["rw [\u2190 <a>basis_ground_iff</a>, <a>comap_basis_iff</a>]", [{"full_name": "Matroid.basis_ground_iff", "def_path": "Mathlib/Data/Matroid/Basic.lean", "def_pos": [874, 17], "def_end_pos": [874, 33]}, {"full_name": "Matroid.comap_basis_iff", "def_path": "Mathlib/Data/Matroid/Map.lean", "def_pos": [194, 15], "def_end_pos": [194, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nB : Set \u03b1\n\u22a2 (N.comap f).Base B \u2194 N.Basis (f '' B) (f '' (f \u207b\u00b9' N.E)) \u2227 InjOn f B \u2227 B \u2286 f \u207b\u00b9' N.E", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nB : Set \u03b1\n\u22a2 N.Basis (f '' B) (f '' (N.comap f).E) \u2227 InjOn f B \u2227 B \u2286 (N.comap f).E \u2194\n    N.Basis (f '' B) (f '' (f \u207b\u00b9' N.E)) \u2227 InjOn f B \u2227 B \u2286 f \u207b\u00b9' N.E"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nB : Set \u03b1\n\u22a2 N.Basis (f '' B) (f '' (N.comap f).E) \u2227 InjOn f B \u2227 B \u2286 (N.comap f).E \u2194\n    N.Basis (f '' B) (f '' (f \u207b\u00b9' N.E)) \u2227 InjOn f B \u2227 B \u2286 f \u207b\u00b9' N.E", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_finset_subset_smul_finset", "start": [1771, 1], "end": [1772, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/FixedPoints.lean", "full_name": "MulAction.set_mem_fixedBy_of_subset_fixedBy", "start": [141, 1], "end": [148, 49], "traced_tactics": [{"tactic": "rw [\u2190 fixedBy_inv]", "annotated_tactic": ["rw [\u2190 <a>fixedBy_inv</a>]", [{"full_name": "MulAction.fixedBy_inv", "def_path": "Mathlib/GroupTheory/GroupAction/FixedPoints.lean", "def_pos": [60, 9], "def_end_pos": [60, 20]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\n\u22a2 s \u2208 fixedBy (Set \u03b1) g", "state_after": "\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\n\u22a2 s \u2208 fixedBy (Set \u03b1) g\u207b\u00b9"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\n\u22a2 s \u2208 fixedBy (Set \u03b1) g\u207b\u00b9", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\n\u22a2 x \u2208 g\u207b\u00b9 \u2022 s \u2194 x \u2208 s"}, {"tactic": "rw [Set.mem_inv_smul_set_iff]", "annotated_tactic": ["rw [<a>Set.mem_inv_smul_set_iff</a>]", [{"full_name": "Set.mem_inv_smul_set_iff", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [906, 9], "def_end_pos": [906, 29]}]], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\n\u22a2 x \u2208 g\u207b\u00b9 \u2022 s \u2194 x \u2208 s", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\n\u22a2 g \u2022 x \u2208 s \u2194 x \u2208 s"}, {"tactic": "refine \u27e8fun gxs => ?xs, fun xs => (s_ss_fixedBy xs).symm \u25b8 xs\u27e9", "annotated_tactic": ["refine \u27e8fun gxs => ?xs, fun xs => (s_ss_fixedBy xs).<a>symm</a> \u25b8 xs\u27e9", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\n\u22a2 g \u2022 x \u2208 s \u2194 x \u2208 s", "state_after": "case xs\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\ngxs : g \u2022 x \u2208 s\n\u22a2 x \u2208 s"}, {"tactic": "rw [\u2190 fixedBy_inv] at s_ss_fixedBy", "annotated_tactic": ["rw [\u2190 <a>fixedBy_inv</a>] at s_ss_fixedBy", [{"full_name": "MulAction.fixedBy_inv", "def_path": "Mathlib/GroupTheory/GroupAction/FixedPoints.lean", "def_pos": [60, 9], "def_end_pos": [60, 20]}]], "state_before": "case xs\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\nx : \u03b1\ngxs : g \u2022 x \u2208 s\n\u22a2 x \u2208 s", "state_after": "case xs\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\u207b\u00b9\nx : \u03b1\ngxs : g \u2022 x \u2208 s\n\u22a2 x \u2208 s"}, {"tactic": "rwa [\u2190 s_ss_fixedBy gxs, inv_smul_smul] at gxs", "annotated_tactic": ["rwa [\u2190 s_ss_fixedBy gxs, <a>inv_smul_smul</a>] at gxs", [{"full_name": "inv_smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [30, 9], "def_end_pos": [30, 22]}]], "state_before": "case xs\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : Set \u03b1\ng : G\ns_ss_fixedBy : s \u2286 fixedBy \u03b1 g\u207b\u00b9\nx : \u03b1\ngxs : g \u2022 x \u2208 s\n\u22a2 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Block.lean", "full_name": "Matrix.BlockTriangular.det_fintype", "start": [253, 1], "end": [257, 20], "traced_tactics": [{"tactic": "refine h.det.trans (prod_subset (subset_univ _) fun a _ ha => ?_)", "annotated_tactic": ["refine h.det.trans (<a>prod_subset</a> (<a>subset_univ</a> _) fun a _ ha => ?_)", [{"full_name": "Finset.prod_subset", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [595, 9], "def_end_pos": [595, 20]}, {"full_name": "Finset.subset_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : \u03b1 \u2192 Type u_6\nn' : \u03b1 \u2192 Type u_7\nR : Type v\ninst\u271d\u2077 : CommRing R\nM N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrder \u03b1\nh : M.BlockTriangular b\n\u22a2 M.det = \u220f k : \u03b1, (M.toSquareBlock b k).det", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : \u03b1 \u2192 Type u_6\nn' : \u03b1 \u2192 Type u_7\nR : Type v\ninst\u271d\u2077 : CommRing R\nM N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrder \u03b1\nh : M.BlockTriangular b\na : \u03b1\nx\u271d : a \u2208 univ\nha : a \u2209 image b univ\n\u22a2 (M.toSquareBlock b a).det = 1"}, {"tactic": "have : IsEmpty { i // b i = a } := \u27e8fun i => ha <| mem_image.2 \u27e8i, mem_univ _, i.2\u27e9\u27e9", "annotated_tactic": ["have : <a>IsEmpty</a> { i // b i = a } := \u27e8fun i => ha <| <a>mem_image</a>.2 \u27e8i, <a>mem_univ</a> _, i.2\u27e9\u27e9", [{"full_name": "IsEmpty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [23, 7], "def_end_pos": [23, 14]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : \u03b1 \u2192 Type u_6\nn' : \u03b1 \u2192 Type u_7\nR : Type v\ninst\u271d\u2077 : CommRing R\nM N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrder \u03b1\nh : M.BlockTriangular b\na : \u03b1\nx\u271d : a \u2208 univ\nha : a \u2209 image b univ\n\u22a2 (M.toSquareBlock b a).det = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : \u03b1 \u2192 Type u_6\nn' : \u03b1 \u2192 Type u_7\nR : Type v\ninst\u271d\u2077 : CommRing R\nM N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrder \u03b1\nh : M.BlockTriangular b\na : \u03b1\nx\u271d : a \u2208 univ\nha : a \u2209 image b univ\nthis : IsEmpty { i // b i = a }\n\u22a2 (M.toSquareBlock b a).det = 1"}, {"tactic": "exact det_isEmpty", "annotated_tactic": ["exact <a>det_isEmpty</a>", [{"full_name": "Matrix.det_isEmpty", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\no : Type u_5\nm' : \u03b1 \u2192 Type u_6\nn' : \u03b1 \u2192 Type u_7\nR : Type v\ninst\u271d\u2077 : CommRing R\nM N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrder \u03b1\nh : M.BlockTriangular b\na : \u03b1\nx\u271d : a \u2208 univ\nha : a \u2209 image b univ\nthis : IsEmpty { i // b i = a }\n\u22a2 (M.toSquareBlock b a).det = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectSum/Basic.lean", "full_name": "DirectSum.equivCongrLeft_apply", "start": [301, 1], "end": [303, 54], "traced_tactics": [{"tactic": "exact DFinsupp.comapDomain'_apply _ h.right_inv _ _", "annotated_tactic": ["exact <a>DFinsupp.comapDomain'_apply</a> _ h.right_inv _ _", [{"full_name": "DFinsupp.comapDomain'_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1376, 9], "def_end_pos": [1376, 27]}]], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03ba : Type u_1\nh : \u03b9 \u2243 \u03ba\nf : \u2a01 (i : \u03b9), \u03b2 i\nk : \u03ba\n\u22a2 ((equivCongrLeft h) f) k = f (h.symm k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Syntax.lean", "full_name": "FirstOrder.Language.BoundedFormula.IsAtomic.castLE", "start": [701, 1], "end": [702, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.map\u2082_cons", "start": [139, 1], "end": [141, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.prod_sum_index", "start": [2106, 1], "end": [2112, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "IsPrimitiveRoot.orderOf", "start": [453, 11], "end": [454, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "full_name": "GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some", "start": [104, 1], "end": [212, 34], "traced_tactics": [{"tactic": "let g := of v", "annotated_tactic": ["let g := <a>of</a> v", [{"full_name": "GeneralizedContinuedFraction.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [195, 15], "def_end_pos": [195, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr"}, {"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr", "state_after": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v 0 = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_n.fr\n\ncase succ\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v (n + 1) = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_n.fr"}, {"tactic": "intro ifp_zero stream_zero_eq", "annotated_tactic": ["intro ifp_zero stream_zero_eq", []], "state_before": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v 0 = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_n.fr", "state_after": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_zero.fr"}, {"tactic": "have : IntFractPair.of v = ifp_zero := by\n  have : IntFractPair.stream v 0 = some (IntFractPair.of v) := rfl\n  simpa only [Nat.zero_eq, this, Option.some.injEq] using stream_zero_eq", "annotated_tactic": ["have : <a>IntFractPair.of</a> v = ifp_zero := by\n      have : <a>IntFractPair.stream</a> v 0 = <a>some</a> (<a>IntFractPair.of</a> v) := <a>rfl</a>\n      simpa only [<a>Nat.zero_eq</a>, this, Option.some.injEq] using stream_zero_eq", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.stream", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [149, 15], "def_end_pos": [149, 21]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}]], "state_before": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_zero.fr", "state_after": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\nthis : IntFractPair.of v = ifp_zero\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_zero.fr"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case zero\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\nthis : IntFractPair.of v = ifp_zero\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) ifp_zero.fr", "state_after": "case zero.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr"}, {"tactic": "cases' Decidable.em (Int.fract v = 0) with fract_eq_zero fract_ne_zero", "annotated_tactic": ["cases' <a>Decidable.em</a> (<a>Int.fract</a> v = 0) with fract_eq_zero fract_ne_zero", [{"full_name": "Decidable.em", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [876, 9], "def_end_pos": [876, 11]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}]], "state_before": "case zero.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr", "state_after": "case zero.refl.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr\n\ncase zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr"}, {"tactic": "have : IntFractPair.stream v 0 = some (IntFractPair.of v) := rfl", "annotated_tactic": ["have : <a>IntFractPair.stream</a> v 0 = <a>some</a> (<a>IntFractPair.of</a> v) := <a>rfl</a>", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.stream", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [149, 15], "def_end_pos": [149, 21]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\n\u22a2 IntFractPair.of v = ifp_zero", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\nthis : IntFractPair.stream v 0 = some (IntFractPair.of v)\n\u22a2 IntFractPair.of v = ifp_zero"}, {"tactic": "simpa only [Nat.zero_eq, this, Option.some.injEq] using stream_zero_eq", "annotated_tactic": ["simpa only [<a>Nat.zero_eq</a>, this, Option.some.injEq] using stream_zero_eq", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nifp_zero : IntFractPair K\nstream_zero_eq : IntFractPair.stream v 0 = some ifp_zero\nthis : IntFractPair.stream v 0 = some (IntFractPair.of v)\n\u22a2 IntFractPair.of v = ifp_zero", "state_after": "no goals"}, {"tactic": "suffices v = \u230av\u230b by\n  field_simp [nextContinuants, nextNumerator, nextDenominator, compExactValue]\n  have : (IntFractPair.of v).fr = Int.fract v := rfl\n  rwa [this, if_pos fract_eq_zero]", "annotated_tactic": ["suffices v = \u230av\u230b by\n        -- Porting note: was `simpa [continuantsAux, fract_eq_zero, compExactValue]`\n        field_simp [<a>nextContinuants</a>, <a>nextNumerator</a>, <a>nextDenominator</a>, <a>compExactValue</a>]\n        have : (<a>IntFractPair.of</a> v).<a>fr</a> = <a>Int.fract</a> v := <a>rfl</a>\n        rwa [this, <a>if_pos</a> fract_eq_zero]", [{"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.fr", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 5]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}]], "state_before": "case zero.refl.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr", "state_after": "case zero.refl.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 v = \u2191\u230av\u230b"}, {"tactic": "calc\n  v = Int.fract v + \u230av\u230b := by rw [Int.fract_add_floor]\n  _ = \u230av\u230b := by simp [fract_eq_zero]", "annotated_tactic": ["calc\n        v = <a>Int.fract</a> v + \u230av\u230b := by rw [<a>Int.fract_add_floor</a>]\n        _ = \u230av\u230b := by simp [fract_eq_zero]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}, {"full_name": "Int.fract_add_floor", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [888, 9], "def_end_pos": [888, 24]}]], "state_before": "case zero.refl.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 v = \u2191\u230av\u230b", "state_after": "no goals"}, {"tactic": "field_simp [nextContinuants, nextNumerator, nextDenominator, compExactValue]", "annotated_tactic": ["field_simp [<a>nextContinuants</a>, <a>nextNumerator</a>, <a>nextDenominator</a>, <a>compExactValue</a>]", [{"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\nthis : v = \u2191\u230av\u230b\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\nthis : v = \u2191\u230av\u230b\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr"}, {"tactic": "have : (IntFractPair.of v).fr = Int.fract v := rfl", "annotated_tactic": ["have : (<a>IntFractPair.of</a> v).<a>fr</a> = <a>Int.fract</a> v := <a>rfl</a>", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.fr", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 5]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\nthis : v = \u2191\u230av\u230b\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\nthis\u271d : v = \u2191\u230av\u230b\nthis : (IntFractPair.of v).fr = Int.fract v\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr"}, {"tactic": "rwa [this, if_pos fract_eq_zero]", "annotated_tactic": ["rwa [this, <a>if_pos</a> fract_eq_zero]", [{"full_name": "if_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [932, 9], "def_end_pos": [932, 15]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\nthis\u271d : v = \u2191\u230av\u230b\nthis : (IntFractPair.of v).fr = Int.fract v\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr", "state_after": "no goals"}, {"tactic": "rw [Int.fract_add_floor]", "annotated_tactic": ["rw [<a>Int.fract_add_floor</a>]", [{"full_name": "Int.fract_add_floor", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [888, 9], "def_end_pos": [888, 24]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 v = Int.fract v + \u2191\u230av\u230b", "state_after": "no goals"}, {"tactic": "simp [fract_eq_zero]", "annotated_tactic": ["simp [fract_eq_zero]", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_eq_zero : Int.fract v = 0\n\u22a2 Int.fract v + \u2191\u230av\u230b = \u2191\u230av\u230b", "state_after": "no goals"}, {"tactic": "field_simp [continuantsAux, nextContinuants, nextNumerator, nextDenominator,\n  of_h_eq_floor, compExactValue]", "annotated_tactic": ["field_simp [<a>continuantsAux</a>, <a>nextContinuants</a>, <a>nextNumerator</a>, <a>nextDenominator</a>,\n        <a>of_h_eq_floor</a>, <a>compExactValue</a>]", [{"full_name": "GeneralizedContinuedFraction.continuantsAux", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 19]}, {"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}, {"full_name": "GeneralizedContinuedFraction.of_h_eq_floor", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [170, 9], "def_end_pos": [170, 22]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "case zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\n\u22a2 v = compExactValue ((of v).continuantsAux 0) ((of v).continuantsAux (0 + 1)) (IntFractPair.of v).fr", "state_after": "case zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr"}, {"tactic": "have : (IntFractPair.of v).fr = Int.fract v := rfl", "annotated_tactic": ["have : (<a>IntFractPair.of</a> v).<a>fr</a> = <a>Int.fract</a> v := <a>rfl</a>", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.fr", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 5]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr", "state_after": "case zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\nthis : (IntFractPair.of v).fr = Int.fract v\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr"}, {"tactic": "rw [this, if_neg fract_ne_zero, Int.floor_add_fract]", "annotated_tactic": ["rw [this, <a>if_neg</a> fract_ne_zero, <a>Int.floor_add_fract</a>]", [{"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}, {"full_name": "Int.floor_add_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [883, 9], "def_end_pos": [883, 24]}]], "state_before": "case zero.refl.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nstream_zero_eq : IntFractPair.stream v 0 = some (IntFractPair.of v)\nfract_ne_zero : \u00acInt.fract v = 0\nthis : (IntFractPair.of v).fr = Int.fract v\n\u22a2 v = if (IntFractPair.of v).fr = 0 then \u2191\u230av\u230b else \u2191\u230av\u230b + (IntFractPair.of v).fr", "state_after": "no goals"}, {"tactic": "intro ifp_succ_n succ_nth_stream_eq", "annotated_tactic": ["intro ifp_succ_n succ_nth_stream_eq", []], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\n\u22a2 \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v (n + 1) = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_n.fr", "state_after": "case succ\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "obtain \u27e8ifp_n, nth_stream_eq, nth_fract_ne_zero, -\u27e9 :\n  \u2203 ifp_n, IntFractPair.stream v n = some ifp_n \u2227\n    ifp_n.fr \u2260 0 \u2227 IntFractPair.of ifp_n.fr\u207b\u00b9 = ifp_succ_n :=\n  IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq", "annotated_tactic": ["obtain \u27e8ifp_n, nth_stream_eq, nth_fract_ne_zero, -\u27e9 :\n      \u2203 ifp_n, <a>IntFractPair.stream</a> v n = <a>some</a> ifp_n \u2227\n        ifp_n.fr \u2260 0 \u2227 <a>IntFractPair.of</a> ifp_n.fr\u207b\u00b9 = ifp_succ_n :=\n      <a>IntFractPair.succ_nth_stream_eq_some_iff</a>.1 succ_nth_stream_eq", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.stream", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [149, 15], "def_end_pos": [149, 21]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_eq_some_iff", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [87, 9], "def_end_pos": [87, 36]}]], "state_before": "case succ\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "let conts := g.continuantsAux (n + 2)", "annotated_tactic": ["let conts := g.continuantsAux (n + 2)", []], "state_before": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "set pconts := g.continuantsAux (n + 1) with pconts_eq", "annotated_tactic": ["set pconts := g.continuantsAux (n + 1) with pconts_eq", []], "state_before": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux (n + 1)) ifp_n.fr\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\n\u22a2 v = compExactValue ((of v).continuantsAux (n + 1)) ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ((of v).continuantsAux n) pconts ifp_n.fr\npconts_eq : pconts = g.continuantsAux (n + 1)\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "set ppconts := g.continuantsAux n with ppconts_eq", "annotated_tactic": ["set ppconts := g.continuantsAux n with ppconts_eq", []], "state_before": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\nIH :\n  \u2200 {ifp_n : IntFractPair K},\n    IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ((of v).continuantsAux n) pconts ifp_n.fr\npconts_eq : pconts = g.continuantsAux (n + 1)\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "cases' Decidable.em (ifp_succ_n.fr = 0) with ifp_succ_n_fr_eq_zero ifp_succ_n_fr_ne_zero", "annotated_tactic": ["cases' <a>Decidable.em</a> (ifp_succ_n.fr = 0) with ifp_succ_n_fr_eq_zero ifp_succ_n_fr_ne_zero", [{"full_name": "Decidable.em", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [876, 9], "def_end_pos": [876, 11]}]], "state_before": "case succ.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr\n\ncase succ.intro.intro.intro.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "suffices v = conts.a / conts.b by simpa [compExactValue, ifp_succ_n_fr_eq_zero]", "annotated_tactic": ["suffices v = conts.a / conts.b by simpa [<a>compExactValue</a>, ifp_succ_n_fr_eq_zero]", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "case succ.intro.intro.intro.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\n\u22a2 v = conts.a / conts.b"}, {"tactic": "obtain \u27e8ifp_n', nth_stream_eq', ifp_n_fract_inv_eq_floor\u27e9 :\n    \u2203 ifp_n, IntFractPair.stream v n = some ifp_n \u2227 ifp_n.fr\u207b\u00b9 = \u230aifp_n.fr\u207b\u00b9\u230b :=\n  IntFractPair.exists_succ_nth_stream_of_fr_zero succ_nth_stream_eq ifp_succ_n_fr_eq_zero", "annotated_tactic": ["obtain \u27e8ifp_n', nth_stream_eq', ifp_n_fract_inv_eq_floor\u27e9 :\n          \u2203 ifp_n, <a>IntFractPair.stream</a> v n = <a>some</a> ifp_n \u2227 ifp_n.fr\u207b\u00b9 = \u230aifp_n.fr\u207b\u00b9\u230b :=\n        <a>IntFractPair.exists_succ_nth_stream_of_fr_zero</a> succ_nth_stream_eq ifp_succ_n_fr_eq_zero", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.stream", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [149, 15], "def_end_pos": [149, 21]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.exists_succ_nth_stream_of_fr_zero", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [112, 9], "def_end_pos": [112, 42]}]], "state_before": "case succ.intro.intro.intro.inl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_inv_eq_floor : ifp_n'.fr\u207b\u00b9 = \u2191\u230aifp_n'.fr\u207b\u00b9\u230b\n\u22a2 v = conts.a / conts.b"}, {"tactic": "have : ifp_n' = ifp_n := by injection Eq.trans nth_stream_eq'.symm nth_stream_eq", "annotated_tactic": ["have : ifp_n' = ifp_n := by injection <a>Eq.trans</a> nth_stream_eq'.symm nth_stream_eq", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "case succ.intro.intro.intro.inl.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_inv_eq_floor : ifp_n'.fr\u207b\u00b9 = \u2191\u230aifp_n'.fr\u207b\u00b9\u230b\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_inv_eq_floor : ifp_n'.fr\u207b\u00b9 = \u2191\u230aifp_n'.fr\u207b\u00b9\u230b\nthis : ifp_n' = ifp_n\n\u22a2 v = conts.a / conts.b"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case succ.intro.intro.intro.inl.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_inv_eq_floor : ifp_n'.fr\u207b\u00b9 = \u2191\u230aifp_n'.fr\u207b\u00b9\u230b\nthis : ifp_n' = ifp_n\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\n\u22a2 v = conts.a / conts.b"}, {"tactic": "have s_nth_eq : g.s.get? n = some \u27e81, \u230aifp_n.fr\u207b\u00b9\u230b\u27e9 :=\n  get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero nth_stream_eq nth_fract_ne_zero", "annotated_tactic": ["have s_nth_eq : g.s.get? n = <a>some</a> \u27e81, \u230aifp_n.fr\u207b\u00b9\u230b\u27e9 :=\n        <a>get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero</a> nth_stream_eq nth_fract_ne_zero", [{"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [250, 9], "def_end_pos": [250, 63]}]], "state_before": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\n\u22a2 v = conts.a / conts.b"}, {"tactic": "rw [\u2190 ifp_n_fract_inv_eq_floor] at s_nth_eq", "annotated_tactic": ["rw [\u2190 ifp_n_fract_inv_eq_floor] at s_nth_eq", []], "state_before": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := ifp_n.fr\u207b\u00b9 }\n\u22a2 v = conts.a / conts.b"}, {"tactic": "suffices v = compExactValue ppconts pconts ifp_n.fr by\n  simpa [conts, continuantsAux, s_nth_eq, compExactValue, nth_fract_ne_zero] using this", "annotated_tactic": ["suffices v = <a>compExactValue</a> ppconts pconts ifp_n.fr by\n        simpa [conts, <a>continuantsAux</a>, s_nth_eq, <a>compExactValue</a>, nth_fract_ne_zero] using this", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.continuantsAux", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 19]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := ifp_n.fr\u207b\u00b9 }\n\u22a2 v = conts.a / conts.b", "state_after": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := ifp_n.fr\u207b\u00b9 }\n\u22a2 v = compExactValue ppconts pconts ifp_n.fr"}, {"tactic": "exact IH nth_stream_eq", "annotated_tactic": ["exact IH nth_stream_eq", []], "state_before": "case succ.intro.intro.intro.inl.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := ifp_n.fr\u207b\u00b9 }\n\u22a2 v = compExactValue ppconts pconts ifp_n.fr", "state_after": "no goals"}, {"tactic": "simpa [compExactValue, ifp_succ_n_fr_eq_zero]", "annotated_tactic": ["simpa [<a>compExactValue</a>, ifp_succ_n_fr_eq_zero]", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nthis : v = conts.a / conts.b\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "no goals"}, {"tactic": "injection Eq.trans nth_stream_eq'.symm nth_stream_eq", "annotated_tactic": ["injection <a>Eq.trans</a> nth_stream_eq'.symm nth_stream_eq", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_inv_eq_floor : ifp_n'.fr\u207b\u00b9 = \u2191\u230aifp_n'.fr\u207b\u00b9\u230b\n\u22a2 ifp_n' = ifp_n", "state_after": "no goals"}, {"tactic": "simpa [conts, continuantsAux, s_nth_eq, compExactValue, nth_fract_ne_zero] using this", "annotated_tactic": ["simpa [conts, <a>continuantsAux</a>, s_nth_eq, <a>compExactValue</a>, nth_fract_ne_zero] using this", [{"full_name": "GeneralizedContinuedFraction.continuantsAux", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 19]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_eq_zero : ifp_succ_n.fr = 0\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_inv_eq_floor : ifp_n.fr\u207b\u00b9 = \u2191\u230aifp_n.fr\u207b\u00b9\u230b\ns_nth_eq : g.s.get? n = some { a := 1, b := ifp_n.fr\u207b\u00b9 }\nthis : v = compExactValue ppconts pconts ifp_n.fr\n\u22a2 v = conts.a / conts.b", "state_after": "no goals"}, {"tactic": "suffices\n  compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr by\n  have : v = compExactValue ppconts pconts ifp_n.fr := IH nth_stream_eq\n  conv_lhs => rw [this]\n  assumption", "annotated_tactic": ["suffices\n        <a>compExactValue</a> ppconts pconts ifp_n.fr = <a>compExactValue</a> pconts conts ifp_succ_n.fr by\n        have : v = <a>compExactValue</a> ppconts pconts ifp_n.fr := IH nth_stream_eq\n        conv_lhs => rw [this]\n        assumption", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "case succ.intro.intro.intro.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr"}, {"tactic": "obtain \u27e8ifp_n', nth_stream_eq', ifp_n_fract_ne_zero, \u27e8refl\u27e9\u27e9 :\n  \u2203 ifp_n, IntFractPair.stream v n = some ifp_n \u2227\n    ifp_n.fr \u2260 0 \u2227 IntFractPair.of ifp_n.fr\u207b\u00b9 = ifp_succ_n :=\n  IntFractPair.succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq", "annotated_tactic": ["obtain \u27e8ifp_n', nth_stream_eq', ifp_n_fract_ne_zero, \u27e8refl\u27e9\u27e9 :\n        \u2203 ifp_n, <a>IntFractPair.stream</a> v n = <a>some</a> ifp_n \u2227\n          ifp_n.fr \u2260 0 \u2227 <a>IntFractPair.of</a> ifp_n.fr\u207b\u00b9 = ifp_succ_n :=\n        <a>IntFractPair.succ_nth_stream_eq_some_iff</a>.1 succ_nth_stream_eq", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.stream", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [149, 15], "def_end_pos": [149, 21]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_eq_some_iff", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [87, 9], "def_end_pos": [87, 36]}]], "state_before": "case succ.intro.intro.intro.inr\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_ne_zero : ifp_n'.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n'.fr\u207b\u00b9).fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n'.fr\u207b\u00b9).fr"}, {"tactic": "have : ifp_n' = ifp_n := by injection Eq.trans nth_stream_eq'.symm nth_stream_eq", "annotated_tactic": ["have : ifp_n' = ifp_n := by injection <a>Eq.trans</a> nth_stream_eq'.symm nth_stream_eq", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_ne_zero : ifp_n'.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n'.fr\u207b\u00b9).fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n'.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_ne_zero : ifp_n'.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n'.fr\u207b\u00b9).fr = 0\nthis : ifp_n' = ifp_n\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n'.fr\u207b\u00b9).fr"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_ne_zero : ifp_n'.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n'.fr\u207b\u00b9).fr = 0\nthis : ifp_n' = ifp_n\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n'.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have s_nth_eq : g.s.get? n = some \u27e81, (\u230aifp_n.fr\u207b\u00b9\u230b : K)\u27e9 :=\n  get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero nth_stream_eq ifp_n_fract_ne_zero", "annotated_tactic": ["have s_nth_eq : g.s.get? n = <a>some</a> \u27e81, (\u230aifp_n.fr\u207b\u00b9\u230b : K)\u27e9 :=\n        <a>get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero</a> nth_stream_eq ifp_n_fract_ne_zero", [{"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "GeneralizedContinuedFraction.get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [250, 9], "def_end_pos": [250, 63]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "let ppA := ppconts.a", "annotated_tactic": ["let ppA := ppconts.a", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "let ppB := ppconts.b", "annotated_tactic": ["let ppB := ppconts.b", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "let pA := pconts.a", "annotated_tactic": ["let pA := pconts.a", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "let pB := pconts.b", "annotated_tactic": ["let pB := pconts.b", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have : compExactValue ppconts pconts ifp_n.fr =\n    (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) := by\n  field_simp [ifp_n_fract_ne_zero, compExactValue, nextContinuants, nextNumerator,\n    nextDenominator, ppA, ppB]\n  ac_rfl", "annotated_tactic": ["have : <a>compExactValue</a> ppconts pconts ifp_n.fr =\n          (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) := by\n        -- unfold compExactValue and the convergent computation once\n        field_simp [ifp_n_fract_ne_zero, <a>compExactValue</a>, <a>nextContinuants</a>, <a>nextNumerator</a>,\n          <a>nextDenominator</a>, ppA, ppB]\n        ac_rfl", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have tmp_calc :=\n  compExactValue_correctness_of_stream_eq_some_aux_comp pA ppA ifp_succ_n_fr_ne_zero", "annotated_tactic": ["have tmp_calc :=\n        <a>compExactValue_correctness_of_stream_eq_some_aux_comp</a> pA ppA ifp_succ_n_fr_ne_zero", [{"full_name": "GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some_aux_comp", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [78, 19], "def_end_pos": [78, 72]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have tmp_calc' :=\n  compExactValue_correctness_of_stream_eq_some_aux_comp pB ppB ifp_succ_n_fr_ne_zero", "annotated_tactic": ["have tmp_calc' :=\n        <a>compExactValue_correctness_of_stream_eq_some_aux_comp</a> pB ppB ifp_succ_n_fr_ne_zero", [{"full_name": "GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some_aux_comp", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [78, 19], "def_end_pos": [78, 72]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "let f := Int.fract (1 / ifp_n.fr)", "annotated_tactic": ["let f := <a>Int.fract</a> (1 / ifp_n.fr)", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\nf : K := Int.fract (1 / ifp_n.fr)\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have f_ne_zero : f \u2260 0 := by simpa [f] using ifp_succ_n_fr_ne_zero", "annotated_tactic": ["have f_ne_zero : f \u2260 0 := by simpa [f] using ifp_succ_n_fr_ne_zero", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\nf : K := Int.fract (1 / ifp_n.fr)\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "rw [inv_eq_one_div] at tmp_calc tmp_calc'", "annotated_tactic": ["rw [<a>inv_eq_one_div</a>] at tmp_calc tmp_calc'", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have hA : (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) + pA * f = pA * (1 / ifp_n.fr) + ppA := by\n  have := congrFun (congrArg HMul.hMul tmp_calc) f\n  rwa [right_distrib, div_mul_cancel\u2080 (h := f_ne_zero),\n    div_mul_cancel\u2080 (h := f_ne_zero)] at this", "annotated_tactic": ["have hA : (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) + pA * f = pA * (1 / ifp_n.fr) + ppA := by\n        have := <a>congrFun</a> (<a>congrArg</a> <a>HMul.hMul</a> tmp_calc) f\n        rwa [<a>right_distrib</a>, <a>div_mul_cancel\u2080</a> (h := f_ne_zero),\n          <a>div_mul_cancel\u2080</a> (h := f_ne_zero)] at this", [{"full_name": "congrFun", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [376, 9], "def_end_pos": [376, 17]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "HMul.hMul", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1199, 3], "def_end_pos": [1199, 7]}, {"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "have hB : (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) + pB * f = pB * (1 / ifp_n.fr) + ppB := by\n  have := congrFun (congrArg HMul.hMul tmp_calc') f\n  rwa [right_distrib, div_mul_cancel\u2080 (h := f_ne_zero),\n    div_mul_cancel\u2080 (h := f_ne_zero)] at this", "annotated_tactic": ["have hB : (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) + pB * f = pB * (1 / ifp_n.fr) + ppB := by\n        have := <a>congrFun</a> (<a>congrArg</a> <a>HMul.hMul</a> tmp_calc') f\n        rwa [<a>right_distrib</a>, <a>div_mul_cancel\u2080</a> (h := f_ne_zero),\n          <a>div_mul_cancel\u2080</a> (h := f_ne_zero)] at this", [{"full_name": "congrFun", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [376, 9], "def_end_pos": [376, 17]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "HMul.hMul", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1199, 3], "def_end_pos": [1199, 7]}, {"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "dsimp only [conts, pconts, ppconts]", "annotated_tactic": ["dsimp only [conts, pconts, ppconts]", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) = compExactValue pconts conts (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) =\n    compExactValue (g.continuantsAux (n + 1)) (g.continuantsAux (n + 2)) (IntFractPair.of ifp_n.fr\u207b\u00b9).fr"}, {"tactic": "field_simp [compExactValue, continuantsAux_recurrence s_nth_eq ppconts_eq pconts_eq,\n  nextContinuants, nextNumerator, nextDenominator]", "annotated_tactic": ["field_simp [<a>compExactValue</a>, <a>continuantsAux_recurrence</a> s_nth_eq ppconts_eq pconts_eq,\n        <a>nextContinuants</a>, <a>nextNumerator</a>, <a>nextDenominator</a>]", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.continuantsAux_recurrence", "def_path": "Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean", "def_pos": [26, 9], "def_end_pos": [26, 34]}, {"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB) =\n    compExactValue (g.continuantsAux (n + 1)) (g.continuantsAux (n + 2)) (IntFractPair.of ifp_n.fr\u207b\u00b9).fr", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    if (IntFractPair.of (1 / ifp_n.fr)).fr = 0 then\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b)\n    else\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a +\n          (g.continuantsAux (n + 1)).a * (IntFractPair.of (1 / ifp_n.fr)).fr) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b +\n          (g.continuantsAux (n + 1)).b * (IntFractPair.of (1 / ifp_n.fr)).fr)"}, {"tactic": "have hfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f := rfl", "annotated_tactic": ["have hfr : (<a>IntFractPair.of</a> (1 / ifp_n.fr)).<a>fr</a> = f := <a>rfl</a>", [{"full_name": "GeneralizedContinuedFraction.IntFractPair.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [131, 15], "def_end_pos": [131, 17]}, {"full_name": "GeneralizedContinuedFraction.IntFractPair.fr", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 5]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    if (IntFractPair.of (1 / ifp_n.fr)).fr = 0 then\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b)\n    else\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a +\n          (g.continuantsAux (n + 1)).a * (IntFractPair.of (1 / ifp_n.fr)).fr) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b +\n          (g.continuantsAux (n + 1)).b * (IntFractPair.of (1 / ifp_n.fr)).fr)", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    if (IntFractPair.of (1 / ifp_n.fr)).fr = 0 then\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b)\n    else\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a +\n          (g.continuantsAux (n + 1)).a * (IntFractPair.of (1 / ifp_n.fr)).fr) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b +\n          (g.continuantsAux (n + 1)).b * (IntFractPair.of (1 / ifp_n.fr)).fr)"}, {"tactic": "rw [one_div, if_neg _, \u2190 one_div, hfr]", "annotated_tactic": ["rw [<a>one_div</a>, <a>if_neg</a> _, \u2190 <a>one_div</a>, hfr]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}]], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    if (IntFractPair.of (1 / ifp_n.fr)).fr = 0 then\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b)\n    else\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a +\n          (g.continuantsAux (n + 1)).a * (IntFractPair.of (1 / ifp_n.fr)).fr) /\n        (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b +\n          (g.continuantsAux (n + 1)).b * (IntFractPair.of (1 / ifp_n.fr)).fr)", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a + (g.continuantsAux (n + 1)).a * f) /\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b + (g.continuantsAux (n + 1)).b * f)\n\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0"}, {"tactic": "have : v = compExactValue ppconts pconts ifp_n.fr := IH nth_stream_eq", "annotated_tactic": ["have : v = <a>compExactValue</a> ppconts pconts ifp_n.fr := IH nth_stream_eq", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\nthis : compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr\nthis : v = compExactValue ppconts pconts ifp_n.fr\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "conv_lhs => rw [this]", "annotated_tactic": ["conv_lhs => rw [this]", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr\nthis : v = compExactValue ppconts pconts ifp_n.fr\n\u22a2 v = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr\nthis : v = compExactValue ppconts pconts ifp_n.fr\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_succ_n : IntFractPair K\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_succ_n_fr_ne_zero : \u00acifp_succ_n.fr = 0\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = compExactValue pconts conts ifp_succ_n.fr\nthis : v = compExactValue ppconts pconts ifp_n.fr\n\u22a2 compExactValue ppconts pconts ifp_n.fr = compExactValue pconts ((of v).continuantsAux (n + 1 + 1)) ifp_succ_n.fr", "state_after": "no goals"}, {"tactic": "injection Eq.trans nth_stream_eq'.symm nth_stream_eq", "annotated_tactic": ["injection <a>Eq.trans</a> nth_stream_eq'.symm nth_stream_eq", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nifp_n' : IntFractPair K\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n'\nifp_n_fract_ne_zero : ifp_n'.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n'.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n'.fr\u207b\u00b9).fr = 0\n\u22a2 ifp_n' = ifp_n", "state_after": "no goals"}, {"tactic": "field_simp [ifp_n_fract_ne_zero, compExactValue, nextContinuants, nextNumerator,\n  nextDenominator, ppA, ppB]", "annotated_tactic": ["field_simp [ifp_n_fract_ne_zero, <a>compExactValue</a>, <a>nextContinuants</a>, <a>nextNumerator</a>,\n          <a>nextDenominator</a>, ppA, ppB]", [{"full_name": "GeneralizedContinuedFraction.compExactValue", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [66, 15], "def_end_pos": [66, 29]}, {"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\n\u22a2 compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\n\u22a2 (pconts.a + ppconts.a * ifp_n.fr) / (pconts.b + ppconts.b * ifp_n.fr) =\n    (ppconts.a * ifp_n.fr + pA) / (ppconts.b * ifp_n.fr + pB)"}, {"tactic": "ac_rfl", "annotated_tactic": ["ac_rfl", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\n\u22a2 (pconts.a + ppconts.a * ifp_n.fr) / (pconts.b + ppconts.b * ifp_n.fr) =\n    (ppconts.a * ifp_n.fr + pA) / (ppconts.b * ifp_n.fr + pB)", "state_after": "no goals"}, {"tactic": "simpa [f] using ifp_succ_n_fr_ne_zero", "annotated_tactic": ["simpa [f] using ifp_succ_n_fr_ne_zero", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pA + ppA) / Int.fract ifp_n.fr\u207b\u00b9 + pA = (pA * ifp_n.fr\u207b\u00b9 + ppA) / Int.fract ifp_n.fr\u207b\u00b9\ntmp_calc' : (\u2191\u230aifp_n.fr\u207b\u00b9\u230b * pB + ppB) / Int.fract ifp_n.fr\u207b\u00b9 + pB = (pB * ifp_n.fr\u207b\u00b9 + ppB) / Int.fract ifp_n.fr\u207b\u00b9\nf : K := Int.fract (1 / ifp_n.fr)\n\u22a2 f \u2260 0", "state_after": "no goals"}, {"tactic": "have := congrFun (congrArg HMul.hMul tmp_calc) f", "annotated_tactic": ["have := <a>congrFun</a> (<a>congrArg</a> <a>HMul.hMul</a> tmp_calc) f", [{"full_name": "congrFun", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [376, 9], "def_end_pos": [376, 17]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "HMul.hMul", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1199, 3], "def_end_pos": [1199, 7]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nthis :\n  ((\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA) * f =\n    (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr) * f\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA"}, {"tactic": "rwa [right_distrib, div_mul_cancel\u2080 (h := f_ne_zero),\n  div_mul_cancel\u2080 (h := f_ne_zero)] at this", "annotated_tactic": ["rwa [<a>right_distrib</a>, <a>div_mul_cancel\u2080</a> (h := f_ne_zero),\n          <a>div_mul_cancel\u2080</a> (h := f_ne_zero)] at this", [{"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nthis :\n  ((\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA) * f =\n    (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr) * f\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA", "state_after": "no goals"}, {"tactic": "have := congrFun (congrArg HMul.hMul tmp_calc') f", "annotated_tactic": ["have := <a>congrFun</a> (<a>congrArg</a> <a>HMul.hMul</a> tmp_calc') f", [{"full_name": "congrFun", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [376, 9], "def_end_pos": [376, 17]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "HMul.hMul", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1199, 3], "def_end_pos": [1199, 7]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB", "state_after": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nthis :\n  ((\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB) * f =\n    (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr) * f\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB"}, {"tactic": "rwa [right_distrib, div_mul_cancel\u2080 (h := f_ne_zero),\n  div_mul_cancel\u2080 (h := f_ne_zero)] at this", "annotated_tactic": ["rwa [<a>right_distrib</a>, <a>div_mul_cancel\u2080</a> (h := f_ne_zero),\n          <a>div_mul_cancel\u2080</a> (h := f_ne_zero)] at this", [{"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}, {"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis\u271d : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nthis :\n  ((\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB) * f =\n    (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr) * f\n\u22a2 \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB", "state_after": "no goals"}, {"tactic": "field_simp [hA, hB]", "annotated_tactic": ["field_simp [hA, hB]", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) =\n    (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).a + (g.continuantsAux n).a + (g.continuantsAux (n + 1)).a * f) /\n      (\u2191\u230a1 / ifp_n.fr\u230b * (g.continuantsAux (n + 1)).b + (g.continuantsAux n).b + (g.continuantsAux (n + 1)).b * f)", "state_after": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) = (pA + ppA * ifp_n.fr) / (pB + ppB * ifp_n.fr)"}, {"tactic": "ac_rfl", "annotated_tactic": ["ac_rfl", []], "state_before": "case succ.intro.intro.intro.inr.intro.intro.intro.refl.refl\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 (ppA * ifp_n.fr + pA) / (ppB * ifp_n.fr + pB) = (pA + ppA * ifp_n.fr) / (pB + ppB * ifp_n.fr)", "state_after": "no goals"}, {"tactic": "rwa [inv_eq_one_div, hfr]", "annotated_tactic": ["rwa [<a>inv_eq_one_div</a>, hfr]", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\nv : K\nn\u271d : \u2115\ninst\u271d : FloorRing K\ng : GeneralizedContinuedFraction K := of v\nn : \u2115\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr \u2260 0\nconts : Pair K := g.continuantsAux (n + 2)\npconts : Pair K := g.continuantsAux (n + 1)\npconts_eq : pconts = g.continuantsAux (n + 1)\nppconts : Pair K := g.continuantsAux n\nIH : \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192 v = compExactValue ppconts pconts ifp_n.fr\nppconts_eq : ppconts = g.continuantsAux n\nnth_stream_eq' : IntFractPair.stream v n = some ifp_n\nifp_n_fract_ne_zero : ifp_n.fr \u2260 0\nsucc_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr\u207b\u00b9)\nifp_succ_n_fr_ne_zero : \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0\ns_nth_eq : g.s.get? n = some { a := 1, b := \u2191\u230aifp_n.fr\u207b\u00b9\u230b }\nppA : K := ppconts.a\nppB : K := ppconts.b\npA : K := pconts.a\npB : K := pconts.b\nthis : compExactValue ppconts pconts ifp_n.fr = (ppA + ifp_n.fr\u207b\u00b9 * pA) / (ppB + ifp_n.fr\u207b\u00b9 * pB)\ntmp_calc :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pA + ppA) / Int.fract (1 / ifp_n.fr) + pA = (pA * (1 / ifp_n.fr) + ppA) / Int.fract (1 / ifp_n.fr)\ntmp_calc' :\n  (\u2191\u230a1 / ifp_n.fr\u230b * pB + ppB) / Int.fract (1 / ifp_n.fr) + pB = (pB * (1 / ifp_n.fr) + ppB) / Int.fract (1 / ifp_n.fr)\nf : K := Int.fract (1 / ifp_n.fr)\nf_ne_zero : f \u2260 0\nhA : \u2191\u230a1 / ifp_n.fr\u230b * pA + ppA + pA * f = pA * (1 / ifp_n.fr) + ppA\nhB : \u2191\u230a1 / ifp_n.fr\u230b * pB + ppB + pB * f = pB * (1 / ifp_n.fr) + ppB\nhfr : (IntFractPair.of (1 / ifp_n.fr)).fr = f\n\u22a2 \u00ac(IntFractPair.of ifp_n.fr\u207b\u00b9).fr = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjointed.lean", "full_name": "disjointed_le", "start": [70, 1], "end": [71, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Integral.lean", "full_name": "ConvexOn.average_mem_epigraph", "start": [112, 1], "end": [119, 79], "traced_tactics": [{"tactic": "have ht_mem : \u2200\u1d50 x \u2202\u03bc, (f x, g (f x)) \u2208 {p : E \u00d7 \u211d | p.1 \u2208 s \u2227 g p.1 \u2264 p.2} :=\n  hfs.mono fun x hx => \u27e8hx, le_rfl\u27e9", "annotated_tactic": ["have ht_mem : \u2200\u1d50 x \u2202\u03bc, (f x, g (f x)) \u2208 {p : E \u00d7 \u211d | p.1 \u2208 s \u2227 g p.1 \u2264 p.2} :=\n    hfs.mono fun x hx => \u27e8hx, <a>le_rfl</a>\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\nt : Set \u03b1\nf : \u03b1 \u2192 E\ng : E \u2192 \u211d\nC : \u211d\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NeZero \u03bc\nhg : ConvexOn \u211d s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 s\nhfi : Integrable f \u03bc\nhgi : Integrable (g \u2218 f) \u03bc\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc, \u2a0d (x : \u03b1), g (f x) \u2202\u03bc) \u2208 {p | p.1 \u2208 s \u2227 g p.1 \u2264 p.2}", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\nt : Set \u03b1\nf : \u03b1 \u2192 E\ng : E \u2192 \u211d\nC : \u211d\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NeZero \u03bc\nhg : ConvexOn \u211d s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 s\nhfi : Integrable f \u03bc\nhgi : Integrable (g \u2218 f) \u03bc\nht_mem : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (f x, g (f x)) \u2208 {p | p.1 \u2208 s \u2227 g p.1 \u2264 p.2}\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc, \u2a0d (x : \u03b1), g (f x) \u2202\u03bc) \u2208 {p | p.1 \u2208 s \u2227 g p.1 \u2264 p.2}"}, {"tactic": "exact average_pair hfi hgi \u25b8\n  hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)", "annotated_tactic": ["exact <a>average_pair</a> hfi hgi \u25b8\n    hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)", [{"full_name": "MeasureTheory.average_pair", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [383, 9], "def_end_pos": [383, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\nt : Set \u03b1\nf : \u03b1 \u2192 E\ng : E \u2192 \u211d\nC : \u211d\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : NeZero \u03bc\nhg : ConvexOn \u211d s g\nhgc : ContinuousOn g s\nhsc : IsClosed s\nhfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 s\nhfi : Integrable f \u03bc\nhgi : Integrable (g \u2218 f) \u03bc\nht_mem : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (f x, g (f x)) \u2208 {p | p.1 \u2208 s \u2227 g p.1 \u2264 p.2}\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc, \u2a0d (x : \u03b1), g (f x) \u2202\u03bc) \u2208 {p | p.1 \u2208 s \u2227 g p.1 \u2264 p.2}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Jensen.lean", "full_name": "StrictConcaveOn.eq_of_map_sum_eq", "start": [170, 1], "end": [180, 65], "traced_tactics": [{"tactic": "by_contra!", "annotated_tactic": ["by_contra!", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nt : Finset \u03b9\nw : \u03b9 \u2192 \ud835\udd5c\np : \u03b9 \u2192 E\nv : \ud835\udd5c\nq : E\nhf : StrictConcaveOn \ud835\udd5c s f\nh\u2080 : \u2200 i \u2208 t, 0 < w i\nh\u2081 : \u2211 i \u2208 t, w i = 1\nhmem : \u2200 i \u2208 t, p i \u2208 s\nh_eq : f (\u2211 i \u2208 t, w i \u2022 p i) \u2264 \u2211 i \u2208 t, w i \u2022 f (p i)\n\u22a2 \u2200 \u2983j : \u03b9\u2984, j \u2208 t \u2192 \u2200 \u2983k : \u03b9\u2984, k \u2208 t \u2192 p j = p k", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nt : Finset \u03b9\nw : \u03b9 \u2192 \ud835\udd5c\np : \u03b9 \u2192 E\nv : \ud835\udd5c\nq : E\nhf : StrictConcaveOn \ud835\udd5c s f\nh\u2080 : \u2200 i \u2208 t, 0 < w i\nh\u2081 : \u2211 i \u2208 t, w i = 1\nhmem : \u2200 i \u2208 t, p i \u2208 s\nh_eq : f (\u2211 i \u2208 t, w i \u2022 p i) \u2264 \u2211 i \u2208 t, w i \u2022 f (p i)\nthis : \u2203 j \u2208 t, \u2203 k \u2208 t, p j \u2260 p k\n\u22a2 False"}, {"tactic": "exact h_eq.not_lt <| hf.lt_map_sum h\u2080 h\u2081 hmem this", "annotated_tactic": ["exact h_eq.not_lt <| hf.lt_map_sum h\u2080 h\u2081 hmem this", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nt : Finset \u03b9\nw : \u03b9 \u2192 \ud835\udd5c\np : \u03b9 \u2192 E\nv : \ud835\udd5c\nq : E\nhf : StrictConcaveOn \ud835\udd5c s f\nh\u2080 : \u2200 i \u2208 t, 0 < w i\nh\u2081 : \u2211 i \u2208 t, w i = 1\nhmem : \u2200 i \u2208 t, p i \u2208 s\nh_eq : f (\u2211 i \u2208 t, w i \u2022 p i) \u2264 \u2211 i \u2208 t, w i \u2022 f (p i)\nthis : \u2203 j \u2208 t, \u2203 k \u2208 t, p j \u2260 p k\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Interval.lean", "full_name": "Finsupp.card_Ico", "start": [113, 1], "end": [114, 46], "traced_tactics": [{"tactic": "rw [card_Ico_eq_card_Icc_sub_one, card_Icc]", "annotated_tactic": ["rw [<a>card_Ico_eq_card_Icc_sub_one</a>, <a>card_Icc</a>]", [{"full_name": "Finset.card_Ico_eq_card_Icc_sub_one", "def_path": "Mathlib/Order/Interval/Finset/Basic.lean", "def_pos": [651, 9], "def_end_pos": [651, 37]}, {"full_name": "Finsupp.card_Icc", "def_path": "Mathlib/Data/Finsupp/Interval.lean", "def_pos": [108, 9], "def_end_pos": [108, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 (Ico f g).card = \u220f i \u2208 f.support \u222a g.support, (Icc (f i) (g i)).card - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectLimit.lean", "full_name": "Module.DirectLimit.of_f", "start": [122, 1], "end": [123, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "Differentiable.sub_const", "start": [622, 1], "end": [623, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometryEquiv.coe_mk", "start": [346, 1], "end": [347, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Commute.lean", "full_name": "neg_one_sq", "start": [200, 1], "end": [200, 65], "traced_tactics": [{"tactic": "simp [neg_sq, one_pow]", "annotated_tactic": ["simp [<a>neg_sq</a>, <a>one_pow</a>]", [{"full_name": "neg_sq", "def_path": "Mathlib/Algebra/Ring/Commute.lean", "def_pos": [196, 7], "def_end_pos": [196, 13]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\n\u22a2 (-1) ^ 2 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq", "start": [944, 1], "end": [962, 20], "traced_tactics": [{"tactic": "refine \u27e8?_, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8?_, fun h => ?_\u27e9", []], "state_before": "\u03b8 \u03c8 : Angle\n\u22a2 \u03b8 = \u03c8 \u2194 \u03b8.sign = \u03c8.sign \u2227 |\u03b8.toReal| = |\u03c8.toReal|", "state_after": "case refine_1\n\u03b8 \u03c8 : Angle\n\u22a2 \u03b8 = \u03c8 \u2192 \u03b8.sign = \u03c8.sign \u2227 |\u03b8.toReal| = |\u03c8.toReal|\n\ncase refine_2\n\u03b8 \u03c8 : Angle\nh : \u03b8.sign = \u03c8.sign \u2227 |\u03b8.toReal| = |\u03c8.toReal|\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rcases h with \u27e8hs, hr\u27e9", "annotated_tactic": ["rcases h with \u27e8hs, hr\u27e9", []], "state_before": "case refine_2\n\u03b8 \u03c8 : Angle\nh : \u03b8.sign = \u03c8.sign \u2227 |\u03b8.toReal| = |\u03c8.toReal|\n\u22a2 \u03b8 = \u03c8", "state_after": "case refine_2.intro\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : |\u03b8.toReal| = |\u03c8.toReal|\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rw [abs_eq_abs] at hr", "annotated_tactic": ["rw [<a>abs_eq_abs</a>] at hr", [{"full_name": "abs_eq_abs", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [255, 3], "def_end_pos": [255, 14]}]], "state_before": "case refine_2.intro\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : |\u03b8.toReal| = |\u03c8.toReal|\n\u22a2 \u03b8 = \u03c8", "state_after": "case refine_2.intro\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = \u03c8.toReal \u2228 \u03b8.toReal = -\u03c8.toReal\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rcases hr with (hr | hr)", "annotated_tactic": ["rcases hr with (hr | hr)", []], "state_before": "case refine_2.intro\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = \u03c8.toReal \u2228 \u03b8.toReal = -\u03c8.toReal\n\u22a2 \u03b8 = \u03c8", "state_after": "case refine_2.intro.inl\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = \u03c8.toReal\n\u22a2 \u03b8 = \u03c8\n\ncase refine_2.intro.inr\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case refine_1\n\u03b8 \u03c8 : Angle\n\u22a2 \u03b8 = \u03c8 \u2192 \u03b8.sign = \u03c8.sign \u2227 |\u03b8.toReal| = |\u03c8.toReal|", "state_after": "case refine_1\n\u03b8 : Angle\n\u22a2 \u03b8.sign = \u03b8.sign \u2227 |\u03b8.toReal| = |\u03b8.toReal|"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "annotated_tactic": ["exact \u27e8<a>rfl</a>, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case refine_1\n\u03b8 : Angle\n\u22a2 \u03b8.sign = \u03b8.sign \u2227 |\u03b8.toReal| = |\u03b8.toReal|", "state_after": "no goals"}, {"tactic": "exact toReal_injective hr", "annotated_tactic": ["exact <a>toReal_injective</a> hr", [{"full_name": "Real.Angle.toReal_injective", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [540, 9], "def_end_pos": [540, 25]}]], "state_before": "case refine_2.intro.inl\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = \u03c8.toReal\n\u22a2 \u03b8 = \u03c8", "state_after": "no goals"}, {"tactic": "by_cases h : \u03b8 = \u03c0", "annotated_tactic": ["by_cases h : \u03b8 = \u03c0", []], "state_before": "case refine_2.intro.inr\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\n\u22a2 \u03b8 = \u03c8", "state_after": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8\n\ncase neg\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rw [h, toReal_pi, \u2190 neg_eq_iff_eq_neg] at hr", "annotated_tactic": ["rw [h, <a>toReal_pi</a>, \u2190 <a>neg_eq_iff_eq_neg</a>] at hr", [{"full_name": "Real.Angle.toReal_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [598, 9], "def_end_pos": [598, 18]}, {"full_name": "neg_eq_iff_eq_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [412, 3], "def_end_pos": [412, 14]}]], "state_before": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : -\u03c0 = \u03c8.toReal\nh : \u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "exact False.elim ((neg_pi_lt_toReal \u03c8).ne hr)", "annotated_tactic": ["exact <a>False.elim</a> ((<a>neg_pi_lt_toReal</a> \u03c8).<a>ne</a> hr)", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}, {"full_name": "Real.Angle.neg_pi_lt_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : -\u03c0 = \u03c8.toReal\nh : \u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "no goals"}, {"tactic": "by_cases h' : \u03c8 = \u03c0", "annotated_tactic": ["by_cases h' : \u03c8 = \u03c0", []], "state_before": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8\n\ncase neg\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rw [h', toReal_pi] at hr", "annotated_tactic": ["rw [h', <a>toReal_pi</a>] at hr", [{"full_name": "Real.Angle.toReal_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [598, 9], "def_end_pos": [598, 18]}]], "state_before": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c0\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "exact False.elim ((neg_pi_lt_toReal \u03b8).ne hr.symm)", "annotated_tactic": ["exact <a>False.elim</a> ((<a>neg_pi_lt_toReal</a> \u03b8).<a>ne</a> hr.symm)", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}, {"full_name": "Real.Angle.neg_pi_lt_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "case pos\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c0\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "no goals"}, {"tactic": "rw [\u2190 sign_toReal h, \u2190 sign_toReal h', hr, Left.sign_neg, SignType.neg_eq_self_iff,\n  _root_.sign_eq_zero_iff, toReal_eq_zero_iff] at hs", "annotated_tactic": ["rw [\u2190 <a>sign_toReal</a> h, \u2190 <a>sign_toReal</a> h', hr, <a>Left.sign_neg</a>, <a>SignType.neg_eq_self_iff</a>,\n          <a>_root_.sign_eq_zero_iff</a>, <a>toReal_eq_zero_iff</a>] at hs", [{"full_name": "Real.Angle.sign_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [923, 9], "def_end_pos": [923, 20]}, {"full_name": "Real.Angle.sign_toReal", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [923, 9], "def_end_pos": [923, 20]}, {"full_name": "Left.sign_neg", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [479, 9], "def_end_pos": [479, 22]}, {"full_name": "SignType.neg_eq_self_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [223, 9], "def_end_pos": [223, 24]}, {"full_name": "sign_eq_zero_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [384, 9], "def_end_pos": [384, 25]}, {"full_name": "Real.Angle.toReal_eq_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [592, 9], "def_end_pos": [592, 27]}]], "state_before": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03b8.sign = \u03c8.sign\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03c8 = 0\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rw [hs, toReal_zero, neg_zero, toReal_eq_zero_iff] at hr", "annotated_tactic": ["rw [hs, <a>toReal_zero</a>, <a>neg_zero</a>, <a>toReal_eq_zero_iff</a>] at hr", [{"full_name": "Real.Angle.toReal_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [586, 9], "def_end_pos": [586, 20]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}, {"full_name": "Real.Angle.toReal_eq_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [592, 9], "def_end_pos": [592, 27]}]], "state_before": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03c8 = 0\nhr : \u03b8.toReal = -\u03c8.toReal\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03c8 = 0\nhr : \u03b8 = 0\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8"}, {"tactic": "rw [hr, hs]", "annotated_tactic": ["rw [hr, hs]", []], "state_before": "case neg\n\u03b8 \u03c8 : Angle\nhs : \u03c8 = 0\nhr : \u03b8 = 0\nh : \u00ac\u03b8 = \u2191\u03c0\nh' : \u00ac\u03c8 = \u2191\u03c0\n\u22a2 \u03b8 = \u03c8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.restrictScalars_mul", "start": [316, 1], "end": [324, 57], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\n\u22a2 restrictScalars A (I * J) = restrictScalars A I * restrictScalars A J", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\n\u22a2 restrictScalars A (I * J) \u2264 restrictScalars A I * restrictScalars A J\n\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\n\u22a2 restrictScalars A I * restrictScalars A J \u2264 restrictScalars A (I * J)"}, {"tactic": "intro x (hx : x \u2208 I * J)", "annotated_tactic": ["intro x (hx : x \u2208 I * J)", []], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\n\u22a2 restrictScalars A (I * J) \u2264 restrictScalars A I * restrictScalars A J", "state_after": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 x \u2208 restrictScalars A I * restrictScalars A J"}, {"tactic": "refine Submodule.mul_induction_on hx ?_ ?_", "annotated_tactic": ["refine <a>Submodule.mul_induction_on</a> hx ?_ ?_", [{"full_name": "Submodule.mul_induction_on", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [176, 19], "def_end_pos": [176, 35]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 x \u2208 restrictScalars A I * restrictScalars A J", "state_after": "case a.refine_1\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 \u2200 m \u2208 I, \u2200 n \u2208 J, m * n \u2208 restrictScalars A I * restrictScalars A J\n\ncase a.refine_2\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 \u2200 (x y : C),\n    x \u2208 restrictScalars A I * restrictScalars A J \u2192\n      y \u2208 restrictScalars A I * restrictScalars A J \u2192 x + y \u2208 restrictScalars A I * restrictScalars A J"}, {"tactic": "exact fun m hm n hn \u21a6 mul_mem_mul hm hn", "annotated_tactic": ["exact fun m hm n hn \u21a6 <a>mul_mem_mul</a> hm hn", [{"full_name": "Submodule.mul_mem_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case a.refine_1\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 \u2200 m \u2208 I, \u2200 n \u2208 J, m * n \u2208 restrictScalars A I * restrictScalars A J", "state_after": "no goals"}, {"tactic": "exact fun _ _ \u21a6 add_mem", "annotated_tactic": ["exact fun _ _ \u21a6 <a>add_mem</a>", [{"full_name": "AddMemClass.add_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}]], "state_before": "case a.refine_2\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\nx : C\nhx : x \u2208 I * J\n\u22a2 \u2200 (x y : C),\n    x \u2208 restrictScalars A I * restrictScalars A J \u2192\n      y \u2208 restrictScalars A I * restrictScalars A J \u2192 x + y \u2208 restrictScalars A I * restrictScalars A J", "state_after": "no goals"}, {"tactic": "exact mul_le.mpr (fun _ hm _ hn \u21a6 mul_mem_mul hm hn)", "annotated_tactic": ["exact mul_le.mpr (fun _ hm _ hn \u21a6 <a>mul_mem_mul</a> hm hn)", [{"full_name": "Submodule.mul_mem_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case a\n\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u2079 : CommSemiring R\nA\u271d : Type v\ninst\u271d\u2078 : Semiring A\u271d\ninst\u271d\u2077 : Algebra R A\u271d\nS T : Set A\u271d\nM N P Q : Submodule R A\u271d\nm n : A\u271d\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring A\ninst\u271d\u2075 : CommSemiring B\ninst\u271d\u2074 : Semiring C\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nI J : Submodule B C\n\u22a2 restrictScalars A I * restrictScalars A J \u2264 restrictScalars A (I * J)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "ContDiffOn.sinh", "start": [1180, 1], "end": [1182, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Directed.lean", "full_name": "Subalgebra.iSupLift_of_mem", "start": [96, 1], "end": [99, 29], "traced_tactics": [{"tactic": "dsimp [iSupLift, inclusion]", "annotated_tactic": ["dsimp [<a>iSupLift</a>, <a>inclusion</a>]", [{"full_name": "Subalgebra.iSupLift", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Directed.lean", "def_pos": [50, 19], "def_end_pos": [50, 27]}, {"full_name": "Subalgebra.inclusion", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [1014, 5], "def_end_pos": [1014, 14]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\nS : Subalgebra R A\n\u03b9 : Type u_4\ninst\u271d : Nonempty \u03b9\nK : \u03b9 \u2192 Subalgebra R A\ndir : Directed (fun x x_1 => x \u2264 x_1) K\nf : (i : \u03b9) \u2192 \u21a5(K i) \u2192\u2090[R] B\nhf : \u2200 (i j : \u03b9) (h : K i \u2264 K j), f i = (f j).comp (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : \u03b9\nx : \u21a5T\nhx : \u2191x \u2208 K i\n\u22a2 (iSupLift K dir f hf T hT) x = (f i) \u27e8\u2191x, hx\u27e9", "state_after": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\nS : Subalgebra R A\n\u03b9 : Type u_4\ninst\u271d : Nonempty \u03b9\nK : \u03b9 \u2192 Subalgebra R A\ndir : Directed (fun x x_1 => x \u2264 x_1) K\nf : (i : \u03b9) \u2192 \u21a5(K i) \u2192\u2090[R] B\nhf : \u2200 (i j : \u03b9) (h : K i \u2264 K j), f i = (f j).comp (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : \u03b9\nx : \u21a5T\nhx : \u2191x \u2208 K i\n\u22a2 Set.iUnionLift (fun i => \u2191(K i)) (fun i x => (f i) x) \u22ef \u2191T \u22ef x = (f i) \u27e8\u2191x, hx\u27e9"}, {"tactic": "rw [Set.iUnionLift_of_mem]", "annotated_tactic": ["rw [<a>Set.iUnionLift_of_mem</a>]", [{"full_name": "Set.iUnionLift_of_mem", "def_path": "Mathlib/Data/Set/UnionLift.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\nS : Subalgebra R A\n\u03b9 : Type u_4\ninst\u271d : Nonempty \u03b9\nK : \u03b9 \u2192 Subalgebra R A\ndir : Directed (fun x x_1 => x \u2264 x_1) K\nf : (i : \u03b9) \u2192 \u21a5(K i) \u2192\u2090[R] B\nhf : \u2200 (i j : \u03b9) (h : K i \u2264 K j), f i = (f j).comp (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : \u03b9\nx : \u21a5T\nhx : \u2191x \u2208 K i\n\u22a2 Set.iUnionLift (fun i => \u2191(K i)) (fun i x => (f i) x) \u22ef \u2191T \u22ef x = (f i) \u27e8\u2191x, hx\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "full_name": "Cardinal.lift_le_continuum", "start": [51, 1], "end": [52, 40], "traced_tactics": [{"tactic": "rw [\u2190 lift_continuum.{v, u}, lift_le]", "annotated_tactic": ["rw [\u2190 <a>lift_continuum</a>.{v, u}, <a>lift_le</a>]", [{"full_name": "Cardinal.lift_continuum", "def_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "def_pos": [41, 9], "def_end_pos": [41, 23]}, {"full_name": "Cardinal.lift_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}]], "state_before": "c : Cardinal.{u}\n\u22a2 lift.{v, u} c \u2264 \ud835\udd20 \u2194 c \u2264 \ud835\udd20", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.fract_add_int", "start": [893, 1], "end": [895, 7], "traced_tactics": [{"tactic": "rw [fract]", "annotated_tactic": ["rw [<a>fract</a>]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 fract (a + \u2191m) = fract a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 a + \u2191m - \u2191\u230aa + \u2191m\u230b = fract a"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 a + \u2191m - \u2191\u230aa + \u2191m\u230b = fract a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Squarefree/Basic.lean", "full_name": "Prime.squarefree", "start": [76, 1], "end": [77, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Partial.lean", "full_name": "Filter.ptendsto_def", "start": [226, 1], "end": [228, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Inseparable.lean", "full_name": "SeparationQuotient.lift_comp_mk", "start": [513, 1], "end": [514, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "full_name": "exists_mem_frontier_infDist_compl_eq_dist", "start": [651, 1], "end": [658, 18], "traced_tactics": [{"tactic": "rcases Metric.exists_mem_closure_infDist_eq_dist (nonempty_compl.2 hs) x with \u27e8y, hys, hyd\u27e9", "annotated_tactic": ["rcases <a>Metric.exists_mem_closure_infDist_eq_dist</a> (<a>nonempty_compl</a>.2 hs) x with \u27e8y, hys, hyd\u27e9", [{"full_name": "Metric.exists_mem_closure_infDist_eq_dist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [665, 9], "def_end_pos": [665, 43]}, {"full_name": "Set.nonempty_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1660, 9], "def_end_pos": [1660, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\n\u22a2 \u2203 y \u2208 frontier s, infDist x s\u1d9c = dist x y", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 closure s\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y \u2208 frontier s, infDist x s\u1d9c = dist x y"}, {"tactic": "rw [closure_compl] at hys", "annotated_tactic": ["rw [<a>closure_compl</a>] at hys", [{"full_name": "closure_compl", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [551, 9], "def_end_pos": [551, 22]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 closure s\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y \u2208 frontier s, infDist x s\u1d9c = dist x y", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y \u2208 frontier s, infDist x s\u1d9c = dist x y"}, {"tactic": "refine \u27e8y, \u27e8Metric.closedBall_infDist_compl_subset_closure hx <|\n  Metric.mem_closedBall.2 <| ge_of_eq ?_, hys\u27e9, hyd\u27e9", "annotated_tactic": ["refine \u27e8y, \u27e8<a>Metric.closedBall_infDist_compl_subset_closure</a> hx <|\n    <a>Metric.mem_closedBall</a>.2 <| <a>ge_of_eq</a> ?_, hys\u27e9, hyd\u27e9", [{"full_name": "Metric.closedBall_infDist_compl_subset_closure", "def_path": "Mathlib/Analysis/NormedSpace/RieszLemma.lean", "def_pos": [108, 9], "def_end_pos": [108, 55]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [468, 17], "def_end_pos": [468, 31]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [330, 9], "def_end_pos": [330, 17]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y \u2208 frontier s, infDist x s\u1d9c = dist x y", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 infDist x s\u1d9c = dist y x"}, {"tactic": "rwa [dist_comm]", "annotated_tactic": ["rwa [<a>dist_comm</a>]", [{"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : infDist x s\u1d9c = dist x y\n\u22a2 infDist x s\u1d9c = dist y x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.le_def", "start": [155, 1], "end": [155, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "subset_antisymm_iff", "start": [664, 1], "end": [665, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean", "full_name": "iSup_eq_iSup_subseq_of_monotone", "start": [326, 1], "end": [332, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.product_cons", "start": [301, 1], "end": [302, 30], "traced_tactics": [{"tactic": "simp [SProd.sprod, product]", "annotated_tactic": ["simp [<a>SProd.sprod</a>, <a>product</a>]", [{"full_name": "SProd.sprod", "def_path": "Mathlib/Data/SProd.lean", "def_pos": [29, 3], "def_end_pos": [29, 8]}, {"full_name": "Multiset.product", "def_path": "Mathlib/Data/Multiset/Bind.lean", "def_pos": [272, 5], "def_end_pos": [272, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\n\u22a2 s \u00d7\u02e2 (b ::\u2098 t) = map (fun a => (a, b)) s + s \u00d7\u02e2 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.deriv_char_of_ne", "start": [201, 1], "end": [202, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean", "full_name": "integrable_rpow_mul_exp_neg_mul_sq", "start": [110, 1], "end": [127, 64], "traced_tactics": [{"tactic": "rw [\u2190 integrableOn_univ, \u2190 @Iio_union_Ici _ _ (0 : \u211d), integrableOn_union,\n  integrableOn_Ici_iff_integrableOn_Ioi]", "annotated_tactic": ["rw [\u2190 <a>integrableOn_univ</a>, \u2190 @<a>Iio_union_Ici</a> _ _ (0 : \u211d), <a>integrableOn_union</a>,\n    <a>integrableOn_Ici_iff_integrableOn_Ioi</a>]", [{"full_name": "MeasureTheory.integrableOn_univ", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [103, 9], "def_end_pos": [103, 26]}, {"full_name": "Set.Iio_union_Ici", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [1267, 9], "def_end_pos": [1267, 22]}, {"full_name": "MeasureTheory.integrableOn_union", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [191, 9], "def_end_pos": [191, 27]}, {"full_name": "integrableOn_Ici_iff_integrableOn_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [782, 9], "def_end_pos": [782, 46]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 Integrable (fun x => x ^ s * rexp (-b * x ^ 2)) volume", "state_after": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume \u2227\n    IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume"}, {"tactic": "refine \u27e8?_, integrableOn_rpow_mul_exp_neg_mul_sq hb hs\u27e9", "annotated_tactic": ["refine \u27e8?_, <a>integrableOn_rpow_mul_exp_neg_mul_sq</a> hb hs\u27e9", [{"full_name": "integrableOn_rpow_mul_exp_neg_mul_sq", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean", "def_pos": [104, 9], "def_end_pos": [104, 45]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume \u2227\n    IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume", "state_after": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume"}, {"tactic": "rw [\u2190 (Measure.measurePreserving_neg (volume : Measure \u211d)).integrableOn_comp_preimage\n    (Homeomorph.neg \u211d).measurableEmbedding]", "annotated_tactic": ["rw [\u2190 (<a>Measure.measurePreserving_neg</a> (<a>volume</a> : <a>Measure</a> \u211d)).<a>integrableOn_comp_preimage</a>\n      (<a>Homeomorph.neg</a> \u211d).<a>measurableEmbedding</a>]", [{"full_name": "MeasureTheory.Measure.measurePreserving_neg", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [431, 3], "def_end_pos": [431, 14]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [367, 3], "def_end_pos": [367, 9]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [77, 11], "def_end_pos": [77, 18]}, {"full_name": "MeasureTheory.MeasurePreserving.integrableOn_comp_preimage", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [256, 9], "def_end_pos": [256, 53]}, {"full_name": "Homeomorph.neg", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [353, 3], "def_end_pos": [353, 14]}, {"full_name": "Homeomorph.measurableEmbedding", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [577, 7], "def_end_pos": [577, 37]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Iio 0) volume", "state_after": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn ((fun x => x ^ s * rexp (-b * x ^ 2)) \u2218 Neg.neg) (Neg.neg \u207b\u00b9' Iio 0) volume"}, {"tactic": "simp only [Function.comp, neg_sq, neg_preimage, preimage_neg_Iio, neg_neg, neg_zero]", "annotated_tactic": ["simp only [<a>Function.comp</a>, <a>neg_sq</a>, <a>neg_preimage</a>, <a>preimage_neg_Iio</a>, <a>neg_neg</a>, <a>neg_zero</a>]", [{"full_name": "Function.comp", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "neg_sq", "def_path": "Mathlib/Algebra/Ring/Commute.lean", "def_pos": [196, 7], "def_end_pos": [196, 13]}, {"full_name": "Set.neg_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [185, 3], "def_end_pos": [185, 14]}, {"full_name": "Set.preimage_neg_Iio", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [232, 9], "def_end_pos": [232, 25]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn ((fun x => x ^ s * rexp (-b * x ^ 2)) \u2218 Neg.neg) (Neg.neg \u207b\u00b9' Iio 0) volume", "state_after": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume"}, {"tactic": "apply Integrable.mono' (integrableOn_rpow_mul_exp_neg_mul_sq hb hs)", "annotated_tactic": ["apply <a>Integrable.mono'</a> (<a>integrableOn_rpow_mul_exp_neg_mul_sq</a> hb hs)", [{"full_name": "MeasureTheory.Integrable.mono'", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [465, 9], "def_end_pos": [465, 25]}, {"full_name": "integrableOn_rpow_mul_exp_neg_mul_sq", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean", "def_pos": [104, 9], "def_end_pos": [104, 45]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 IntegrableOn (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume", "state_after": "case hf\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 AEStronglyMeasurable (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (volume.restrict (Ioi 0))\n\ncase h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202volume.restrict (Ioi 0), \u2016(-a) ^ s * rexp (-b * a ^ 2)\u2016 \u2264 a ^ s * rexp (-b * a ^ 2)"}, {"tactic": "apply Measurable.aestronglyMeasurable", "annotated_tactic": ["apply <a>Measurable.aestronglyMeasurable</a>", [{"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 9], "def_end_pos": [1322, 47]}]], "state_before": "case hf\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 AEStronglyMeasurable (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (volume.restrict (Ioi 0))", "state_after": "case hf.hf\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 Measurable fun x => (-x) ^ s * rexp (-b * x ^ 2)"}, {"tactic": "exact (measurable_id'.neg.pow measurable_const).mul\n  ((measurable_id'.pow measurable_const).const_mul (-b)).exp", "annotated_tactic": ["exact (measurable_id'.neg.pow <a>measurable_const</a>).<a>mul</a>\n      ((measurable_id'.pow <a>measurable_const</a>).<a>const_mul</a> (-b)).<a>exp</a>", [{"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [586, 9], "def_end_pos": [586, 25]}, {"full_name": "Measurable.mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [140, 9], "def_end_pos": [140, 23]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [586, 9], "def_end_pos": [586, 25]}, {"full_name": "Measurable.const_mul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [105, 9], "def_end_pos": [105, 29]}, {"full_name": "Measurable.exp", "def_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "def_pos": [154, 9], "def_end_pos": [154, 23]}]], "state_before": "case hf.hf\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 Measurable fun x => (-x) ^ s * rexp (-b * x ^ 2)", "state_after": "no goals"}, {"tactic": "have : MeasurableSet (Ioi (0 : \u211d)) := measurableSet_Ioi", "annotated_tactic": ["have : <a>MeasurableSet</a> (<a>Ioi</a> (0 : \u211d)) := <a>measurableSet_Ioi</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [187, 9], "def_end_pos": [187, 26]}]], "state_before": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202volume.restrict (Ioi 0), \u2016(-a) ^ s * rexp (-b * a ^ 2)\u2016 \u2264 a ^ s * rexp (-b * a ^ 2)", "state_after": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202volume.restrict (Ioi 0), \u2016(-a) ^ s * rexp (-b * a ^ 2)\u2016 \u2264 a ^ s * rexp (-b * a ^ 2)"}, {"tactic": "filter_upwards [ae_restrict_mem this] with x hx", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> this] with x hx", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [653, 9], "def_end_pos": [653, 24]}]], "state_before": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202volume.restrict (Ioi 0), \u2016(-a) ^ s * rexp (-b * a ^ 2)\u2016 \u2264 a ^ s * rexp (-b * a ^ 2)", "state_after": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2016(-x) ^ s * rexp (-b * x ^ 2)\u2016 \u2264 x ^ s * rexp (-b * x ^ 2)"}, {"tactic": "have h'x : 0 \u2264 x := le_of_lt hx", "annotated_tactic": ["have h'x : 0 \u2264 x := <a>le_of_lt</a> hx", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2016(-x) ^ s * rexp (-b * x ^ 2)\u2016 \u2264 x ^ s * rexp (-b * x ^ 2)", "state_after": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 \u2016(-x) ^ s * rexp (-b * x ^ 2)\u2016 \u2264 x ^ s * rexp (-b * x ^ 2)"}, {"tactic": "rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (exp_pos _).le]", "annotated_tactic": ["rw [<a>Real.norm_eq_abs</a>, <a>abs_mul</a>, <a>abs_of_nonneg</a> (<a>exp_pos</a> _).<a>le</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [48, 7], "def_end_pos": [48, 14]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 \u2016(-x) ^ s * rexp (-b * x ^ 2)\u2016 \u2264 x ^ s * rexp (-b * x ^ 2)", "state_after": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 |(-x) ^ s| * rexp (-b * x ^ 2) \u2264 x ^ s * rexp (-b * x ^ 2)"}, {"tactic": "apply mul_le_mul_of_nonneg_right _ (exp_pos _).le", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_right</a> _ (<a>exp_pos</a> _).<a>le</a>", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [210, 9], "def_end_pos": [210, 35]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case h\nb : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 |(-x) ^ s| * rexp (-b * x ^ 2) \u2264 x ^ s * rexp (-b * x ^ 2)", "state_after": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 |(-x) ^ s| \u2264 x ^ s"}, {"tactic": "simpa [abs_of_nonneg h'x] using abs_rpow_le_abs_rpow (-x) s", "annotated_tactic": ["simpa [<a>abs_of_nonneg</a> h'x] using <a>abs_rpow_le_abs_rpow</a> (-x) s", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Real.abs_rpow_le_abs_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [179, 9], "def_end_pos": [179, 29]}]], "state_before": "b : \u211d\nhb : 0 < b\ns : \u211d\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : \u211d\nhx : x \u2208 Ioi 0\nh'x : 0 \u2264 x\n\u22a2 |(-x) ^ s| \u2264 x ^ s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "OrderTop.tendsto_atTop_nhds", "start": [1012, 1], "end": [1014, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/Sites.lean", "full_name": "TopCat.Presheaf.covering_presieve_eq_self", "start": [90, 1], "end": [94, 94], "traced_tactics": [{"tactic": "funext Z", "annotated_tactic": ["funext Z", []], "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R"}, {"tactic": "ext f", "annotated_tactic": ["ext f", []], "state_before": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R"}, {"tactic": "exact \u27e8fun \u27e8\u27e8_, f', h\u27e9, rfl\u27e9 => by rwa [Subsingleton.elim f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, rfl\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun \u27e8\u27e8_, f', h\u27e9, <a>rfl</a>\u27e9 => by rwa [<a>Subsingleton.elim</a> f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, <a>rfl</a>\u27e9\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R", "state_after": "no goals"}, {"tactic": "rwa [Subsingleton.elim f f']", "annotated_tactic": ["rwa [<a>Subsingleton.elim</a> f f']", [{"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}]], "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\nx\u271d : f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y\nf' : Z \u27f6 Y\nh : R f'\n\u22a2 f \u2208 R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "full_name": "Finset.prod_eq_prod_Ico_succ_bot", "start": [118, 1], "end": [121, 54], "traced_tactics": [{"tactic": "have ha : a \u2209 Ico (a + 1) b := by simp", "annotated_tactic": ["have ha : a \u2209 <a>Ico</a> (a + 1) b := by simp", [{"full_name": "Finset.Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [296, 5], "def_end_pos": [296, 8]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 M\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 M\n\u22a2 \u220f k \u2208 Ico a b, f k = f a * \u220f k \u2208 Ico (a + 1) b, f k", "state_after": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 M\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 M\nha : a \u2209 Ico (a + 1) b\n\u22a2 \u220f k \u2208 Ico a b, f k = f a * \u220f k \u2208 Ico (a + 1) b, f k"}, {"tactic": "rw [\u2190 prod_insert ha, Nat.Ico_insert_succ_left hab]", "annotated_tactic": ["rw [\u2190 <a>prod_insert</a> ha, <a>Nat.Ico_insert_succ_left</a> hab]", [{"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [351, 9], "def_end_pos": [351, 20]}, {"full_name": "Nat.Ico_insert_succ_left", "def_path": "Mathlib/Order/Interval/Finset/Nat.lean", "def_pos": [192, 9], "def_end_pos": [192, 29]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 M\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 M\nha : a \u2209 Ico (a + 1) b\n\u22a2 \u220f k \u2208 Ico a b, f k = f a * \u220f k \u2208 Ico (a + 1) b, f k", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 M\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 M\n\u22a2 a \u2209 Ico (a + 1) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.translationNumber_conj_eq'", "start": [763, 1], "end": [765, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean", "full_name": "Orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two", "start": [46, 1], "end": [50, 66], "traced_tactics": [{"tactic": "rw [\u2190 neg_inj, oangle_rev, \u2190 oangle_neg_orientation_eq_neg, neg_inj] at h \u22a2", "annotated_tactic": ["rw [\u2190 <a>neg_inj</a>, <a>oangle_rev</a>, \u2190 <a>oangle_neg_orientation_eq_neg</a>, <a>neg_inj</a>] at h \u22a2", [{"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}, {"full_name": "Orientation.oangle_rev", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "def_pos": [193, 9], "def_end_pos": [193, 19]}, {"full_name": "Orientation.oangle_neg_orientation_eq_neg", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 38]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : o.oangle x y = \u2191(\u03c0 / 2)\n\u22a2 o.oangle (x + y) y = \u2191(Real.arccos (\u2016y\u2016 / \u2016x + y\u2016))", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : (-o).oangle y x = \u2191(\u03c0 / 2)\n\u22a2 (-o).oangle y (x + y) = \u2191(Real.arccos (\u2016y\u2016 / \u2016x + y\u2016))"}, {"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : (-o).oangle y x = \u2191(\u03c0 / 2)\n\u22a2 (-o).oangle y (x + y) = \u2191(Real.arccos (\u2016y\u2016 / \u2016x + y\u2016))", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : (-o).oangle y x = \u2191(\u03c0 / 2)\n\u22a2 (-o).oangle y (y + x) = \u2191(Real.arccos (\u2016y\u2016 / \u2016y + x\u2016))"}, {"tactic": "exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h", "annotated_tactic": ["exact (-o).<a>oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two</a> h", [{"full_name": "Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean", "def_pos": [36, 9], "def_end_pos": [36, 59]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : (-o).oangle y x = \u2191(\u03c0 / 2)\n\u22a2 (-o).oangle y (y + x) = \u2191(Real.arccos (\u2016y\u2016 / \u2016y + x\u2016))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "full_name": "PartialEquiv.source_inter_preimage_target_inter", "start": [525, 1], "end": [527, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Sqrt.lean", "full_name": "Real.mul_self_sqrt", "start": [166, 1], "end": [167, 80], "traced_tactics": [{"tactic": "rw [Real.sqrt, \u2190 NNReal.coe_mul, NNReal.mul_self_sqrt, Real.coe_toNNReal _ h]", "annotated_tactic": ["rw [<a>Real.sqrt</a>, \u2190 <a>NNReal.coe_mul</a>, <a>NNReal.mul_self_sqrt</a>, <a>Real.coe_toNNReal</a> _ h]", [{"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [129, 19], "def_end_pos": [129, 23]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [185, 19], "def_end_pos": [185, 26]}, {"full_name": "NNReal.mul_self_sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [56, 15], "def_end_pos": [56, 28]}, {"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 33]}]], "state_before": "x y : \u211d\nh : 0 \u2264 x\n\u22a2 \u221ax * \u221ax = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Asymptotics.lean", "full_name": "Asymptotics.IsBigO.integrable", "start": [47, 1], "end": [50, 64], "traced_tactics": [{"tactic": "rewrite [\u2190 integrableAtFilter_top] at *", "annotated_tactic": ["rewrite [\u2190 <a>integrableAtFilter_top</a>] at *", [{"full_name": "MeasureTheory.integrableAtFilter_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [490, 9], "def_end_pos": [490, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\na b : \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b1\nhfm : AEStronglyMeasurable f \u03bc\nhf : f =O[\u22a4] g\nhg : Integrable g \u03bc\n\u22a2 Integrable f \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\na b : \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b1\nhfm : AEStronglyMeasurable f \u03bc\nhf : f =O[\u22a4] g\nhg : IntegrableAtFilter g \u22a4 \u03bc\n\u22a2 IntegrableAtFilter f \u22a4 \u03bc"}, {"tactic": "exact hf.integrableAtFilter \u27e8univ, univ_mem, hfm.restrict\u27e9 hg", "annotated_tactic": ["exact hf.integrableAtFilter \u27e8<a>univ</a>, <a>univ_mem</a>, hfm.restrict\u27e9 hg", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "Filter.univ_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [147, 9], "def_end_pos": [147, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\na b : \u03b1\n\u03bc : Measure \u03b1\nl : Filter \u03b1\nhfm : AEStronglyMeasurable f \u03bc\nhf : f =O[\u22a4] g\nhg : IntegrableAtFilter g \u22a4 \u03bc\n\u22a2 IntegrableAtFilter f \u22a4 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.frequently_iff_forall_eventually_exists_and", "start": [1340, 1], "end": [1343, 60], "traced_tactics": [{"tactic": "simpa only [and_not_self_iff, exists_false] using H hp", "annotated_tactic": ["simpa only [<a>and_not_self_iff</a>, <a>exists_false</a>] using H hp", [{"full_name": "and_not_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [36, 9], "def_end_pos": [36, 25]}, {"full_name": "exists_false", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [263, 17], "def_end_pos": [263, 29]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\np : \u03b1 \u2192 Prop\nf : Filter \u03b1\nH : \u2200 {q : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in f, q x) \u2192 \u2203 x, p x \u2227 q x\nhp : \u2200\u1da0 (x : \u03b1) in f, \u00ac(fun x => p x) x\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.extraction_forall_of_eventually'", "start": [558, 1], "end": [560, 66], "traced_tactics": [{"tactic": "simp [eventually_atTop, h]", "annotated_tactic": ["simp [<a>eventually_atTop</a>, h]", [{"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nP : \u2115 \u2192 \u2115 \u2192 Prop\nh : \u2200 (n : \u2115), \u2203 N, \u2200 k \u2265 N, P n k\n\u22a2 \u2200 (n : \u2115), \u2200\u1da0 (k : \u2115) in atTop, P n k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Restrict.lean", "full_name": "Matroid.restrict_eq_restrict_iff", "start": [198, 1], "end": [203, 88], "traced_tactics": [{"tactic": "refine \u27e8fun h I hIX \u21a6 ?_, fun h \u21a6 eq_of_indep_iff_indep_forall rfl fun I (hI : I \u2286 X) \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h I hIX \u21a6 ?_, fun h \u21a6 <a>eq_of_indep_iff_indep_forall</a> <a>rfl</a> fun I (hI : I \u2286 X) \u21a6 ?_\u27e9", [{"full_name": "Matroid.eq_of_indep_iff_indep_forall", "def_path": "Mathlib/Data/Matroid/Basic.lean", "def_pos": [652, 9], "def_end_pos": [652, 37]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I J X\u271d Y : Set \u03b1\nM M' : Matroid \u03b1\nX : Set \u03b1\n\u22a2 M \u21be X = M' \u21be X \u2194 \u2200 I \u2286 X, M.Indep I \u2194 M'.Indep I", "state_after": "case refine_1\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I\u271d J X\u271d Y : Set \u03b1\nM M' : Matroid \u03b1\nX : Set \u03b1\nh : M \u21be X = M' \u21be X\nI : Set \u03b1\nhIX : I \u2286 X\n\u22a2 M.Indep I \u2194 M'.Indep I\n\ncase refine_2\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I\u271d J X\u271d Y : Set \u03b1\nM M' : Matroid \u03b1\nX : Set \u03b1\nh : \u2200 I \u2286 X, M.Indep I \u2194 M'.Indep I\nI : Set \u03b1\nhI : I \u2286 X\n\u22a2 (M \u21be X).Indep I \u2194 (M' \u21be X).Indep I"}, {"tactic": "rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]", "annotated_tactic": ["rw [<a>restrict_indep_iff</a>, <a>and_iff_left</a> hI, <a>restrict_indep_iff</a>, <a>and_iff_left</a> hI, h _ hI]", [{"full_name": "Matroid.restrict_indep_iff", "def_path": "Mathlib/Data/Matroid/Restrict.lean", "def_pos": [123, 17], "def_end_pos": [123, 35]}, {"full_name": "and_iff_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}, {"full_name": "Matroid.restrict_indep_iff", "def_path": "Mathlib/Data/Matroid/Restrict.lean", "def_pos": [123, 17], "def_end_pos": [123, 35]}, {"full_name": "and_iff_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I\u271d J X\u271d Y : Set \u03b1\nM M' : Matroid \u03b1\nX : Set \u03b1\nh : \u2200 I \u2286 X, M.Indep I \u2194 M'.Indep I\nI : Set \u03b1\nhI : I \u2286 X\n\u22a2 (M \u21be X).Indep I \u2194 (M' \u21be X).Indep I", "state_after": "no goals"}, {"tactic": "rw [\u2190 and_iff_left (a := (M.Indep I)) hIX, \u2190 and_iff_left (a := (M'.Indep I)) hIX,\n  \u2190 restrict_indep_iff, h, restrict_indep_iff]", "annotated_tactic": ["rw [\u2190 <a>and_iff_left</a> (a := (M.Indep I)) hIX, \u2190 <a>and_iff_left</a> (a := (M'.Indep I)) hIX,\n      \u2190 <a>restrict_indep_iff</a>, h, <a>restrict_indep_iff</a>]", [{"full_name": "and_iff_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}, {"full_name": "and_iff_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [69, 9], "def_end_pos": [69, 21]}, {"full_name": "Matroid.restrict_indep_iff", "def_path": "Mathlib/Data/Matroid/Restrict.lean", "def_pos": [123, 17], "def_end_pos": [123, 35]}, {"full_name": "Matroid.restrict_indep_iff", "def_path": "Mathlib/Data/Matroid/Restrict.lean", "def_pos": [123, 17], "def_end_pos": [123, 35]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I\u271d J X\u271d Y : Set \u03b1\nM M' : Matroid \u03b1\nX : Set \u03b1\nh : M \u21be X = M' \u21be X\nI : Set \u03b1\nhIX : I \u2286 X\n\u22a2 M.Indep I \u2194 M'.Indep I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.SameCycle.rfl", "start": [60, 1], "end": [61, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.some_inter_some", "start": [840, 1], "end": [841, 19], "traced_tactics": [{"tactic": "simp [inter_def]", "annotated_tactic": ["simp [<a>inter_def</a>]", [{"full_name": "Part.inter_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [706, 9], "def_end_pos": [706, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Inter \u03b1\na b : \u03b1\n\u22a2 some a \u2229 some b = some (a \u2229 b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "full_name": "FractionalIdeal.coeIdeal_le_coeIdeal'", "start": [283, 1], "end": [285, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mem_inv_smul_finset_iff", "start": [2103, 1], "end": [2104, 51], "traced_tactics": [{"tactic": "rw [\u2190 smul_mem_smul_finset_iff a, smul_inv_smul]", "annotated_tactic": ["rw [\u2190 <a>smul_mem_smul_finset_iff</a> a, <a>smul_inv_smul</a>]", [{"full_name": "Finset.smul_mem_smul_finset_iff", "def_path": "Mathlib/Data/Finset/Pointwise.lean", "def_pos": [2091, 9], "def_end_pos": [2091, 33]}, {"full_name": "smul_inv_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [35, 9], "def_end_pos": [35, 22]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns t : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 a\u207b\u00b9 \u2022 s \u2194 a \u2022 b \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "integral_comp_neg_Iic", "start": [85, 1], "end": [91, 90], "traced_tactics": [{"tactic": "have A : MeasurableEmbedding fun x : \u211d => -x :=\n  (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding", "annotated_tactic": ["have A : <a>MeasurableEmbedding</a> fun x : \u211d => -x :=\n    (<a>Homeomorph.neg</a> \u211d).closedEmbedding.measurableEmbedding", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean", "def_pos": [55, 11], "def_end_pos": [55, 30]}, {"full_name": "Homeomorph.neg", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [353, 3], "def_end_pos": [353, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "have := MeasurableEmbedding.setIntegral_map (\u03bc := volume) A f (Ici (-c))", "annotated_tactic": ["have := <a>MeasurableEmbedding.setIntegral_map</a> (\u03bc := <a>volume</a>) A f (<a>Ici</a> (-c))", [{"full_name": "MeasurableEmbedding.setIntegral_map", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [577, 9], "def_end_pos": [577, 51]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [367, 3], "def_end_pos": [367, 9]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "rw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this", "annotated_tactic": ["rw [<a>Measure.map_neg_eq_self</a> (<a>volume</a> : <a>Measure</a> \u211d)] at this", [{"full_name": "MeasureTheory.Measure.map_neg_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [423, 3], "def_end_pos": [423, 14]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [367, 3], "def_end_pos": [367, 9]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [77, 11], "def_end_pos": [77, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "simp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>integral_Ici_eq_integral_Ioi</a>, this, <a>neg_preimage</a>, <a>preimage_neg_Ici</a>, <a>neg_neg</a>]", [{"full_name": "MeasureTheory.integral_Ici_eq_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [798, 9], "def_end_pos": [798, 37]}, {"full_name": "Set.neg_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [185, 3], "def_end_pos": [185, 14]}, {"full_name": "Set.preimage_neg_Ici", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [217, 9], "def_end_pos": [217, 25]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Surjective.of_comp", "start": [167, 1], "end": [169, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Fintype.lean", "full_name": "Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range", "start": [85, 1], "end": [87, 82], "traced_tactics": [{"tactic": "rwa [Equiv.Perm.viaFintypeEmbedding, Equiv.Perm.extendDomain_apply_not_subtype]", "annotated_tactic": ["rwa [<a>Equiv.Perm.viaFintypeEmbedding</a>, <a>Equiv.Perm.extendDomain_apply_not_subtype</a>]", [{"full_name": "Equiv.Perm.viaFintypeEmbedding", "def_path": "Mathlib/Logic/Equiv/Fintype.lean", "def_pos": [67, 5], "def_end_pos": [67, 35]}, {"full_name": "Equiv.Perm.extendDomain_apply_not_subtype", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1540, 9], "def_end_pos": [1540, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\ne : Perm \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\nh : b \u2209 Set.range \u21d1f\n\u22a2 (e.viaFintypeEmbedding f) b = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/SpecificGroups/Cyclic.lean", "full_name": "IsCyclic.unique_zpow_zmod", "start": [386, 1], "end": [394, 14], "traced_tactics": [{"tactic": "obtain \u27e8n, rfl\u27e9 := ha x", "annotated_tactic": ["obtain \u27e8n, rfl\u27e9 := ha x", []], "state_before": "\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nx : \u03b1\n\u22a2 \u2203! n, x = a ^ n.val", "state_after": "case intro\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\n\u22a2 \u2203! n_1, (fun x => a ^ x) n = a ^ n_1.val"}, {"tactic": "refine \u27e8n, (?_ : a ^ n = _), fun y (hy : a ^ n = _) \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8n, (?_ : a ^ n = _), fun y (hy : a ^ n = _) \u21a6 ?_\u27e9", []], "state_before": "case intro\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\n\u22a2 \u2203! n_1, (fun x => a ^ x) n = a ^ n_1.val", "state_after": "case intro.refine_1\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\n\u22a2 a ^ n = a ^ (\u2191n).val\n\ncase intro.refine_2\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\ny : ZMod (Fintype.card \u03b1)\nhy : a ^ n = a ^ y.val\n\u22a2 y = \u2191n"}, {"tactic": "rw [\u2190 zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,\n  Int.modEq_comm, Int.modEq_iff_add_fac, \u2190 ZMod.intCast_eq_iff]", "annotated_tactic": ["rw [\u2190 <a>zpow_natCast</a>, <a>zpow_eq_zpow_iff_modEq</a>, <a>orderOf_eq_card_of_forall_mem_zpowers</a> ha,\n      <a>Int.modEq_comm</a>, <a>Int.modEq_iff_add_fac</a>, \u2190 <a>ZMod.intCast_eq_iff</a>]", [{"full_name": "zpow_natCast", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}, {"full_name": "zpow_eq_zpow_iff_modEq", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [961, 9], "def_end_pos": [961, 31]}, {"full_name": "orderOf_eq_card_of_forall_mem_zpowers", "def_path": "Mathlib/GroupTheory/SpecificGroups/Cyclic.lean", "def_pos": [189, 9], "def_end_pos": [189, 46]}, {"full_name": "Int.modEq_comm", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "Int.modEq_iff_add_fac", "def_path": "Mathlib/Data/Int/ModEq.lean", "def_pos": [98, 9], "def_end_pos": [98, 26]}, {"full_name": "ZMod.intCast_eq_iff", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 23]}]], "state_before": "case intro.refine_1\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\n\u22a2 a ^ n = a ^ (\u2191n).val", "state_after": "no goals"}, {"tactic": "rw [\u2190 zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,\n  \u2190 ZMod.intCast_eq_intCast_iff] at hy", "annotated_tactic": ["rw [\u2190 <a>zpow_natCast</a>, <a>zpow_eq_zpow_iff_modEq</a>, <a>orderOf_eq_card_of_forall_mem_zpowers</a> ha,\n      \u2190 <a>ZMod.intCast_eq_intCast_iff</a>] at hy", [{"full_name": "zpow_natCast", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}, {"full_name": "zpow_eq_zpow_iff_modEq", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [961, 9], "def_end_pos": [961, 31]}, {"full_name": "orderOf_eq_card_of_forall_mem_zpowers", "def_path": "Mathlib/GroupTheory/SpecificGroups/Cyclic.lean", "def_pos": [189, 9], "def_end_pos": [189, 46]}, {"full_name": "ZMod.intCast_eq_intCast_iff", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 31]}]], "state_before": "case intro.refine_2\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\ny : ZMod (Fintype.card \u03b1)\nhy : a ^ n = a ^ y.val\n\u22a2 y = \u2191n", "state_after": "case intro.refine_2\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\ny : ZMod (Fintype.card \u03b1)\nhy : \u2191n = \u2191\u2191y.val\n\u22a2 y = \u2191n"}, {"tactic": "simp [hy]", "annotated_tactic": ["simp [hy]", []], "state_before": "case intro.refine_2\n\u03b1 : Type u\na : \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nha : \u2200 (x : \u03b1), x \u2208 zpowers a\nn : \u2124\ny : ZMod (Fintype.card \u03b1)\nhy : \u2191n = \u2191\u2191y.val\n\u22a2 y = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/TietzeExtension.lean", "full_name": "BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_closedEmbedding", "start": [314, 1], "end": [417, 32], "traced_tactics": [{"tactic": "inhabit X", "annotated_tactic": ["inhabit X", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "obtain \u27e8a, ha\u27e9 : \u2203 a, IsGLB (range f) a := \u27e8_, isGLB_ciInf f.isBounded_range.bddBelow\u27e9", "annotated_tactic": ["obtain \u27e8a, ha\u27e9 : \u2203 a, <a>IsGLB</a> (<a>range</a> f) a := \u27e8_, <a>isGLB_ciInf</a> f.isBounded_range.bddBelow\u27e9", [{"full_name": "IsGLB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [81, 5], "def_end_pos": [81, 10]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "isGLB_ciInf", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [519, 9], "def_end_pos": [519, 20]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "obtain \u27e8b, hb\u27e9 : \u2203 b, IsLUB (range f) b := \u27e8_, isLUB_ciSup f.isBounded_range.bddAbove\u27e9", "annotated_tactic": ["obtain \u27e8b, hb\u27e9 : \u2203 b, <a>IsLUB</a> (<a>range</a> f) b := \u27e8_, <a>isLUB_ciSup</a> f.isBounded_range.bddAbove\u27e9", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "isLUB_ciSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 20]}]], "state_before": "case intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "have hmem : \u2200 x, f x \u2208 Icc a b := fun x => \u27e8ha.1 \u27e8x, rfl\u27e9, hb.1 \u27e8x, rfl\u27e9\u27e9", "annotated_tactic": ["have hmem : \u2200 x, f x \u2208 <a>Icc</a> a b := fun x => \u27e8ha.1 \u27e8x, <a>rfl</a>\u27e9, hb.1 \u27e8x, <a>rfl</a>\u27e9\u27e9", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "have hle : a \u2264 b := (hmem default).1.trans (hmem default).2", "annotated_tactic": ["have hle : a \u2264 b := (hmem <a>default</a>).1.<a>trans</a> (hmem <a>default</a>).2", [{"full_name": "Inhabited.default", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [697, 3], "def_end_pos": [697, 10]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "Inhabited.default", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [697, 3], "def_end_pos": [697, 10]}]], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "rcases hle.eq_or_lt with (rfl | hlt)", "annotated_tactic": ["rcases hle.eq_or_lt with (rfl | hlt)", []], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f\n\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "set c := (a + b) / 2", "annotated_tactic": ["set c := (a + b) / 2", []], "state_before": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "have hac : a < c := left_lt_add_div_two.2 hlt", "annotated_tactic": ["have hac : a < c := <a>left_lt_add_div_two</a>.2 hlt", [{"full_name": "left_lt_add_div_two", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [480, 9], "def_end_pos": [480, 28]}]], "state_before": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "have hcb : c < b := add_div_two_lt_right.2 hlt", "annotated_tactic": ["have hcb : c < b := <a>add_div_two_lt_right</a>.2 hlt", [{"full_name": "add_div_two_lt_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [483, 9], "def_end_pos": [483, 29]}]], "state_before": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "have hsub : c - a = b - c := by\n  field_simp [c]\n  ring", "annotated_tactic": ["have hsub : c - a = b - c := by\n    field_simp [c]\n    ring", []], "state_before": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "choose xl hxl hgb using hg_mem", "annotated_tactic": ["choose xl hxl hgb using hg_mem", []], "state_before": "case intro.intro.inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b\nhgf : \u21d1g \u2218 e = \u21d1f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "rcases em (\u2203 x, f x = b) with (\u27e8x, rfl\u27e9 | hb')", "annotated_tactic": ["rcases <a>em</a> (\u2203 x, f x = b) with (\u27e8x, rfl\u27e9 | hb')", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}]], "state_before": "case intro.intro.inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nx : X\nhb : IsLUB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc a (f x)\nhle : a \u2264 f x\nhlt : a < f x\nc : \u211d := (a + f x) / 2\nhac : a < c\nhcb : c < f x\nhsub : c - a = f x - c\nhgb : \u2200 (y : Y), g y \u2264 f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f\n\ncase intro.intro.inr.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "rcases exists_bounded_mem_Icc_of_closed_of_le\n    (he.isClosed_range.union <| isClosed_Iic.preimage g.continuous)\n    (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with\n  \u27e8dg, dg0, dgb, dgmem\u27e9", "annotated_tactic": ["rcases <a>exists_bounded_mem_Icc_of_closed_of_le</a>\n      (he.isClosed_range.union <| isClosed_Iic.preimage g.continuous)\n      (isClosed_singleton.preimage g.continuous) hd (<a>sub_nonneg</a>.2 hcb.le) with\n    \u27e8dg, dg0, dgb, dgmem\u27e9", [{"full_name": "exists_bounded_mem_Icc_of_closed_of_le", "def_path": "Mathlib/Topology/UrysohnsBounded.lean", "def_pos": [49, 9], "def_end_pos": [49, 47]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case intro.intro.inr.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "replace hgf : \u2200 x, (g - dg) (e x) = f x := by\n  intro x\n  simp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]", "annotated_tactic": ["replace hgf : \u2200 x, (g - dg) (e x) = f x := by\n    intro x\n    simp [dg0 (<a>Or.inl</a> <| <a>mem_range_self</a> _), \u2190 hgf]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "refine \u27e8g - dg, fun y => ?_, funext hgf\u27e9", "annotated_tactic": ["refine \u27e8g - dg, fun y => ?_, <a>funext</a> hgf\u27e9", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "rcases hb.exists_between hyb with \u27e8_, \u27e8xu, rfl\u27e9, hyxu, _\u27e9", "annotated_tactic": ["rcases hb.exists_between hyb with \u27e8_, \u27e8xu, rfl\u27e9, hyxu, _\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "cases' lt_or_le c (g y) with hc hc", "annotated_tactic": ["cases' <a>lt_or_le</a> c (g y) with hc hc", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [338, 9], "def_end_pos": [338, 17]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)\n\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : g y \u2264 c\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "have : \u2200 x, f x = a := by simpa using hmem", "annotated_tactic": ["have : \u2200 x, f x = a := by simpa using hmem", []], "state_before": "case intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), f x = a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "use const Y a", "annotated_tactic": ["use <a>const</a> Y a", [{"full_name": "BoundedContinuousFunction.const", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [293, 5], "def_end_pos": [293, 10]}]], "state_before": "case intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), f x = a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), f x = a\n\u22a2 (\u2200 (y : Y), \u2203 x\u2081 x\u2082, (const Y a) y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1(const Y a) \u2218 e = \u21d1f"}, {"tactic": "simp [this, Function.funext_iff]", "annotated_tactic": ["simp [this, <a>Function.funext_iff</a>]", [{"full_name": "Function.funext_iff", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [63, 9], "def_end_pos": [63, 28]}]], "state_before": "case h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), f x = a\n\u22a2 (\u2200 (y : Y), \u2203 x\u2081 x\u2082, (const Y a) y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1(const Y a) \u2218 e = \u21d1f", "state_after": "no goals"}, {"tactic": "simpa using hmem", "annotated_tactic": ["simpa using hmem", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nhb : IsLUB (range \u21d1f) a\nhmem : \u2200 (x : X), f x \u2208 Icc a a\nhle : a \u2264 a\n\u22a2 \u2200 (x : X), f x = a", "state_after": "no goals"}, {"tactic": "field_simp [c]", "annotated_tactic": ["field_simp [c]", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 c - a = b - c", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 a + b - 2 * a = b * 2 - (a + b)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 a + b - 2 * a = b * 2 - (a + b)", "state_after": "no goals"}, {"tactic": "rcases exists_extension_forall_mem_Icc_of_closedEmbedding f hmem hle he with \u27e8g, hg_mem, hgf\u27e9", "annotated_tactic": ["rcases <a>exists_extension_forall_mem_Icc_of_closedEmbedding</a> f hmem hle he with \u27e8g, hg_mem, hgf\u27e9", [{"full_name": "BoundedContinuousFunction.exists_extension_forall_mem_Icc_of_closedEmbedding", "def_path": "Mathlib/Topology/TietzeExtension.lean", "def_pos": [293, 9], "def_end_pos": [293, 59]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "rcases em (\u2203 x, f x = a) with (\u27e8x, rfl\u27e9 | ha')", "annotated_tactic": ["rcases <a>em</a> (\u2203 x, f x = a) with (\u27e8x, rfl\u27e9 | ha')", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}]], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhg_mem : \u2200 (y : Y), g y \u2208 Icc (f x) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f\n\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "rcases exists_bounded_mem_Icc_of_closed_of_le\n    (he.isClosed_range.union <| isClosed_Ici.preimage g.continuous)\n    (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with\n  \u27e8dg, dg0, dga, dgmem\u27e9", "annotated_tactic": ["rcases <a>exists_bounded_mem_Icc_of_closed_of_le</a>\n        (he.isClosed_range.union <| isClosed_Ici.preimage g.continuous)\n        (isClosed_singleton.preimage g.continuous) hd (<a>sub_nonneg</a>.2 hac.le) with\n      \u27e8dg, dg0, dga, dgmem\u27e9", [{"full_name": "exists_bounded_mem_Icc_of_closed_of_le", "def_path": "Mathlib/Topology/UrysohnsBounded.lean", "def_pos": [49, 9], "def_end_pos": [49, 47]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "case intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "replace hgf : \u2200 x, (g + dg) (e x) = f x := by\n  intro x\n  simp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]", "annotated_tactic": ["replace hgf : \u2200 x, (g + dg) (e x) = f x := by\n      intro x\n      simp [dg0 (<a>Or.inl</a> <| <a>mem_range_self</a> _), \u2190 hgf]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}]], "state_before": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f"}, {"tactic": "refine \u27e8g + dg, fun y => ?_, funext hgf\u27e9", "annotated_tactic": ["refine \u27e8g + dg, fun y => ?_, <a>funext</a> hgf\u27e9", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\n\u22a2 \u2203 x, (g + dg) y \u2208 Icc (f x) b"}, {"tactic": "rcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9", "annotated_tactic": ["rcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\n\u22a2 \u2203 x, (g + dg) y \u2208 Icc (f x) b", "state_after": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\n\u22a2 \u2203 x, (g + dg) y \u2208 Icc (f x) b"}, {"tactic": "refine \u27e8x, hxy.le, ?_\u27e9", "annotated_tactic": ["refine \u27e8x, hxy.le, ?_\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\n\u22a2 \u2203 x, (g + dg) y \u2208 Icc (f x) b", "state_after": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\n\u22a2 (g + dg) y \u2264 b"}, {"tactic": "rcases le_total c (g y) with hc | hc", "annotated_tactic": ["rcases <a>le_total</a> c (g y) with hc | hc", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [297, 9], "def_end_pos": [297, 17]}]], "state_before": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\n\u22a2 (g + dg) y \u2264 b", "state_after": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\nhc : c \u2264 g y\n\u22a2 (g + dg) y \u2264 b\n\ncase intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\nhc : g y \u2264 c\n\u22a2 (g + dg) y \u2264 b"}, {"tactic": "exact \u27e8g, fun y => \u27e8x, hg_mem _\u27e9, hgf\u27e9", "annotated_tactic": ["exact \u27e8g, fun y => \u27e8x, hg_mem _\u27e9, hgf\u27e9", []], "state_before": "case intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhg_mem : \u2200 (y : Y), g y \u2208 Icc (f x) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, g y \u2208 Icc (f x) b) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "no goals"}, {"tactic": "refine disjoint_union_left.2 \u27e8?_, Disjoint.preimage _ ?_\u27e9", "annotated_tactic": ["refine <a>disjoint_union_left</a>.2 \u27e8?_, <a>Disjoint.preimage</a> _ ?_\u27e9", [{"full_name": "Set.disjoint_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1533, 7], "def_end_pos": [1533, 26]}, {"full_name": "Disjoint.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1572, 9], "def_end_pos": [1572, 26]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})", "state_after": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 Disjoint (range e) (\u21d1g \u207b\u00b9' {a})\n\ncase refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 Disjoint (Ici c) {a}"}, {"tactic": "rw [Set.disjoint_left]", "annotated_tactic": ["rw [<a>Set.disjoint_left</a>]", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1495, 9], "def_end_pos": [1495, 22]}]], "state_before": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 Disjoint (range e) (\u21d1g \u207b\u00b9' {a})", "state_after": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 \u2200 \u2983a_1 : Y\u2984, a_1 \u2208 range e \u2192 a_1 \u2209 \u21d1g \u207b\u00b9' {a}"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = a)", "annotated_tactic": ["rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = a)", []], "state_before": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 \u2200 \u2983a_1 : Y\u2984, a_1 \u2208 range e \u2192 a_1 \u2209 \u21d1g \u207b\u00b9' {a}", "state_after": "case refine_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nx : X\nha : IsGLB (range \u21d1f) (g (e x))\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (g (e x)) b\nhle : g (e x) \u2264 b\nhlt : g (e x) < b\nc : \u211d := (g (e x) + b) / 2\nhac : g (e x) < c\nhcb : c < b\nhsub : c - g (e x) = b - c\nhg_mem : \u2200 (y : Y), g y \u2208 Icc (g (e x)) b\nha' : \u00ac\u2203 x_1, f x_1 = g (e x)\n\u22a2 False"}, {"tactic": "exact ha' \u27e8x, (congr_fun hgf x).symm\u27e9", "annotated_tactic": ["exact ha' \u27e8x, (<a>congr_fun</a> hgf x).<a>symm</a>\u27e9", [{"full_name": "congr_fun", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [78, 7], "def_end_pos": [78, 16]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case refine_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nx : X\nha : IsGLB (range \u21d1f) (g (e x))\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (g (e x)) b\nhle : g (e x) \u2264 b\nhlt : g (e x) < b\nc : \u211d := (g (e x) + b) / 2\nhac : g (e x) < c\nhcb : c < b\nhsub : c - g (e x) = b - c\nhg_mem : \u2200 (y : Y), g y \u2208 Icc (g (e x)) b\nha' : \u00ac\u2203 x_1, f x_1 = g (e x)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact Set.disjoint_singleton_right.2 hac.not_le", "annotated_tactic": ["exact <a>Set.disjoint_singleton_right</a>.2 hac.not_le", [{"full_name": "Set.disjoint_singleton_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1573, 7], "def_end_pos": [1573, 31]}]], "state_before": "case refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\n\u22a2 Disjoint (Ici c) {a}", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\n\u22a2 \u2200 (x : X), (g + dg) (e x) = f x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nx : X\n\u22a2 (g + dg) (e x) = f x"}, {"tactic": "simp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]", "annotated_tactic": ["simp [dg0 (<a>Or.inl</a> <| <a>mem_range_self</a> _), \u2190 hgf]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nhgf : \u21d1g \u2218 e = \u21d1f\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nx : X\n\u22a2 (g + dg) (e x) = f x", "state_after": "no goals"}, {"tactic": "rcases (hg_mem y).1.eq_or_lt with (rfl | hlt)", "annotated_tactic": ["rcases (hg_mem y).1.<a>eq_or_lt</a> with (rfl | hlt)", [{"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [395, 7], "def_end_pos": [395, 21]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\n\u22a2 a < (g + dg) y", "state_after": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nha : IsGLB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc (g y) b\nhle : g y \u2264 b\nhlt : g y < b\nc : \u211d := (g y + b) / 2\nhac : g y < c\nhcb : c < b\nhsub : c - g y = b - c\nhg_mem : \u2200 (y_1 : Y), g y_1 \u2208 Icc (g y) b\nha' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - g y)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - g y)\n\u22a2 g y < (g + dg) y\n\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhlt : a < g y\n\u22a2 a < (g + dg) y"}, {"tactic": "refine (lt_add_iff_pos_right _).2 ?_", "annotated_tactic": ["refine (<a>lt_add_iff_pos_right</a> _).2 ?_", [{"full_name": "lt_add_iff_pos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [563, 30], "def_end_pos": [563, 50]}]], "state_before": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nha : IsGLB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc (g y) b\nhle : g y \u2264 b\nhlt : g y < b\nc : \u211d := (g y + b) / 2\nhac : g y < c\nhcb : c < b\nhsub : c - g y = b - c\nhg_mem : \u2200 (y_1 : Y), g y_1 \u2208 Icc (g y) b\nha' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - g y)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - g y)\n\u22a2 g y < (g + dg) y", "state_after": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nha : IsGLB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc (g y) b\nhle : g y \u2264 b\nhlt : g y < b\nc : \u211d := (g y + b) / 2\nhac : g y < c\nhcb : c < b\nhsub : c - g y = b - c\nhg_mem : \u2200 (y_1 : Y), g y_1 \u2208 Icc (g y) b\nha' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - g y)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - g y)\n\u22a2 0 < dg y"}, {"tactic": "calc\n  0 < c - g y := sub_pos.2 hac\n  _ = dg y := (dga rfl).symm", "annotated_tactic": ["calc\n          0 < c - g y := <a>sub_pos</a>.2 hac\n          _ = dg y := (dga <a>rfl</a>).<a>symm</a>", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nha : IsGLB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc (g y) b\nhle : g y \u2264 b\nhlt : g y < b\nc : \u211d := (g y + b) / 2\nhac : g y < c\nhcb : c < b\nhsub : c - g y = b - c\nhg_mem : \u2200 (y_1 : Y), g y_1 \u2208 Icc (g y) b\nha' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - g y)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - g y)\n\u22a2 0 < dg y", "state_after": "no goals"}, {"tactic": "exact hlt.trans_le (le_add_of_nonneg_right (dgmem y).1)", "annotated_tactic": ["exact hlt.trans_le (<a>le_add_of_nonneg_right</a> (dgmem y).1)", [{"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}]], "state_before": "case inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhlt : a < g y\n\u22a2 a < (g + dg) y", "state_after": "no goals"}, {"tactic": "simp [dg0 (Or.inr hc), (hg_mem y).2]", "annotated_tactic": ["simp [dg0 (<a>Or.inr</a> hc), (hg_mem y).2]", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\nhc : c \u2264 g y\n\u22a2 (g + dg) y \u2264 b", "state_after": "no goals"}, {"tactic": "calc\n  g y + dg y \u2264 c + (c - a) := add_le_add hc (dgmem _).2\n  _ = b := by rw [hsub, add_sub_cancel]", "annotated_tactic": ["calc\n        g y + dg y \u2264 c + (c - a) := <a>add_le_add</a> hc (dgmem _).2\n        _ = b := by rw [hsub, <a>add_sub_cancel</a>]", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1330, 3], "def_end_pos": [1330, 14]}]], "state_before": "case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\nhc : g y \u2264 c\n\u22a2 (g + dg) y \u2264 b", "state_after": "no goals"}, {"tactic": "rw [hsub, add_sub_cancel]", "annotated_tactic": ["rw [hsub, <a>add_sub_cancel</a>]", [{"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1330, 3], "def_end_pos": [1330, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), g y \u2208 Icc a b\nha' : \u00ac\u2203 x, f x = a\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Ici c) (\u21d1g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Ici c)\ndga : EqOn (\u21d1dg) (Function.const Y (c - a)) (\u21d1g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), (g + dg) (e x) = f x\ny : Y\nhay : a < (g + dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g + dg) y\nhc : g y \u2264 c\n\u22a2 c + (c - a) = b", "state_after": "no goals"}, {"tactic": "exact \u27e8g, fun y => \u27e8xl y, x, hxl y, hgb y\u27e9, hgf\u27e9", "annotated_tactic": ["exact \u27e8g, fun y => \u27e8xl y, x, hxl y, hgb y\u27e9, hgf\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nx : X\nhb : IsLUB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc a (f x)\nhle : a \u2264 f x\nhlt : a < f x\nc : \u211d := (a + f x) / 2\nhac : a < c\nhcb : c < f x\nhsub : c - a = f x - c\nhgb : \u2200 (y : Y), g y \u2264 f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, g y \u2208 Icc (f x\u2081) (f x\u2082)) \u2227 \u21d1g \u2218 e = \u21d1f", "state_after": "no goals"}, {"tactic": "refine disjoint_union_left.2 \u27e8?_, Disjoint.preimage _ ?_\u27e9", "annotated_tactic": ["refine <a>disjoint_union_left</a>.2 \u27e8?_, <a>Disjoint.preimage</a> _ ?_\u27e9", [{"full_name": "Set.disjoint_union_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1533, 7], "def_end_pos": [1533, 26]}, {"full_name": "Disjoint.preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1572, 9], "def_end_pos": [1572, 26]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})", "state_after": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 Disjoint (range e) (\u21d1g \u207b\u00b9' {b})\n\ncase refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 Disjoint (Iic c) {b}"}, {"tactic": "rw [Set.disjoint_left]", "annotated_tactic": ["rw [<a>Set.disjoint_left</a>]", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1495, 9], "def_end_pos": [1495, 22]}]], "state_before": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 Disjoint (range e) (\u21d1g \u207b\u00b9' {b})", "state_after": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 \u2200 \u2983a : Y\u2984, a \u2208 range e \u2192 a \u2209 \u21d1g \u207b\u00b9' {b}"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = b)", "annotated_tactic": ["rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = b)", []], "state_before": "case refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 \u2200 \u2983a : Y\u2984, a \u2208 range e \u2192 a \u2209 \u21d1g \u207b\u00b9' {b}", "state_after": "case refine_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nx : X\nhb : IsLUB (range \u21d1f) (g (e x))\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc a (g (e x))\nhle : a \u2264 g (e x)\nhlt : a < g (e x)\nc : \u211d := (a + g (e x)) / 2\nhac : a < c\nhcb : c < g (e x)\nhsub : c - a = g (e x) - c\nhgb : \u2200 (y : Y), g y \u2264 g (e x)\nhb' : \u00ac\u2203 x_1, f x_1 = g (e x)\n\u22a2 False"}, {"tactic": "exact hb' \u27e8x, (congr_fun hgf x).symm\u27e9", "annotated_tactic": ["exact hb' \u27e8x, (<a>congr_fun</a> hgf x).<a>symm</a>\u27e9", [{"full_name": "congr_fun", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [78, 7], "def_end_pos": [78, 16]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case refine_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nx : X\nhb : IsLUB (range \u21d1f) (g (e x))\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc a (g (e x))\nhle : a \u2264 g (e x)\nhlt : a < g (e x)\nc : \u211d := (a + g (e x)) / 2\nhac : a < c\nhcb : c < g (e x)\nhsub : c - a = g (e x) - c\nhgb : \u2200 (y : Y), g y \u2264 g (e x)\nhb' : \u00ac\u2203 x_1, f x_1 = g (e x)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact Set.disjoint_singleton_right.2 hcb.not_le", "annotated_tactic": ["exact <a>Set.disjoint_singleton_right</a>.2 hcb.not_le", [{"full_name": "Set.disjoint_singleton_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1573, 7], "def_end_pos": [1573, 31]}]], "state_before": "case refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\n\u22a2 Disjoint (Iic c) {b}", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\n\u22a2 \u2200 (x : X), (g - dg) (e x) = f x", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nx : X\n\u22a2 (g - dg) (e x) = f x"}, {"tactic": "simp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]", "annotated_tactic": ["simp [dg0 (<a>Or.inl</a> <| <a>mem_range_self</a> _), \u2190 hgf]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [163, 23], "def_end_pos": [163, 37]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u21d1g \u2218 e = \u21d1f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nx : X\n\u22a2 (g - dg) (e x) = f x", "state_after": "no goals"}, {"tactic": "rcases (hgb y).eq_or_lt with (rfl | hlt)", "annotated_tactic": ["rcases (hgb y).<a>eq_or_lt</a> with (rfl | hlt)", [{"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [395, 7], "def_end_pos": [395, 21]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\n\u22a2 (g - dg) y < b", "state_after": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhb : IsLUB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc a (g y)\nhle : a \u2264 g y\nhlt : a < g y\nc : \u211d := (a + g y) / 2\nhac : a < c\nhcb : c < g y\nhsub : c - a = g y - c\nhgb : \u2200 (y_1 : Y), g y_1 \u2264 g y\nhb' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (g y - c)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (g y - c)\n\u22a2 (g - dg) y < g y\n\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhlt : g y < b\n\u22a2 (g - dg) y < b"}, {"tactic": "refine (sub_lt_self_iff _).2 ?_", "annotated_tactic": ["refine (<a>sub_lt_self_iff</a> _).2 ?_", [{"full_name": "sub_lt_self_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [417, 3], "def_end_pos": [417, 14]}]], "state_before": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhb : IsLUB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc a (g y)\nhle : a \u2264 g y\nhlt : a < g y\nc : \u211d := (a + g y) / 2\nhac : a < c\nhcb : c < g y\nhsub : c - a = g y - c\nhgb : \u2200 (y_1 : Y), g y_1 \u2264 g y\nhb' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (g y - c)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (g y - c)\n\u22a2 (g - dg) y < g y", "state_after": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhb : IsLUB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc a (g y)\nhle : a \u2264 g y\nhlt : a < g y\nc : \u211d := (a + g y) / 2\nhac : a < c\nhcb : c < g y\nhsub : c - a = g y - c\nhgb : \u2200 (y_1 : Y), g y_1 \u2264 g y\nhb' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (g y - c)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (g y - c)\n\u22a2 0 < dg y"}, {"tactic": "calc\n  0 < g y - c := sub_pos.2 hcb\n  _ = dg y := (dgb rfl).symm", "annotated_tactic": ["calc\n        0 < g y - c := <a>sub_pos</a>.2 hcb\n        _ = dg y := (dgb <a>rfl</a>).<a>symm</a>", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhb : IsLUB (range \u21d1f) (g y)\nhmem : \u2200 (x : X), f x \u2208 Icc a (g y)\nhle : a \u2264 g y\nhlt : a < g y\nc : \u211d := (a + g y) / 2\nhac : a < c\nhcb : c < g y\nhsub : c - a = g y - c\nhgb : \u2200 (y_1 : Y), g y_1 \u2264 g y\nhb' : \u00ac\u2203 x, f x = g y\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {g y})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (g y - c)) (\u21d1g \u207b\u00b9' {g y})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (g y - c)\n\u22a2 0 < dg y", "state_after": "no goals"}, {"tactic": "exact ((sub_le_self_iff _).2 (dgmem _).1).trans_lt hlt", "annotated_tactic": ["exact ((<a>sub_le_self_iff</a> _).2 (dgmem _).1).<a>trans_lt</a> hlt", [{"full_name": "sub_le_self_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [359, 3], "def_end_pos": [359, 14]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "case inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhlt : g y < b\n\u22a2 (g - dg) y < b", "state_after": "no goals"}, {"tactic": "rcases em (a \u2208 range f) with (\u27e8x, rfl\u27e9 | _)", "annotated_tactic": ["rcases <a>em</a> (a \u2208 <a>range</a> f) with (\u27e8x, rfl\u27e9 | _)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhc : c < g y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)\n\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "refine \u27e8x, xu, ?_, hyxu.le\u27e9", "annotated_tactic": ["refine \u27e8x, xu, ?_, hyxu.le\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhc : c < g y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhc : c < g y\n\u22a2 f x \u2264 (g - dg) y"}, {"tactic": "calc\n  f x = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n  _ \u2264 g y - dg y := sub_le_sub hc.le (dgmem _).2", "annotated_tactic": ["calc\n        f x = c - (b - c) := by rw [\u2190 hsub, <a>sub_sub_cancel</a>]\n        _ \u2264 g y - dg y := <a>sub_le_sub</a> hc.le (dgmem _).2", [{"full_name": "sub_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1304, 3], "def_end_pos": [1304, 14]}, {"full_name": "sub_le_sub", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [853, 32], "def_end_pos": [853, 42]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhc : c < g y\n\u22a2 f x \u2264 (g - dg) y", "state_after": "no goals"}, {"tactic": "rw [\u2190 hsub, sub_sub_cancel]", "annotated_tactic": ["rw [\u2190 hsub, <a>sub_sub_cancel</a>]", [{"full_name": "sub_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1304, 3], "def_end_pos": [1304, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u21d1f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nx : X\nha : IsGLB (range \u21d1f) (f x)\nhmem : \u2200 (x_1 : X), f x_1 \u2208 Icc (f x) b\nhle : f x \u2264 b\nhlt : f x < b\nc : \u211d := (f x + b) / 2\nhac : f x < c\nhcb : c < b\nhsub : c - f x = b - c\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhc : c < g y\n\u22a2 f x = c - (b - c)", "state_after": "no goals"}, {"tactic": "have hay : a < (g - dg) y := by\n  calc\n    a = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n    _ < g y - (b - c) := sub_lt_sub_right hc _\n    _ \u2264 g y - dg y := sub_le_sub_left (dgmem _).2 _", "annotated_tactic": ["have hay : a < (g - dg) y := by\n        calc\n          a = c - (b - c) := by rw [\u2190 hsub, <a>sub_sub_cancel</a>]\n          _ < g y - (b - c) := <a>sub_lt_sub_right</a> hc _\n          _ \u2264 g y - dg y := <a>sub_le_sub_left</a> (dgmem _).2 _", [{"full_name": "sub_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1304, 3], "def_end_pos": [1304, 14]}, {"full_name": "sub_lt_sub_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [880, 32], "def_end_pos": [880, 48]}, {"full_name": "sub_le_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [782, 32], "def_end_pos": [782, 47]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\nhay : a < (g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "rcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9", "annotated_tactic": ["rcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\nhay : a < (g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\nhay : a < (g - dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)"}, {"tactic": "exact \u27e8x, xu, hxy.le, hyxu.le\u27e9", "annotated_tactic": ["exact \u27e8x, xu, hxy.le, hyxu.le\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\nhay : a < (g - dg) y\nx : X\nleft\u271d : a \u2264 f x\nhxy : f x < (g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "no goals"}, {"tactic": "calc\n  a = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n  _ < g y - (b - c) := sub_lt_sub_right hc _\n  _ \u2264 g y - dg y := sub_le_sub_left (dgmem _).2 _", "annotated_tactic": ["calc\n          a = c - (b - c) := by rw [\u2190 hsub, <a>sub_sub_cancel</a>]\n          _ < g y - (b - c) := <a>sub_lt_sub_right</a> hc _\n          _ \u2264 g y - dg y := <a>sub_le_sub_left</a> (dgmem _).2 _", [{"full_name": "sub_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1304, 3], "def_end_pos": [1304, 14]}, {"full_name": "sub_lt_sub_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [880, 32], "def_end_pos": [880, 48]}, {"full_name": "sub_le_sub_left", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [782, 32], "def_end_pos": [782, 47]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\n\u22a2 a < (g - dg) y", "state_after": "no goals"}, {"tactic": "rw [\u2190 hsub, sub_sub_cancel]", "annotated_tactic": ["rw [\u2190 hsub, <a>sub_sub_cancel</a>]", [{"full_name": "sub_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1304, 3], "def_end_pos": [1304, 14]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : c < g y\nh\u271d : a \u2209 range \u21d1f\n\u22a2 a = c - (b - c)", "state_after": "no goals"}, {"tactic": "refine \u27e8xl y, xu, ?_, hyxu.le\u27e9", "annotated_tactic": ["refine \u27e8xl y, xu, ?_, hyxu.le\u27e9", []], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : g y \u2264 c\n\u22a2 \u2203 x\u2081 x\u2082, (g - dg) y \u2208 Icc (f x\u2081) (f x\u2082)", "state_after": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : g y \u2264 c\n\u22a2 f (xl y) \u2264 (g - dg) y"}, {"tactic": "simp [dg0 (Or.inr hc), hxl]", "annotated_tactic": ["simp [dg0 (<a>Or.inr</a> hc), hxl]", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u21d1f) a\nb : \u211d\nhb : IsLUB (range \u21d1f) b\nhmem : \u2200 (x : X), f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), f (xl y) \u2264 g y\nhgb : \u2200 (y : Y), g y \u2264 b\nhb' : \u00ac\u2203 x, f x = b\nhd : Disjoint (range e \u222a \u21d1g \u207b\u00b9' Iic c) (\u21d1g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u21d1dg) (Function.const Y 0) (range e \u222a \u21d1g \u207b\u00b9' Iic c)\ndgb : EqOn (\u21d1dg) (Function.const Y (b - c)) (\u21d1g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), (g - dg) (e x) = f x\ny : Y\nhyb : (g - dg) y < b\nxu : X\nhyxu : (g - dg) y < f xu\nright\u271d : f xu \u2264 b\nhc : g y \u2264 c\n\u22a2 f (xl y) \u2264 (g - dg) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.subset_extend", "start": [1464, 1], "end": [1467, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Finset.eventuallyEq_iUnion", "start": [1836, 1], "end": [1838, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartitionOfUnity.lean", "full_name": "PartitionOfUnity.exists_finset_nhd_support_subset", "start": [303, 1], "end": [306, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.toGlueR_isometry", "start": [529, 1], "end": [530, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_comp_neg", "start": [889, 1], "end": [890, 57], "traced_tactics": [{"tactic": "simpa only [zero_sub] using integral_comp_sub_left f 0", "annotated_tactic": ["simpa only [<a>zero_sub</a>] using <a>integral_comp_sub_left</a> f 0", [{"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "intervalIntegral.integral_comp_sub_left", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [884, 9], "def_end_pos": [884, 31]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in a..b, f (-x) = \u222b (x : \u211d) in -b..-a, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.conjugatesOfSet_subset_normalClosure", "start": [2366, 1], "end": [2367, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean", "full_name": "CategoryTheory.ShortComplex.add_\u03c4\u2082", "start": [59, 1], "end": [59, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/NNReal.lean", "full_name": "NNReal.summable_coe", "start": [197, 1], "end": [200, 48], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\n\u22a2 (Summable fun a => \u2191(f a)) \u2194 Summable f", "state_after": "case mp\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\n\u22a2 (Summable fun a => \u2191(f a)) \u2192 Summable f\n\ncase mpr\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\n\u22a2 Summable f \u2192 Summable fun a => \u2191(f a)"}, {"tactic": "exact fun \u27e8a, ha\u27e9 => \u27e8\u27e8a, ha.nonneg fun x => (f x).2\u27e9, hasSum_coe.1 ha\u27e9", "annotated_tactic": ["exact fun \u27e8a, ha\u27e9 => \u27e8\u27e8a, ha.nonneg fun x => (f x).2\u27e9, <a>hasSum_coe</a>.1 ha\u27e9", [{"full_name": "NNReal.hasSum_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [181, 9], "def_end_pos": [181, 19]}]], "state_before": "case mp\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\n\u22a2 (Summable fun a => \u2191(f a)) \u2192 Summable f", "state_after": "no goals"}, {"tactic": "exact fun \u27e8a, ha\u27e9 => \u27e8a.1, hasSum_coe.2 ha\u27e9", "annotated_tactic": ["exact fun \u27e8a, ha\u27e9 => \u27e8a.1, <a>hasSum_coe</a>.2 ha\u27e9", [{"full_name": "NNReal.hasSum_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [181, 9], "def_end_pos": [181, 19]}]], "state_before": "case mpr\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\n\u22a2 Summable f \u2192 Summable fun a => \u2191(f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "full_name": "RightDerivMeasurableAux.differentiable_set_subset_D", "start": [562, 1], "end": [574, 77], "traced_tactics": [{"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\n\u22a2 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K} \u2286 D f K", "state_after": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\n\u22a2 x \u2208 D f K"}, {"tactic": "rw [D, mem_iInter]", "annotated_tactic": ["rw [<a>D</a>, <a>mem_iInter</a>]", [{"full_name": "RightDerivMeasurableAux.D", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "def_pos": [470, 5], "def_end_pos": [470, 6]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [274, 9], "def_end_pos": [274, 19]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\n\u22a2 x \u2208 D f K", "state_after": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\n\u22a2 \u2200 (i : \u2115), x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)"}, {"tactic": "intro e", "annotated_tactic": ["intro e", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\n\u22a2 \u2200 (i : \u2115), x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)", "state_after": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)"}, {"tactic": "have : (0 : \u211d) < (1 / 2) ^ e := pow_pos (by norm_num) _", "annotated_tactic": ["have : (0 : \u211d) < (1 / 2) ^ e := <a>pow_pos</a> (by norm_num) _", [{"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)", "state_after": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)"}, {"tactic": "rcases mem_A_of_differentiable this hx.1 with \u27e8R, R_pos, hR\u27e9", "annotated_tactic": ["rcases <a>mem_A_of_differentiable</a> this hx.1 with \u27e8R, R_pos, hR\u27e9", [{"full_name": "RightDerivMeasurableAux.mem_A_of_differentiable", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "def_pos": [514, 9], "def_end_pos": [514, 32]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)", "state_after": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2) ^ n < R :=\n  exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : \u211d) / 2 < 1)", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2) ^ n < R :=\n    <a>exists_pow_lt_of_lt_one</a> R_pos (by norm_num : (1 : \u211d) / 2 < 1)", [{"full_name": "exists_pow_lt_of_lt_one", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [242, 9], "def_end_pos": [242, 32]}]], "state_before": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)", "state_after": "case intro.intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)"}, {"tactic": "simp only [mem_iUnion, mem_iInter, B, mem_inter_iff]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>, <a>mem_iInter</a>, <a>B</a>, <a>mem_inter_iff</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [274, 9], "def_end_pos": [274, 19]}, {"full_name": "RightDerivMeasurableAux.B", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "def_pos": [463, 5], "def_end_pos": [463, 6]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [874, 9], "def_end_pos": [874, 22]}]], "state_before": "case intro.intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\n\u22a2 x \u2208 \u22c3 n, \u22c2 p, \u22c2 (_ : p \u2265 n), \u22c2 q, \u22c2 (_ : q \u2265 n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)", "state_after": "case intro.intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\n\u22a2 \u2203 i,\n    \u2200 i_1 \u2265 i,\n      \u2200 i_3 \u2265 i, \u2203 i, \u2203 (_ : i \u2208 K), x \u2208 A f i ((1 / 2) ^ i_1) ((1 / 2) ^ e) \u2227 x \u2208 A f i ((1 / 2) ^ i_3) ((1 / 2) ^ e)"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\n\u22a2 0 < 1 / 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\n\u22a2 1 / 2 < 1", "state_after": "no goals"}, {"tactic": "refine hR _ \u27e8pow_pos (by norm_num) _, lt_of_le_of_lt ?_ hn\u27e9", "annotated_tactic": ["refine hR _ \u27e8<a>pow_pos</a> (by norm_num) _, <a>lt_of_le_of_lt</a> ?_ hn\u27e9", [{"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case intro.intro.intro.refine_2\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 x \u2208 A f (derivWithin f (Ici x) x) ((1 / 2) ^ q) ((1 / 2) ^ e)", "state_after": "case intro.intro.intro.refine_2\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 (1 / 2) ^ q \u2264 (1 / 2) ^ n"}, {"tactic": "exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by assumption)", "annotated_tactic": ["exact <a>pow_le_pow_of_le_one</a> (by norm_num) (by norm_num) (by assumption)", [{"full_name": "pow_le_pow_of_le_one", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [250, 7], "def_end_pos": [250, 27]}]], "state_before": "case intro.intro.intro.refine_2\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 (1 / 2) ^ q \u2264 (1 / 2) ^ n", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 0 < 1 / 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 0 \u2264 1 / 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 1 / 2 \u2264 1", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nx : \u211d\nhx : x \u2208 {x | DifferentiableWithinAt \u211d f (Ici x) x \u2227 derivWithin f (Ici x) x \u2208 K}\ne : \u2115\nthis : 0 < (1 / 2) ^ e\nR : \u211d\nR_pos : R > 0\nhR : \u2200 r \u2208 Ioo 0 R, x \u2208 A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)\nn : \u2115\nhn : (1 / 2) ^ n < R\np : \u2115\nhp : p \u2265 n\nq : \u2115\nhq : q \u2265 n\n\u22a2 n \u2264 q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "full_name": "CompositionAsSet.card_boundaries_pos", "start": [880, 1], "end": [881, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean", "full_name": "CategoryTheory.Monoidal.rightUnitor_inv_app", "start": [147, 1], "end": [149, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/StructurePolynomial.lean", "full_name": "constantCoeff_wittStructureRat_zero", "start": [355, 1], "end": [359, 22], "traced_tactics": [{"tactic": "simp only [wittStructureRat, bind\u2081, map_aeval, xInTermsOfW_zero, constantCoeff_rename,\n  constantCoeff_wittPolynomial, aeval_X, constantCoeff_comp_algebraMap, eval\u2082Hom_zero'_apply,\n  RingHom.id_apply]", "annotated_tactic": ["simp only [<a>wittStructureRat</a>, <a>bind\u2081</a>, <a>map_aeval</a>, <a>xInTermsOfW_zero</a>, <a>constantCoeff_rename</a>,\n    <a>constantCoeff_wittPolynomial</a>, <a>aeval_X</a>, <a>constantCoeff_comp_algebraMap</a>, <a>eval\u2082Hom_zero'_apply</a>,\n    <a>RingHom.id_apply</a>]", [{"full_name": "wittStructureRat", "def_path": "Mathlib/RingTheory/WittVector/StructurePolynomial.lean", "def_pos": [137, 19], "def_end_pos": [137, 35]}, {"full_name": "MvPolynomial.bind\u2081", "def_path": "Mathlib/Algebra/MvPolynomial/Monad.lean", "def_pos": [66, 5], "def_end_pos": [66, 10]}, {"full_name": "MvPolynomial.map_aeval", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1566, 9], "def_end_pos": [1566, 18]}, {"full_name": "xInTermsOfW_zero", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [239, 9], "def_end_pos": [239, 25]}, {"full_name": "MvPolynomial.constantCoeff_rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [331, 9], "def_end_pos": [331, 29]}, {"full_name": "constantCoeff_wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [125, 9], "def_end_pos": [125, 37]}, {"full_name": "MvPolynomial.aeval_X", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 16]}, {"full_name": "MvPolynomial.constantCoeff_comp_algebraMap", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [989, 9], "def_end_pos": [989, 38]}, {"full_name": "MvPolynomial.eval\u2082Hom_zero'_apply", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 29]}, {"full_name": "RingHom.id_apply", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}]], "state_before": "p : \u2115\nR : Type u_1\nidx : Type u_2\ninst\u271d : CommRing R\nhp : Fact (Nat.Prime p)\n\u03a6 : MvPolynomial idx \u211a\n\u22a2 constantCoeff (wittStructureRat p \u03a6 0) = constantCoeff \u03a6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.coe_max", "start": [1327, 1], "end": [1328, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "full_name": "cfc_add", "start": [411, 1], "end": [417, 43], "traced_tactics": [{"tactic": "by_cases ha : p a", "annotated_tactic": ["by_cases ha : p a", []], "state_before": "R : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\n\u22a2 cfc (fun x => f x + g x) a = cfc f a + cfc g a", "state_after": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : p a\n\u22a2 cfc (fun x => f x + g x) a = cfc f a + cfc g a\n\ncase neg\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : \u00acp a\n\u22a2 cfc (fun x => f x + g x) a = cfc f a + cfc g a"}, {"tactic": "rw [cfc_apply f a, cfc_apply g a, \u2190 map_add, cfc_apply _ a]", "annotated_tactic": ["rw [<a>cfc_apply</a> f a, <a>cfc_apply</a> g a, \u2190 <a>map_add</a>, <a>cfc_apply</a> _ a]", [{"full_name": "cfc_apply", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [294, 7], "def_end_pos": [294, 16]}, {"full_name": "cfc_apply", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [294, 7], "def_end_pos": [294, 16]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "cfc_apply", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [294, 7], "def_end_pos": [294, 16]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : p a\n\u22a2 cfc (fun x => f x + g x) a = cfc f a + cfc g a", "state_after": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : p a\n\u22a2 (cfcHom ha) { toFun := (spectrum R a).restrict fun x => f x + g x, continuous_toFun := \u22ef } =\n    (cfcHom ha)\n      ({ toFun := (spectrum R a).restrict f, continuous_toFun := \u22ef } +\n        { toFun := (spectrum R a).restrict g, continuous_toFun := \u22ef })"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case pos\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : p a\n\u22a2 (cfcHom ha) { toFun := (spectrum R a).restrict fun x => f x + g x, continuous_toFun := \u22ef } =\n    (cfcHom ha)\n      ({ toFun := (spectrum R a).restrict f, continuous_toFun := \u22ef } +\n        { toFun := (spectrum R a).restrict g, continuous_toFun := \u22ef })", "state_after": "no goals"}, {"tactic": "simp [cfc_apply_of_not_predicate a ha]", "annotated_tactic": ["simp [<a>cfc_apply_of_not_predicate</a> a ha]", [{"full_name": "cfc_apply_of_not_predicate", "def_path": "Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean", "def_pos": [307, 7], "def_end_pos": [307, 33]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\np : A \u2192 Prop\ninst\u271d\u2079 : CommSemiring R\ninst\u271d\u2078 : StarRing R\ninst\u271d\u2077 : MetricSpace R\ninst\u271d\u2076 : TopologicalSemiring R\ninst\u271d\u2075 : ContinuousStar R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : ContinuousFunctionalCalculus R p\nf\u271d g\u271d : R \u2192 R\na : A\nha\u271d : autoParam (p a) _auto\u271d\nhf\u271d : autoParam (ContinuousOn f\u271d (spectrum R a)) _auto\u271d\nhg\u271d : autoParam (ContinuousOn g\u271d (spectrum R a)) _auto\u271d\nf g : R \u2192 R\nhf : autoParam (ContinuousOn f (spectrum R a)) _auto\u271d\nhg : autoParam (ContinuousOn g (spectrum R a)) _auto\u271d\nha : \u00acp a\n\u22a2 cfc (fun x => f x + g x) a = cfc f a + cfc g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "PadicSeq.equiv_zero_of_val_eq_of_equiv_zero", "start": [145, 1], "end": [148, 45], "traced_tactics": [{"tactic": "simpa [h] using hi _ hj", "annotated_tactic": ["simpa [h] using hi _ hj", []], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : f \u2248 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\ni : \u2115\nhi : \u2200 j \u2265 i, padicNorm p (\u2191(f - 0) j) < \u03b5\nj : \u2115\nhj : j \u2265 i\n\u22a2 padicNorm p (\u2191(g - 0) j) < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_ennreal_pos", "start": [688, 1], "end": [689, 58], "traced_tactics": [{"tactic": "rw [\u2190 coe_ennreal_zero, coe_ennreal_lt_coe_ennreal_iff]", "annotated_tactic": ["rw [\u2190 <a>coe_ennreal_zero</a>, <a>coe_ennreal_lt_coe_ennreal_iff</a>]", [{"full_name": "EReal.coe_ennreal_zero", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [602, 9], "def_end_pos": [602, 25]}, {"full_name": "EReal.coe_ennreal_lt_coe_ennreal_iff", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 39]}]], "state_before": "x : \u211d\u22650\u221e\n\u22a2 0 < \u2191x \u2194 0 < x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.subset_le_of_restrict_le_restrict", "start": [851, 1], "end": [855, 52], "traced_tactics": [{"tactic": "by_cases hj\u2081 : MeasurableSet j", "annotated_tactic": ["by_cases hj\u2081 : <a>MeasurableSet</a> j", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : PartialOrder M\nv w : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nhi\u2082 : v.restrict i \u2264 w.restrict i\nj : Set \u03b1\nhj : j \u2286 i\n\u22a2 \u2191v j \u2264 \u2191w j", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : PartialOrder M\nv w : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nhi\u2082 : v.restrict i \u2264 w.restrict i\nj : Set \u03b1\nhj : j \u2286 i\nhj\u2081 : MeasurableSet j\n\u22a2 \u2191v j \u2264 \u2191w j\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : PartialOrder M\nv w : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nhi\u2082 : v.restrict i \u2264 w.restrict i\nj : Set \u03b1\nhj : j \u2286 i\nhj\u2081 : \u00acMeasurableSet j\n\u22a2 \u2191v j \u2264 \u2191w j"}, {"tactic": "exact (restrict_le_restrict_iff _ _ hi).1 hi\u2082 hj\u2081 hj", "annotated_tactic": ["exact (<a>restrict_le_restrict_iff</a> _ _ hi).1 hi\u2082 hj\u2081 hj", [{"full_name": "MeasureTheory.VectorMeasure.restrict_le_restrict_iff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [843, 9], "def_end_pos": [843, 33]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : PartialOrder M\nv w : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nhi\u2082 : v.restrict i \u2264 w.restrict i\nj : Set \u03b1\nhj : j \u2286 i\nhj\u2081 : MeasurableSet j\n\u22a2 \u2191v j \u2264 \u2191w j", "state_after": "no goals"}, {"tactic": "rw [v.not_measurable hj\u2081, w.not_measurable hj\u2081]", "annotated_tactic": ["rw [v.not_measurable hj\u2081, w.not_measurable hj\u2081]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : PartialOrder M\nv w : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nhi\u2082 : v.restrict i \u2264 w.restrict i\nj : Set \u03b1\nhj : j \u2286 i\nhj\u2081 : \u00acMeasurableSet j\n\u22a2 \u2191v j \u2264 \u2191w j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean", "full_name": "FreeGroupBasis.reindex_apply", "start": [95, 1], "end": [96, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.inv_hom_id_tensor", "start": [684, 1], "end": [686, 56], "traced_tactics": [{"tactic": "rw [\u2190 tensor_comp, f.inv_hom_id]", "annotated_tactic": ["rw [\u2190 <a>tensor_comp</a>, f.inv_hom_id]", [{"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [168, 3], "def_end_pos": [168, 14]}]], "state_before": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f.inv \u2297 g) \u226b (f.hom \u2297 h) = (\ud835\udfd9 W \u2297 g) \u226b (\ud835\udfd9 W \u2297 h)", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 \ud835\udfd9 W \u2297 g \u226b h = (\ud835\udfd9 W \u2297 g) \u226b (\ud835\udfd9 W \u2297 h)"}, {"tactic": "simp [id_tensorHom]", "annotated_tactic": ["simp [<a>id_tensorHom</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.id_tensorHom", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [225, 9], "def_end_pos": [225, 21]}]], "state_before": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 \ud835\udfd9 W \u2297 g \u226b h = (\ud835\udfd9 W \u2297 g) \u226b (\ud835\udfd9 W \u2297 h)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Ico_eq_Ico_iff", "start": [1190, 1], "end": [1200, 36], "traced_tactics": [{"tactic": "simp only [Subset.antisymm_iff] at e", "annotated_tactic": ["simp only [<a>Subset.antisymm_iff</a>] at e", [{"full_name": "Set.Subset.antisymm_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 = Ico a\u2082 b\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082"}, {"tactic": "simp only [le_antisymm_iff]", "annotated_tactic": ["simp only [<a>le_antisymm_iff</a>]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [194, 9], "def_end_pos": [194, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081"}, {"tactic": "cases' h with h h <;>\nsimp only [gt_iff_lt, not_lt, ge_iff_le, Ico_subset_Ico_iff h] at e <;>\n[ rcases e with \u27e8\u27e8h\u2081, h\u2082\u27e9, e'\u27e9; rcases e with \u27e8e', \u27e8h\u2081, h\u2082\u27e9\u27e9 ] <;>\nhave hab := (Ico_subset_Ico_iff <| h\u2081.trans_lt <| h.trans_le h\u2082).1 e' <;>\n[ exact \u27e8\u27e8hab.left, h\u2081\u27e9, \u27e8h\u2082, hab.right\u27e9\u27e9; exact \u27e8\u27e8h\u2081, hab.left\u27e9, \u27e8hab.right, h\u2082\u27e9\u27e9 ]", "annotated_tactic": ["cases' h with h h <;>\n      simp only [<a>gt_iff_lt</a>, <a>not_lt</a>, <a>ge_iff_le</a>, <a>Ico_subset_Ico_iff</a> h] at e <;>\n      [ rcases e with \u27e8\u27e8h\u2081, h\u2082\u27e9, e'\u27e9; rcases e with \u27e8e', \u27e8h\u2081, h\u2082\u27e9\u27e9 ] <;>\n      -- Porting note: restore `tauto`\n      have hab := (<a>Ico_subset_Ico_iff</a> <| h\u2081.trans_lt <| h.trans_le h\u2082).1 e' <;>\n      [ exact \u27e8\u27e8hab.left, h\u2081\u27e9, \u27e8h\u2082, hab.right\u27e9\u27e9; exact \u27e8\u27e8h\u2081, hab.left\u27e9, \u27e8hab.right, h\u2082\u27e9\u27e9 ]", [{"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}, {"full_name": "ge_iff_le", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1947, 17], "def_end_pos": [1947, 26]}, {"full_name": "Set.Ico_subset_Ico_iff", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [1167, 9], "def_end_pos": [1167, 27]}, {"full_name": "Set.Ico_subset_Ico_iff", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [1167, 9], "def_end_pos": [1167, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081", "state_after": "no goals"}, {"tactic": "rw [h\u2081, h\u2082]", "annotated_tactic": ["rw [h\u2081, h\u2082]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\nx\u271d : a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : b\u2081 = b\u2082\n\u22a2 Ico a\u2081 b\u2081 = Ico a\u2082 b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "IsCompact.measure_eq_iInf_isOpen", "start": [788, 1], "end": [798, 91], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u03bc K = \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u03bc K \u2264 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U\n\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U \u2264 \u03bc K"}, {"tactic": "simp only [le_iInf_iff]", "annotated_tactic": ["simp only [<a>le_iInf_iff</a>]", [{"full_name": "le_iInf_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [854, 9], "def_end_pos": [854, 20]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u03bc K \u2264 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2200 (i : Set \u03b1), K \u2286 i \u2192 IsOpen i \u2192 \u03bc K \u2264 \u03bc i"}, {"tactic": "exact fun U KU _ \u21a6 measure_mono KU", "annotated_tactic": ["exact fun U KU _ \u21a6 <a>measure_mono</a> KU", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2200 (i : Set \u03b1), K \u2286 i \u2192 IsOpen i \u2192 \u03bc K \u2264 \u03bc i", "state_after": "no goals"}, {"tactic": "apply le_of_forall_lt'", "annotated_tactic": ["apply <a>le_of_forall_lt'</a>", [{"full_name": "le_of_forall_lt'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [535, 9], "def_end_pos": [535, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U \u2264 \u03bc K", "state_after": "case a.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2200 (c : \u211d\u22650\u221e), \u03bc K < c \u2192 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U < c"}, {"tactic": "simpa only [iInf_lt_iff, exists_prop, exists_and_left] using hK.exists_isOpen_lt_of_lt", "annotated_tactic": ["simpa only [<a>iInf_lt_iff</a>, <a>exists_prop</a>, <a>exists_and_left</a>] using hK.exists_isOpen_lt_of_lt", [{"full_name": "iInf_lt_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [548, 9], "def_end_pos": [548, 20]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "exists_and_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [288, 17], "def_end_pos": [288, 32]}]], "state_before": "case a.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : \u03bc.InnerRegularCompactLTTop\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : R1Space \u03b1\ninst\u271d : BorelSpace \u03b1\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 \u2200 (c : \u211d\u22650\u221e), \u03bc K < c \u2192 \u2a05 U, \u2a05 (_ : K \u2286 U), \u2a05 (_ : IsOpen U), \u03bc U < c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Zlattice/Basic.lean", "full_name": "Zspan.quotientEquiv_apply_mk", "start": [276, 1], "end": [277, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "Antivary.sum_smul_le_sum_comp_perm_smul", "start": [326, 1], "end": [328, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean", "full_name": "CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_subtype", "start": [151, 1], "end": [154, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean", "full_name": "Measurable.im", "start": [53, 1], "end": [54, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/GoldenRatio.lean", "full_name": "one_sub_goldConj", "start": [75, 1], "end": [76, 31], "traced_tactics": [{"tactic": "linarith [gold_add_goldConj]", "annotated_tactic": ["linarith [<a>gold_add_goldConj</a>]", [{"full_name": "gold_add_goldConj", "def_path": "Mathlib/Data/Real/GoldenRatio.lean", "def_pos": [70, 9], "def_end_pos": [70, 26]}]], "state_before": "\u22a2 1 - \u03c6 = \u03c8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "full_name": "HasStrictFDerivAt.continuousMultilinear_apply_const", "start": [178, 1], "end": [180, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/HomEquiv.lean", "full_name": "CategoryTheory.LocalizerMorphism.homMap_apply", "start": [69, 1], "end": [84, 14], "traced_tactics": [{"tactic": "let G' := \u03a6.localizedFunctor L\u2081 L\u2082", "annotated_tactic": ["let G' := \u03a6.localizedFunctor L\u2081 L\u2082", []], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\n\u22a2 \u03a6.homMap L\u2081 L\u2082 f = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\n\u22a2 \u03a6.homMap L\u2081 L\u2082 f = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "let e' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'", "annotated_tactic": ["let e' := <a>CatCommSq.iso</a> \u03a6.functor L\u2081 L\u2082 G'", [{"full_name": "CategoryTheory.CatCommSq.iso", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [42, 5], "def_end_pos": [42, 8]}]], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\n\u22a2 \u03a6.homMap L\u2081 L\u2082 f = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\n\u22a2 \u03a6.homMap L\u2081 L\u2082 f = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "change e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = _", "annotated_tactic": ["change e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = _", []], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\n\u22a2 \u03a6.homMap L\u2081 L\u2082 f = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "letI : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := \u27e8e.symm\u27e9", "annotated_tactic": ["letI : <a>Localization.Lifting</a> L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := \u27e8e.symm\u27e9", [{"full_name": "CategoryTheory.Localization.Lifting", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [291, 7], "def_end_pos": [291, 14]}]], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "let \u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) _ _ e'.symm", "annotated_tactic": ["let \u03b1 : G' \u2245 G := <a>Localization.liftNatIso</a> L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) _ _ e'.symm", [{"full_name": "CategoryTheory.Localization.liftNatIso", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [368, 5], "def_end_pos": [368, 15]}]], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "have : e = e' \u226a\u226b isoWhiskerLeft _ \u03b1 := by\n  ext X\n  dsimp [\u03b1]\n  rw [Localization.liftNatTrans_app]\n  erw [id_comp]\n  rw [Iso.hom_inv_id_app_assoc]\n  rfl", "annotated_tactic": ["have : e = e' \u226a\u226b <a>isoWhiskerLeft</a> _ \u03b1 := by\n    ext X\n    dsimp [\u03b1]\n    rw [<a>Localization.liftNatTrans_app</a>]\n    erw [<a>id_comp</a>]\n    rw [<a>Iso.hom_inv_id_app_assoc</a>]\n    rfl", [{"full_name": "CategoryTheory.isoWhiskerLeft", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [172, 5], "def_end_pos": [172, 19]}, {"full_name": "CategoryTheory.Localization.liftNatTrans_app", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [340, 9], "def_end_pos": [340, 25]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Iso.hom_inv_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [63, 3], "def_end_pos": [63, 25]}]], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis\u271d : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nthis : e = e' \u226a\u226b isoWhiskerLeft L\u2081 \u03b1\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis\u271d : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nthis : e = e' \u226a\u226b isoWhiskerLeft L\u2081 \u03b1\n\u22a2 e'.hom.app X \u226b G'.map f \u226b e'.inv.app Y = e.hom.app X \u226b G.map f \u226b e.inv.app Y", "state_after": "no goals"}, {"tactic": "ext X", "annotated_tactic": ["ext X", []], "state_before": "C : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\n\u22a2 e = e' \u226a\u226b isoWhiskerLeft L\u2081 \u03b1", "state_after": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = (e' \u226a\u226b isoWhiskerLeft L\u2081 \u03b1).hom.app X"}, {"tactic": "dsimp [\u03b1]", "annotated_tactic": ["dsimp [\u03b1]", []], "state_before": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = (e' \u226a\u226b isoWhiskerLeft L\u2081 \u03b1).hom.app X", "state_after": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = e'.hom.app X \u226b (Localization.liftNatTrans L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.inv).app (L\u2081.obj X)"}, {"tactic": "rw [Localization.liftNatTrans_app]", "annotated_tactic": ["rw [<a>Localization.liftNatTrans_app</a>]", [{"full_name": "CategoryTheory.Localization.liftNatTrans_app", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [340, 9], "def_end_pos": [340, 25]}]], "state_before": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = e'.hom.app X \u226b (Localization.liftNatTrans L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.inv).app (L\u2081.obj X)", "state_after": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X =\n    e'.hom.app X \u226b\n      (Localization.Lifting.iso L\u2081 W\u2081 (L\u2081 \u22d9 G') G').hom.app X \u226b\n        e'.inv.app X \u226b (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X"}, {"tactic": "erw [id_comp]", "annotated_tactic": ["erw [<a>id_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X =\n    e'.hom.app X \u226b\n      (Localization.Lifting.iso L\u2081 W\u2081 (L\u2081 \u22d9 G') G').hom.app X \u226b\n        e'.inv.app X \u226b (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X", "state_after": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = e'.hom.app X \u226b e'.inv.app X \u226b (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X"}, {"tactic": "rw [Iso.hom_inv_id_app_assoc]", "annotated_tactic": ["rw [<a>Iso.hom_inv_id_app_assoc</a>]", [{"full_name": "CategoryTheory.Iso.hom_inv_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [63, 3], "def_end_pos": [63, 25]}]], "state_before": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = e'.hom.app X \u226b e'.inv.app X \u226b (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X", "state_after": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case w.w.h\nC : Type u_1\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\nD\u2081 : Type u_5\nD\u2082 : Type u_6\nD\u2083 : Type u_7\ninst\u271d\u2079 : Category.{?u.15870, u_1} C\ninst\u271d\u2078 : Category.{u_10, u_2} C\u2081\ninst\u271d\u2077 : Category.{u_11, u_3} C\u2082\ninst\u271d\u2076 : Category.{?u.15882, u_4} C\u2083\ninst\u271d\u2075 : Category.{u_8, u_5} D\u2081\ninst\u271d\u2074 : Category.{u_9, u_6} D\u2082\ninst\u271d\u00b3 : Category.{?u.15894, u_7} D\u2083\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\n\u03a8 : LocalizerMorphism W\u2082 W\u2083\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nL\u2083 : C\u2083 \u2964 D\u2083\ninst\u271d : L\u2083.IsLocalization W\u2083\nX\u271d Y Z : C\u2081\nG : D\u2081 \u2964 D\u2082\ne : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G\nf : L\u2081.obj X\u271d \u27f6 L\u2081.obj Y\nG' : D\u2081 \u2964 D\u2082 := \u03a6.localizedFunctor L\u2081 L\u2082\ne' : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 G' := CatCommSq.iso \u03a6.functor L\u2081 L\u2082 G'\nthis : Localization.Lifting L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G := { iso' := e.symm }\n\u03b1 : G' \u2245 G := Localization.liftNatIso L\u2081 W\u2081 (L\u2081 \u22d9 G') (\u03a6.functor \u22d9 L\u2082) G' G e'.symm\nX : C\u2081\n\u22a2 e.hom.app X = (Localization.Lifting.iso L\u2081 W\u2081 (\u03a6.functor \u22d9 L\u2082) G).inv.app X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.kroneckerTMul_add", "start": [511, 1], "end": [513, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Box.lean", "full_name": "Int.existsUnique_mem_box", "start": [100, 1], "end": [101, 79], "traced_tactics": [{"tactic": "use max x.1.natAbs x.2.natAbs", "annotated_tactic": ["use <a>max</a> x.1.<a>natAbs</a> x.2.<a>natAbs</a>", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}, {"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}]], "state_before": "n : \u2115\nx\u271d x : \u2124 \u00d7 \u2124\n\u22a2 \u2203! n, x \u2208 box n", "state_after": "case h\nn : \u2115\nx\u271d x : \u2124 \u00d7 \u2124\n\u22a2 (fun n => x \u2208 box n) (max x.1.natAbs x.2.natAbs) \u2227 \u2200 (y : \u2115), (fun n => x \u2208 box n) y \u2192 y = max x.1.natAbs x.2.natAbs"}, {"tactic": "simp only [mem_box, and_self_iff, forall_eq']", "annotated_tactic": ["simp only [<a>mem_box</a>, <a>and_self_iff</a>, <a>forall_eq'</a>]", [{"full_name": "Int.mem_box", "def_path": "Mathlib/Order/Interval/Finset/Box.lean", "def_pos": [88, 15], "def_end_pos": [88, 22]}, {"full_name": "and_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [35, 9], "def_end_pos": [35, 21]}, {"full_name": "forall_eq'", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [276, 17], "def_end_pos": [276, 27]}]], "state_before": "case h\nn : \u2115\nx\u271d x : \u2124 \u00d7 \u2124\n\u22a2 (fun n => x \u2208 box n) (max x.1.natAbs x.2.natAbs) \u2227 \u2200 (y : \u2115), (fun n => x \u2208 box n) y \u2192 y = max x.1.natAbs x.2.natAbs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.iInf_sum", "start": [619, 1], "end": [628, 74], "traced_tactics": [{"tactic": "induction' s using Finset.cons_induction_on with a s ha ih", "annotated_tactic": ["induction' s using <a>Finset.cons_induction_on</a> with a s ha ih", [{"full_name": "Finset.cons_induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1261, 9], "def_end_pos": [1261, 26]}]], "state_before": "\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\n\u22a2 \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a", "state_after": "case h\u2081\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\n\u22a2 \u2a05 i, \u2211 a \u2208 \u2205, f i a = \u2211 a \u2208 \u2205, \u2a05 i, f i a\n\ncase h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\n\u22a2 \u2a05 i, \u2211 a \u2208 Finset.cons a s ha, f i a = \u2211 a \u2208 Finset.cons a s ha, \u2a05 i, f i a"}, {"tactic": "simp only [Finset.sum_empty, ciInf_const]", "annotated_tactic": ["simp only [<a>Finset.sum_empty</a>, <a>ciInf_const</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}, {"full_name": "ciInf_const", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [853, 9], "def_end_pos": [853, 20]}]], "state_before": "case h\u2081\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\n\u22a2 \u2a05 i, \u2211 a \u2208 \u2205, f i a = \u2211 a \u2208 \u2205, \u2a05 i, f i a", "state_after": "no goals"}, {"tactic": "simp only [Finset.sum_cons, \u2190 ih]", "annotated_tactic": ["simp only [<a>Finset.sum_cons</a>, \u2190 ih]", [{"full_name": "Finset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [344, 3], "def_end_pos": [344, 14]}]], "state_before": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\n\u22a2 \u2a05 i, \u2211 a \u2208 Finset.cons a s ha, f i a = \u2211 a \u2208 Finset.cons a s ha, \u2a05 i, f i a", "state_after": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\n\u22a2 \u2a05 i, f i a + \u2211 a \u2208 s, f i a = (\u2a05 i, f i a) + \u2a05 i, \u2211 a \u2208 s, f i a"}, {"tactic": "refine (iInf_add_iInf fun i j => ?_).symm", "annotated_tactic": ["refine (<a>iInf_add_iInf</a> fun i j => ?_).<a>symm</a>", [{"full_name": "ENNReal.iInf_add_iInf", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [610, 9], "def_end_pos": [610, 22]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\n\u22a2 \u2a05 i, f i a + \u2211 a \u2208 s, f i a = (\u2a05 i, f i a) + \u2a05 i, \u2211 a \u2208 s, f i a", "state_after": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j : \u03b9\n\u22a2 \u2203 k, f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a"}, {"tactic": "refine (h (Finset.cons a s ha) i j).imp fun k hk => ?_", "annotated_tactic": ["refine (h (<a>Finset.cons</a> a s ha) i j).<a>imp</a> fun k hk => ?_", [{"full_name": "Finset.cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [868, 5], "def_end_pos": [868, 9]}, {"full_name": "Exists.imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [193, 9], "def_end_pos": [193, 19]}]], "state_before": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j : \u03b9\n\u22a2 \u2203 k, f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a", "state_after": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j k : \u03b9\nhk : \u2200 a_1 \u2208 Finset.cons a s ha, f k a_1 \u2264 f i a_1 \u2227 f k a_1 \u2264 f j a_1\n\u22a2 f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a"}, {"tactic": "rw [Finset.forall_mem_cons] at hk", "annotated_tactic": ["rw [<a>Finset.forall_mem_cons</a>] at hk", [{"full_name": "Finset.forall_mem_cons", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [890, 9], "def_end_pos": [890, 24]}]], "state_before": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j k : \u03b9\nhk : \u2200 a_1 \u2208 Finset.cons a s ha, f k a_1 \u2264 f i a_1 \u2227 f k a_1 \u2264 f j a_1\n\u22a2 f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a", "state_after": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j k : \u03b9\nhk : (f k a \u2264 f i a \u2227 f k a \u2264 f j a) \u2227 \u2200 x \u2208 s, f k x \u2264 f i x \u2227 f k x \u2264 f j x\n\u22a2 f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a"}, {"tactic": "exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)", "annotated_tactic": ["exact <a>add_le_add</a> hk.1.1 (<a>Finset.sum_le_sum</a> fun a ha => (hk.2 a ha).2)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [109, 15], "def_end_pos": [109, 25]}]], "state_before": "case h\u2082\n\u03b9 : Sort u_1\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_2\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 a \u2208 t, f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : a \u2209 s\nih : \u2a05 i, \u2211 a \u2208 s, f i a = \u2211 a \u2208 s, \u2a05 i, f i a\ni j k : \u03b9\nhk : (f k a \u2264 f i a \u2227 f k a \u2264 f j a) \u2227 \u2200 x \u2208 s, f k x \u2264 f i x \u2227 f k x \u2264 f j x\n\u22a2 f k a + \u2211 a \u2208 s, f k a \u2264 f i a + \u2211 a \u2208 s, f j a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.image_subset_image\u2082_right", "start": [94, 1], "end": [95, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/WeakDual.lean", "full_name": "isOpenMap_toWeakSpace_symm", "start": [355, 1], "end": [357, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Ring/Basic.lean", "full_name": "Mathlib.Tactic.Ring.atom_pf", "start": [872, 1], "end": [872, 91], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "u : Lean.Level\nR : Type u_1\n\u03b1 : Q(Type u)\ns\u03b1 : Q(CommSemiring \u00ab$\u03b1\u00bb)\ninst\u271d : CommSemiring R\na : R\n\u22a2 a = a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Segment.lean", "full_name": "Prod.segment_subset", "start": [617, 1], "end": [619, 95], "traced_tactics": [{"tactic": "rintro z \u27e8a, b, ha, hb, hab, hz\u27e9", "annotated_tactic": ["rintro z \u27e8a, b, ha, hb, hab, hz\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E \u00d7 F\n\u22a2 [x-[\ud835\udd5c]y] \u2286 [x.1-[\ud835\udd5c]y.1] \u00d7\u02e2 [x.2-[\ud835\udd5c]y.2]", "state_after": "case intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E \u00d7 F\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 z \u2208 [x.1-[\ud835\udd5c]y.1] \u00d7\u02e2 [x.2-[\ud835\udd5c]y.2]"}, {"tactic": "exact \u27e8\u27e8a, b, ha, hb, hab, congr_arg Prod.fst hz\u27e9, a, b, ha, hb, hab, congr_arg Prod.snd hz\u27e9", "annotated_tactic": ["exact \u27e8\u27e8a, b, ha, hb, hab, <a>congr_arg</a> <a>Prod.fst</a> hz\u27e9, a, b, ha, hb, hab, <a>congr_arg</a> <a>Prod.snd</a> hz\u27e9", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Prod.fst", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [483, 3], "def_end_pos": [483, 6]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Prod.snd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [485, 3], "def_end_pos": [485, 6]}]], "state_before": "case intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E \u00d7 F\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 z \u2208 [x.1-[\ud835\udd5c]y.1] \u00d7\u02e2 [x.2-[\ud835\udd5c]y.2]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "full_name": "IsSigmaCompact.image_of_continuousOn", "start": [88, 1], "end": [93, 93], "traced_tactics": [{"tactic": "rcases hs with \u27e8K, hcompact, hcov\u27e9", "annotated_tactic": ["rcases hs with \u27e8K, hcompact, hcov\u27e9", []], "state_before": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns\u271d t : Set X\nf : X \u2192 Y\ns : Set X\nhs : IsSigmaCompact s\nhf : ContinuousOn f s\n\u22a2 IsSigmaCompact (f '' s)", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns\u271d t : Set X\nf : X \u2192 Y\ns : Set X\nhf : ContinuousOn f s\nK : \u2115 \u2192 Set X\nhcompact : \u2200 (n : \u2115), IsCompact (K n)\nhcov : \u22c3 n, K n = s\n\u22a2 IsSigmaCompact (f '' s)"}, {"tactic": "refine \u27e8fun n \u21a6 f '' K n, ?_, hcov.symm \u25b8 image_iUnion.symm\u27e9", "annotated_tactic": ["refine \u27e8fun n \u21a6 f '' K n, ?_, hcov.symm \u25b8 image_iUnion.symm\u27e9", []], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns\u271d t : Set X\nf : X \u2192 Y\ns : Set X\nhf : ContinuousOn f s\nK : \u2115 \u2192 Set X\nhcompact : \u2200 (n : \u2115), IsCompact (K n)\nhcov : \u22c3 n, K n = s\n\u22a2 IsSigmaCompact (f '' s)", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns\u271d t : Set X\nf : X \u2192 Y\ns : Set X\nhf : ContinuousOn f s\nK : \u2115 \u2192 Set X\nhcompact : \u2200 (n : \u2115), IsCompact (K n)\nhcov : \u22c3 n, K n = s\n\u22a2 \u2200 (n : \u2115), IsCompact ((fun n => f '' K n) n)"}, {"tactic": "exact fun n \u21a6 (hcompact n).image_of_continuousOn (hf.mono (hcov.symm \u25b8 subset_iUnion K n))", "annotated_tactic": ["exact fun n \u21a6 (hcompact n).<a>image_of_continuousOn</a> (hf.mono (hcov.symm \u25b8 <a>subset_iUnion</a> K n))", [{"full_name": "IsCompact.image_of_continuousOn", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [104, 9], "def_end_pos": [104, 40]}, {"full_name": "Set.subset_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [278, 9], "def_end_pos": [278, 22]}]], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns\u271d t : Set X\nf : X \u2192 Y\ns : Set X\nhf : ContinuousOn f s\nK : \u2115 \u2192 Set X\nhcompact : \u2200 (n : \u2115), IsCompact (K n)\nhcov : \u22c3 n, K n = s\n\u22a2 \u2200 (n : \u2115), IsCompact ((fun n => f '' K n) n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Partial.lean", "full_name": "Filter.rtendsto_def", "start": [95, 1], "end": [97, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.smul_imK", "start": [1144, 9], "end": [1144, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Galois.lean", "full_name": "IsGalois.card_aut_eq_finrank", "start": [103, 1], "end": [125, 61], "traced_tactics": [{"tactic": "cases' Field.exists_primitive_element F E with \u03b1 h\u03b1", "annotated_tactic": ["cases' <a>Field.exists_primitive_element</a> F E with \u03b1 h\u03b1", [{"full_name": "Field.exists_primitive_element", "def_path": "Mathlib/FieldTheory/PrimitiveElement.lean", "def_pos": [214, 9], "def_end_pos": [214, 33]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "let iso : F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => e.val\n    invFun := fun e => \u27e8e, by rw [h\u03b1]; exact IntermediateField.mem_top\u27e9\n    left_inv := fun _ => by ext; rfl\n    right_inv := fun _ => rfl\n    map_mul' := fun _ _ => rfl\n    map_add' := fun _ _ => rfl\n    commutes' := fun _ => rfl }", "annotated_tactic": ["let iso : F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n    { toFun := fun e => e.val\n      invFun := fun e => \u27e8e, by rw [h\u03b1]; exact <a>IntermediateField.mem_top</a>\u27e9\n      left_inv := fun _ => by ext; rfl\n      right_inv := fun _ => <a>rfl</a>\n      map_mul' := fun _ _ => <a>rfl</a>\n      map_add' := fun _ _ => <a>rfl</a>\n      commutes' := fun _ => <a>rfl</a> }", [{"full_name": "IntermediateField.mem_top", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [127, 9], "def_end_pos": [127, 16]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "have H : IsIntegral F \u03b1 := IsGalois.integral F \u03b1", "annotated_tactic": ["have H : <a>IsIntegral</a> F \u03b1 := <a>IsGalois.integral</a> F \u03b1", [{"full_name": "IsIntegral", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [61, 5], "def_end_pos": [61, 15]}, {"full_name": "IsGalois.integral", "def_path": "Mathlib/FieldTheory/Galois.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "have h_sep : (minpoly F \u03b1).Separable := IsGalois.separable F \u03b1", "annotated_tactic": ["have h_sep : (<a>minpoly</a> F \u03b1).<a>Separable</a> := <a>IsGalois.separable</a> F \u03b1", [{"full_name": "minpoly", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [38, 19], "def_end_pos": [38, 26]}, {"full_name": "Polynomial.Separable", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [40, 5], "def_end_pos": [40, 14]}, {"full_name": "IsGalois.separable", "def_path": "Mathlib/FieldTheory/Galois.lean", "def_pos": [78, 9], "def_end_pos": [78, 18]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "have h_splits : (minpoly F \u03b1).Splits (algebraMap F E) := IsGalois.splits F \u03b1", "annotated_tactic": ["have h_splits : (<a>minpoly</a> F \u03b1).<a>Splits</a> (<a>algebraMap</a> F E) := <a>IsGalois.splits</a> F \u03b1", [{"full_name": "minpoly", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [38, 19], "def_end_pos": [38, 26]}, {"full_name": "Polynomial.Splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [47, 5], "def_end_pos": [47, 11]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IsGalois.splits", "def_path": "Mathlib/FieldTheory/Galois.lean", "def_pos": [82, 9], "def_end_pos": [82, 15]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F E) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "replace h_splits : Polynomial.Splits (algebraMap F F\u27ee\u03b1\u27ef) (minpoly F \u03b1) := by\n  simpa using\n    Polynomial.splits_comp_of_splits (algebraMap F E) iso.symm.toAlgHom.toRingHom h_splits", "annotated_tactic": ["replace h_splits : <a>Polynomial.Splits</a> (<a>algebraMap</a> F F\u27ee\u03b1\u27ef) (<a>minpoly</a> F \u03b1) := by\n    simpa using\n      <a>Polynomial.splits_comp_of_splits</a> (<a>algebraMap</a> F E) iso.symm.toAlgHom.toRingHom h_splits", [{"full_name": "Polynomial.Splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [47, 5], "def_end_pos": [47, 11]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "minpoly", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [38, 19], "def_end_pos": [38, 26]}, {"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [440, 9], "def_end_pos": [440, 30]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F E) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E"}, {"tactic": "rw [\u2190 LinearEquiv.finrank_eq iso.toLinearEquiv]", "annotated_tactic": ["rw [\u2190 <a>LinearEquiv.finrank_eq</a> iso.toLinearEquiv]", [{"full_name": "LinearEquiv.finrank_eq", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [113, 9], "def_end_pos": [113, 19]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F E", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F \u21a5F\u27ee\u03b1\u27ef"}, {"tactic": "rw [\u2190 IntermediateField.AdjoinSimple.card_aut_eq_finrank F E H h_sep h_splits]", "annotated_tactic": ["rw [\u2190 <a>IntermediateField.AdjoinSimple.card_aut_eq_finrank</a> F E H h_sep h_splits]", [{"full_name": "IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrank", "def_path": "Mathlib/FieldTheory/Galois.lean", "def_pos": [93, 9], "def_end_pos": [93, 59]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = finrank F \u21a5F\u27ee\u03b1\u27ef", "state_after": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = Fintype.card (\u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef)"}, {"tactic": "apply Fintype.card_congr", "annotated_tactic": ["apply <a>Fintype.card_congr</a>", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [151, 9], "def_end_pos": [151, 19]}]], "state_before": "case intro\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Fintype.card (E \u2243\u2090[F] E) = Fintype.card (\u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef)", "state_after": "case intro.f\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 (E \u2243\u2090[F] E) \u2243 \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef"}, {"tactic": "apply Equiv.mk (fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) fun \u03d5 => iso.symm.trans (\u03d5.trans iso)", "annotated_tactic": ["apply <a>Equiv.mk</a> (fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) fun \u03d5 => iso.symm.trans (\u03d5.trans iso)", [{"full_name": "Equiv.mk", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [63, 11], "def_end_pos": [63, 16]}]], "state_before": "case intro.f\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 (E \u2243\u2090[F] E) \u2243 \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef", "state_after": "case intro.f.left_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Function.LeftInverse (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) fun \u03d5 => iso.trans (\u03d5.trans iso.symm)\n\ncase intro.f.right_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Function.RightInverse (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) fun \u03d5 => iso.trans (\u03d5.trans iso.symm)"}, {"tactic": "rw [h\u03b1]", "annotated_tactic": ["rw [h\u03b1]", []], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\ne : E\n\u22a2 e \u2208 F\u27ee\u03b1\u27ef", "state_after": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\ne : E\n\u22a2 e \u2208 \u22a4"}, {"tactic": "exact IntermediateField.mem_top", "annotated_tactic": ["exact <a>IntermediateField.mem_top</a>", [{"full_name": "IntermediateField.mem_top", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [127, 9], "def_end_pos": [127, 16]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\ne : E\n\u22a2 e \u2208 \u22a4", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\nx\u271d : \u21a5F\u27ee\u03b1\u27ef\n\u22a2 (fun e => \u27e8e, \u22ef\u27e9) ((fun e => \u2191e) x\u271d) = x\u271d", "state_after": "case a\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\nx\u271d : \u21a5F\u27ee\u03b1\u27ef\n\u22a2 \u2191((fun e => \u27e8e, \u22ef\u27e9) ((fun e => \u2191e) x\u271d)) = \u2191x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\nx\u271d : \u21a5F\u27ee\u03b1\u27ef\n\u22a2 \u2191((fun e => \u27e8e, \u22ef\u27e9) ((fun e => \u2191e) x\u271d)) = \u2191x\u271d", "state_after": "no goals"}, {"tactic": "simpa using\n  Polynomial.splits_comp_of_splits (algebraMap F E) iso.symm.toAlgHom.toRingHom h_splits", "annotated_tactic": ["simpa using\n      <a>Polynomial.splits_comp_of_splits</a> (<a>algebraMap</a> F E) iso.symm.toAlgHom.toRingHom h_splits", [{"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [440, 9], "def_end_pos": [440, 30]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}]], "state_before": "F : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F E) (minpoly F \u03b1)\n\u22a2 Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)", "state_after": "no goals"}, {"tactic": "intro \u03d5", "annotated_tactic": ["intro \u03d5", []], "state_before": "case intro.f.left_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Function.LeftInverse (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) fun \u03d5 => iso.trans (\u03d5.trans iso.symm)", "state_after": "case intro.f.left_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : E \u2243\u2090[F] E\n\u22a2 (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) \u03d5) = \u03d5"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case intro.f.left_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : E \u2243\u2090[F] E\n\u22a2 (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) \u03d5) = \u03d5", "state_after": "case intro.f.left_inv.h\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : E \u2243\u2090[F] E\na\u271d : E\n\u22a2 ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) \u03d5)) a\u271d = \u03d5 a\u271d"}, {"tactic": "simp only [trans_apply, apply_symm_apply]", "annotated_tactic": ["simp only [<a>trans_apply</a>, <a>apply_symm_apply</a>]", [{"full_name": "AlgEquiv.trans_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [428, 9], "def_end_pos": [428, 20]}, {"full_name": "AlgEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [407, 9], "def_end_pos": [407, 25]}]], "state_before": "case intro.f.left_inv.h\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : E \u2243\u2090[F] E\na\u271d : E\n\u22a2 ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) \u03d5)) a\u271d = \u03d5 a\u271d", "state_after": "no goals"}, {"tactic": "intro \u03d5", "annotated_tactic": ["intro \u03d5", []], "state_before": "case intro.f.right_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u22a2 Function.RightInverse (fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) fun \u03d5 => iso.trans (\u03d5.trans iso.symm)", "state_after": "case intro.f.right_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef\n\u22a2 (fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) \u03d5) = \u03d5"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "case intro.f.right_inv\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef\n\u22a2 (fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) \u03d5) = \u03d5", "state_after": "case intro.f.right_inv.h\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef\na\u271d : \u21a5F\u27ee\u03b1\u27ef\n\u22a2 ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) \u03d5)) a\u271d = \u03d5 a\u271d"}, {"tactic": "simp only [trans_apply, symm_apply_apply]", "annotated_tactic": ["simp only [<a>trans_apply</a>, <a>symm_apply_apply</a>]", [{"full_name": "AlgEquiv.trans_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [428, 9], "def_end_pos": [428, 20]}, {"full_name": "AlgEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [412, 9], "def_end_pos": [412, 25]}]], "state_before": "case intro.f.right_inv.h\nF : Type u_1\ninst\u271d\u2074 : Field F\nE : Type u_2\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : FiniteDimensional F E\ninst\u271d : IsGalois F E\n\u03b1 : E\nh\u03b1 : F\u27ee\u03b1\u27ef = \u22a4\niso : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] E :=\n  { toFun := fun e => \u2191e, invFun := fun e => \u27e8e, \u22ef\u27e9, left_inv := \u22ef, right_inv := \u22ef, map_mul' := \u22ef, map_add' := \u22ef,\n    commutes' := \u22ef }\nH : IsIntegral F \u03b1\nh_sep : (minpoly F \u03b1).Separable\nh_splits : Polynomial.Splits (algebraMap F \u21a5F\u27ee\u03b1\u27ef) (minpoly F \u03b1)\n\u03d5 : \u21a5F\u27ee\u03b1\u27ef \u2243\u2090[F] \u21a5F\u27ee\u03b1\u27ef\na\u271d : \u21a5F\u27ee\u03b1\u27ef\n\u22a2 ((fun \u03d5 => iso.trans (\u03d5.trans iso.symm)) ((fun \u03d5 => iso.symm.trans (\u03d5.trans iso)) \u03d5)) a\u271d = \u03d5 a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Eigenspace/Basic.lean", "full_name": "Module.End.HasEigenvector.apply_eq_smul", "start": [109, 1], "end": [111, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Basic.lean", "full_name": "BoxIntegral.Prepartition.le_def", "start": [161, 1], "end": [161, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Comma/Presheaf.lean", "full_name": "CategoryTheory.OverPresheafAux.MakesOverArrow.of_arrow", "start": [101, 1], "end": [103, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/NNReal.lean", "full_name": "NNReal.hasSum_coe", "start": [181, 1], "end": [182, 45], "traced_tactics": [{"tactic": "simp only [HasSum, \u2190 coe_sum, tendsto_coe]", "annotated_tactic": ["simp only [<a>HasSum</a>, \u2190 <a>coe_sum</a>, <a>tendsto_coe</a>]", [{"full_name": "HasSum", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Defs.lean", "def_pos": [74, 3], "def_end_pos": [74, 14]}, {"full_name": "NNReal.coe_sum", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "NNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/NNReal.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}]], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\nr : \u211d\u22650\n\u22a2 HasSum (fun a => \u2191(f a)) \u2191r \u2194 HasSum f r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.erase_add", "start": [1127, 1], "end": [1130, 67], "traced_tactics": [{"tactic": "ext s", "annotated_tactic": ["ext s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\n\u22a2 erase a (f + f') = erase a f + erase a f'", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s"}, {"tactic": "by_cases hs : s = a", "annotated_tactic": ["by_cases hs : s = a", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : s = a\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : \u00acs = a\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s"}, {"tactic": "rw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply]", "annotated_tactic": ["rw [<a>add_apply</a>, <a>erase_ne</a> hs, <a>erase_ne</a> hs, <a>erase_ne</a> hs, <a>add_apply</a>]", [{"full_name": "Finsupp.add_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 18]}, {"full_name": "Finsupp.erase_ne", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [650, 9], "def_end_pos": [650, 17]}, {"full_name": "Finsupp.erase_ne", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [650, 9], "def_end_pos": [650, 17]}, {"full_name": "Finsupp.erase_ne", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [650, 9], "def_end_pos": [650, 17]}, {"full_name": "Finsupp.add_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : \u00acs = a\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s", "state_after": "no goals"}, {"tactic": "rw [hs, add_apply, erase_same, erase_same, erase_same, add_zero]", "annotated_tactic": ["rw [hs, <a>add_apply</a>, <a>erase_same</a>, <a>erase_same</a>, <a>erase_same</a>, <a>add_zero</a>]", [{"full_name": "Finsupp.add_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [1014, 9], "def_end_pos": [1014, 18]}, {"full_name": "Finsupp.erase_same", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 19]}, {"full_name": "Finsupp.erase_same", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 19]}, {"full_name": "Finsupp.erase_same", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 19]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : s = a\n\u22a2 (erase a (f + f')) s = (erase a f + erase a f') s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "full_name": "Filter.EventuallyEq.contDiffWithinAt_iff", "start": [474, 1], "end": [477, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_mkRat", "start": [232, 1], "end": [233, 84], "traced_tactics": [{"tactic": "if z : d = 0 then simp [z]; rfl else simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["if z : d = 0 then simp [z]; rfl else simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [98, 9], "def_end_pos": [98, 27]}, {"full_name": "Rat.neg_normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [229, 9], "def_end_pos": [229, 22]}]], "state_before": "n : Int\nd : Nat\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}, {"tactic": "simp [z]", "annotated_tactic": ["simp [z]", []], "state_before": "n : Int\nd : Nat\nz : d = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "n : Int\nd : Nat\nz : d = 0\n\u22a2 -0 = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "n : Int\nd : Nat\nz : d = 0\n\u22a2 -0 = 0", "state_after": "no goals"}, {"tactic": "simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [98, 9], "def_end_pos": [98, 27]}, {"full_name": "Rat.neg_normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [229, 9], "def_end_pos": [229, 22]}]], "state_before": "n : Int\nd : Nat\nz : \u00acd = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "full_name": "ContDiff.dense_compl_range_of_finrank_lt_finrank", "start": [643, 1], "end": [645, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.Measure.FiniteAtFilter.inf_of_right", "start": [1421, 1], "end": [1422, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.moveLeft_lf", "start": [599, 1], "end": [600, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "full_name": "PFunctor.M.ichildren_mk", "start": [547, 1], "end": [550, 15], "traced_tactics": [{"tactic": "dsimp only [ichildren, PFunctor.Obj.iget]", "annotated_tactic": ["dsimp only [<a>ichildren</a>, <a>PFunctor.Obj.iget</a>]", [{"full_name": "PFunctor.M.ichildren", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [260, 5], "def_end_pos": [260, 14]}, {"full_name": "PFunctor.Obj.iget", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [149, 5], "def_end_pos": [149, 13]}]], "state_before": "F : PFunctor.{u}\nX : Type u_1\nf : X \u2192 \u2191F X\ninst\u271d\u00b9 : DecidableEq F.A\ninst\u271d : Inhabited F.M\nx : \u2191F F.M\ni : F.Idx\n\u22a2 ichildren i (M.mk x) = x.iget i", "state_after": "F : PFunctor.{u}\nX : Type u_1\nf : X \u2192 \u2191F X\ninst\u271d\u00b9 : DecidableEq F.A\ninst\u271d : Inhabited F.M\nx : \u2191F F.M\ni : F.Idx\n\u22a2 (if H' : i.fst = (M.mk x).head then (M.mk x).children (cast \u22ef i.snd) else default) =\n    if h : i.fst = x.fst then x.snd (cast \u22ef i.snd) else default"}, {"tactic": "congr with h", "annotated_tactic": ["congr with h", []], "state_before": "F : PFunctor.{u}\nX : Type u_1\nf : X \u2192 \u2191F X\ninst\u271d\u00b9 : DecidableEq F.A\ninst\u271d : Inhabited F.M\nx : \u2191F F.M\ni : F.Idx\n\u22a2 (if H' : i.fst = (M.mk x).head then (M.mk x).children (cast \u22ef i.snd) else default) =\n    if h : i.fst = x.fst then x.snd (cast \u22ef i.snd) else default", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "full_name": "Ordinal.log_mono_right", "start": [358, 1], "end": [364, 67], "traced_tactics": [{"tactic": "simp only [hx, log_zero_right, Ordinal.zero_le]", "annotated_tactic": ["simp only [hx, <a>log_zero_right</a>, <a>Ordinal.zero_le</a>]", [{"full_name": "Ordinal.log_zero_right", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [284, 9], "def_end_pos": [284, 23]}, {"full_name": "Ordinal.zero_le", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [386, 19], "def_end_pos": [386, 26]}]], "state_before": "b x y : Ordinal.{u_1}\nxy : x \u2264 y\nhx : x = 0\n\u22a2 log b x \u2264 log b y", "state_after": "no goals"}, {"tactic": "simp only [log_of_not_one_lt_left hb, Ordinal.zero_le]", "annotated_tactic": ["simp only [<a>log_of_not_one_lt_left</a> hb, <a>Ordinal.zero_le</a>]", [{"full_name": "Ordinal.log_of_not_one_lt_left", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [270, 9], "def_end_pos": [270, 31]}, {"full_name": "Ordinal.zero_le", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [386, 19], "def_end_pos": [386, 26]}]], "state_before": "b x y : Ordinal.{u_1}\nxy : x \u2264 y\nhx : \u00acx = 0\nhb : \u00ac1 < b\n\u22a2 log b x \u2264 log b y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.biUnion_range", "start": [1675, 1], "end": [1676, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.add_conj", "start": [375, 1], "end": [378, 56], "traced_tactics": [{"tactic": "rw [re_add_im, conj_eq_re_sub_im]", "annotated_tactic": ["rw [<a>re_add_im</a>, <a>conj_eq_re_sub_im</a>]", [{"full_name": "RCLike.re_add_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 18]}, {"full_name": "RCLike.conj_eq_re_sub_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [358, 9], "def_end_pos": [358, 26]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nz : K\n\u22a2 z + (starRingEnd K) z = \u2191(re z) + \u2191(im z) * I + (\u2191(re z) - \u2191(im z) * I)", "state_after": "no goals"}, {"tactic": "rw [add_add_sub_cancel, two_mul]", "annotated_tactic": ["rw [<a>add_add_sub_cancel</a>, <a>two_mul</a>]", [{"full_name": "add_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1350, 3], "def_end_pos": [1350, 14]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [179, 9], "def_end_pos": [179, 16]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nz : K\n\u22a2 \u2191(re z) + \u2191(im z) * I + (\u2191(re z) - \u2191(im z) * I) = 2 * \u2191(re z)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.inductionOn", "start": [127, 1], "end": [128, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "mul_le_one", "start": [313, 1], "end": [314, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "compact_open_separated_mul_left", "start": [1772, 1], "end": [1779, 45], "traced_tactics": [{"tactic": "rcases compact_open_separated_mul_right (hK.image continuous_op) (opHomeomorph.isOpenMap U hU)\n    (image_subset op hKU) with\n  \u27e8V, hV : V \u2208 \ud835\udcdd (op (1 : G)), hV' : op '' K * V \u2286 op '' U\u27e9", "annotated_tactic": ["rcases <a>compact_open_separated_mul_right</a> (hK.image <a>continuous_op</a>) (opHomeomorph.isOpenMap U hU)\n      (<a>image_subset</a> <a>op</a> hKU) with\n    \u27e8V, hV : V \u2208 \ud835\udcdd (<a>op</a> (1 : G)), hV' : <a>op</a> '' K * V \u2286 <a>op</a> '' U\u27e9", [{"full_name": "compact_open_separated_mul_right", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1744, 9], "def_end_pos": [1744, 41]}, {"full_name": "MulOpposite.continuous_op", "def_path": "Mathlib/Topology/Algebra/Constructions.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}, {"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}, {"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}, {"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}]], "state_before": "G : Type w\nH : Type x\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K \u2286 U\n\u22a2 \u2203 V \u2208 \ud835\udcdd 1, V * K \u2286 U", "state_after": "case intro.intro\nG : Type w\nH : Type x\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K \u2286 U\nV : Set G\u1d50\u1d52\u1d56\nhV : V \u2208 \ud835\udcdd (op 1)\nhV' : op '' K * V \u2286 op '' U\n\u22a2 \u2203 V \u2208 \ud835\udcdd 1, V * K \u2286 U"}, {"tactic": "refine \u27e8op \u207b\u00b9' V, continuous_op.continuousAt hV, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>op</a> \u207b\u00b9' V, continuous_op.continuousAt hV, ?_\u27e9", [{"full_name": "MulOpposite.op", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [78, 5], "def_end_pos": [78, 7]}]], "state_before": "case intro.intro\nG : Type w\nH : Type x\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K \u2286 U\nV : Set G\u1d50\u1d52\u1d56\nhV : V \u2208 \ud835\udcdd (op 1)\nhV' : op '' K * V \u2286 op '' U\n\u22a2 \u2203 V \u2208 \ud835\udcdd 1, V * K \u2286 U", "state_after": "case intro.intro\nG : Type w\nH : Type x\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K \u2286 U\nV : Set G\u1d50\u1d52\u1d56\nhV : V \u2208 \ud835\udcdd (op 1)\nhV' : op '' K * V \u2286 op '' U\n\u22a2 op \u207b\u00b9' V * K \u2286 U"}, {"tactic": "rwa [\u2190 image_preimage_eq V op_surjective, \u2190 image_op_mul, image_subset_iff,\n  preimage_image_eq _ op_injective] at hV'", "annotated_tactic": ["rwa [\u2190 <a>image_preimage_eq</a> V <a>op_surjective</a>, \u2190 <a>image_op_mul</a>, <a>image_subset_iff</a>,\n    <a>preimage_image_eq</a> _ <a>op_injective</a>] at hV'", [{"full_name": "Set.image_preimage_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [499, 9], "def_end_pos": [499, 26]}, {"full_name": "MulOpposite.op_surjective", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}, {"full_name": "Set.image_op_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [573, 9], "def_end_pos": [573, 21]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [481, 9], "def_end_pos": [481, 25]}, {"full_name": "Set.preimage_image_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [494, 9], "def_end_pos": [494, 26]}, {"full_name": "MulOpposite.op_injective", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [143, 9], "def_end_pos": [143, 21]}]], "state_before": "case intro.intro\nG : Type w\nH : Type x\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K \u2286 U\nV : Set G\u1d50\u1d52\u1d56\nhV : V \u2208 \ud835\udcdd (op 1)\nhV' : op '' K * V \u2286 op '' U\n\u22a2 op \u207b\u00b9' V * K \u2286 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "full_name": "HomologicalComplex.pOpcyclesIso_hom_inv_id", "start": [527, 1], "end": [530, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.Iic_le", "start": [1337, 1], "end": [1337, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/ModEq.lean", "full_name": "Int.modEq_sub", "start": [247, 1], "end": [248, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "contDiffAt_map_inverse", "start": [1849, 1], "end": [1864, 21], "traced_tactics": [{"tactic": "nontriviality E", "annotated_tactic": ["nontriviality E", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e"}, {"tactic": "let O\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp (e.symm : F \u2192L[\ud835\udd5c] E)", "annotated_tactic": ["let O\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp (e.symm : F \u2192L[\ud835\udd5c] E)", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e"}, {"tactic": "let O\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (e.symm : F \u2192L[\ud835\udd5c] E).comp f", "annotated_tactic": ["let O\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (e.symm : F \u2192L[\ud835\udd5c] E).<a>comp</a> f", [{"full_name": "ContinuousLinearMap.comp", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [779, 5], "def_end_pos": [779, 9]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e"}, {"tactic": "have : ContinuousLinearMap.inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082 := funext (to_ring_inverse e)", "annotated_tactic": ["have : <a>ContinuousLinearMap.inverse</a> = O\u2081 \u2218 <a>Ring.inverse</a> \u2218 O\u2082 := <a>funext</a> (<a>to_ring_inverse</a> e)", [{"full_name": "ContinuousLinearMap.inverse", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2606, 19], "def_end_pos": [2606, 26]}, {"full_name": "Ring.inverse", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [93, 19], "def_end_pos": [93, 26]}, {"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}, {"full_name": "ContinuousLinearMap.to_ring_inverse", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [2643, 9], "def_end_pos": [2643, 24]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n inverse \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e"}, {"tactic": "have h\u2081 : ContDiff \ud835\udd5c n O\u2081 := contDiff_id.clm_comp contDiff_const", "annotated_tactic": ["have h\u2081 : <a>ContDiff</a> \ud835\udd5c n O\u2081 := contDiff_id.clm_comp <a>contDiff_const</a>", [{"full_name": "ContDiff", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1443, 5], "def_end_pos": [1443, 13]}, {"full_name": "contDiff_const", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e"}, {"tactic": "have h\u2082 : ContDiff \ud835\udd5c n O\u2082 := contDiff_const.clm_comp contDiff_id", "annotated_tactic": ["have h\u2082 : <a>ContDiff</a> \ud835\udd5c n O\u2082 := contDiff_const.clm_comp <a>contDiff_id</a>", [{"full_name": "ContDiff", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1443, 5], "def_end_pos": [1443, 13]}, {"full_name": "contDiff_id", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [183, 9], "def_end_pos": [183, 20]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e"}, {"tactic": "refine h\u2081.contDiffAt.comp _ (ContDiffAt.comp _ ?_ h\u2082.contDiffAt)", "annotated_tactic": ["refine h\u2081.contDiffAt.comp _ (<a>ContDiffAt.comp</a> _ ?_ h\u2082.contDiffAt)", [{"full_name": "ContDiffAt.comp", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [730, 16], "def_end_pos": [730, 31]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n (O\u2081 \u2218 Ring.inverse \u2218 O\u2082) \u2191e", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n Ring.inverse (O\u2082 \u2191e)"}, {"tactic": "convert contDiffAt_ring_inverse \ud835\udd5c (1 : (E \u2192L[\ud835\udd5c] E)\u02e3)", "annotated_tactic": ["convert <a>contDiffAt_ring_inverse</a> \ud835\udd5c (1 : (E \u2192L[\ud835\udd5c] E)\u02e3)", [{"full_name": "contDiffAt_ring_inverse", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [1758, 9], "def_end_pos": [1758, 32]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 ContDiffAt \ud835\udd5c n Ring.inverse (O\u2082 \u2191e)", "state_after": "case h.e'_11\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 O\u2082 \u2191e = \u21911"}, {"tactic": "simp [O\u2082, one_def]", "annotated_tactic": ["simp [O\u2082, <a>one_def</a>]", [{"full_name": "ContinuousLinearMap.one_def", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}]], "state_before": "case h.e'_11\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2070 : NormedAddCommGroup D\ninst\u271d\u2079 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\na\u271d : Nontrivial E\nO\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 F \u2192L[\ud835\udd5c] E := fun f => f.comp \u2191e.symm\nO\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (\u2191e.symm).comp f\nthis : inverse = O\u2081 \u2218 Ring.inverse \u2218 O\u2082\nh\u2081 : ContDiff \ud835\udd5c n O\u2081\nh\u2082 : ContDiff \ud835\udd5c n O\u2082\n\u22a2 O\u2082 \u2191e = \u21911", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Matrix.lean", "full_name": "Matrix.nnnorm_def", "start": [86, 1], "end": [86, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean", "full_name": "CategoryTheory.FreeMonoidalCategory.inclusion_obj", "start": [86, 1], "end": [88, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Exposed.lean", "full_name": "exposedPoints_subset", "start": [211, 1], "end": [211, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max'_mem", "start": [1584, 1], "end": [1585, 70], "traced_tactics": [{"tactic": "simp only [max', Finset.max, id_eq, coe_sup']", "annotated_tactic": ["simp only [<a>max'</a>, <a>Finset.max</a>, <a>id_eq</a>, <a>coe_sup'</a>]", [{"full_name": "Finset.max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1552, 5], "def_end_pos": [1552, 9]}, {"full_name": "Finset.max", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1371, 15], "def_end_pos": [1371, 18]}, {"full_name": "id_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [297, 17], "def_end_pos": [297, 22]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [821, 9], "def_end_pos": [821, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : s.Nonempty\nx : \u03b1\n\u22a2 s.max = \u2191(s.max' H)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : s.Nonempty\nx : \u03b1\n\u22a2 s.sup WithBot.some = s.sup (WithBot.some \u2218 fun x => x)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : s.Nonempty\nx : \u03b1\n\u22a2 s.sup WithBot.some = s.sup (WithBot.some \u2218 fun x => x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Agrees.push", "start": [91, 1], "end": [101, 13], "traced_tactics": [{"tactic": "cases H", "annotated_tactic": ["cases H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nn : \u2115\nm : Fin n \u2192 \u03b2\nH : Agrees arr f m\nk : \u2115\nhk : k = n + 1\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < n), m' i = m \u27e8\u2191i, h\u27e9\nhm\u2082 : \u2200 (h : n < k), f x = m' \u27e8n, h\u27e9\n\u22a2 Agrees (arr.push x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\n\u22a2 Agrees (arr.push x) f m'"}, {"tactic": "have : k = (arr.push x).size := by simp [hk]", "annotated_tactic": ["have : k = (arr.push x).<a>size</a> := by simp [hk]", [{"full_name": "Array.size", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2586, 5], "def_end_pos": [2586, 15]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\n\u22a2 Agrees (arr.push x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\n\u22a2 Agrees (arr.push x) f m'"}, {"tactic": "refine mk' this fun i h\u2081 h\u2082 \u21a6 ?_", "annotated_tactic": ["refine <a>mk'</a> this fun i h\u2081 h\u2082 \u21a6 ?_", [{"full_name": "UFModel.Agrees.mk'", "def_path": "Mathlib/Data/UnionFind.lean", "def_pos": [73, 9], "def_end_pos": [73, 12]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\n\u22a2 Agrees (arr.push x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\n\u22a2 f ((arr.push x).get \u27e8i, h\u2081\u27e9) = m' \u27e8i, h\u2082\u27e9"}, {"tactic": "simp [Array.get_push]", "annotated_tactic": ["simp [<a>Array.get_push</a>]", [{"full_name": "Array.get_push", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [127, 9], "def_end_pos": [127, 17]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\n\u22a2 f ((arr.push x).get \u27e8i, h\u2081\u27e9) = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\n\u22a2 f (if h : i < arr.size then arr[i] else x) = m' \u27e8i, h\u2082\u27e9"}, {"tactic": "split <;> (rename_i h; simp at hm\u2081 \u22a2)", "annotated_tactic": ["split <;> (rename_i h; simp at hm\u2081 \u22a2)", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\n\u22a2 f (if h : i < arr.size then arr[i] else x) = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk.isTrue\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : i < arr.size\n\u22a2 f arr[i] = m' \u27e8i, h\u2082\u27e9\n\ncase mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9"}, {"tactic": "simp [hk]", "annotated_tactic": ["simp [hk]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\n\u22a2 k = (arr.push x).size", "state_after": "no goals"}, {"tactic": "rename_i h", "annotated_tactic": ["rename_i h", []], "state_before": "case mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh\u271d : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9"}, {"tactic": "simp at hm\u2081 \u22a2", "annotated_tactic": ["simp at hm\u2081 \u22a2", []], "state_before": "case mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = (fun i => f (arr.get i)) \u27e8\u2191i, h\u27e9\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9"}, {"tactic": "rw [\u2190 hm\u2081 \u27e8i, h\u2082\u27e9]", "annotated_tactic": ["rw [\u2190 hm\u2081 \u27e8i, h\u2082\u27e9]", []], "state_before": "case mk.isTrue\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : i < arr.size\n\u22a2 f arr[i] = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk.isTrue\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : i < arr.size\n\u22a2 \u2191\u27e8i, h\u2082\u27e9 < arr.size"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case mk.isTrue\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : i < arr.size\n\u22a2 \u2191\u27e8i, h\u2082\u27e9 < arr.size", "state_after": "no goals"}, {"tactic": "cases show i = arr.size by apply Nat.le_antisymm <;> simp_all [Nat.lt_succ]", "annotated_tactic": ["cases show i = arr.size by apply <a>Nat.le_antisymm</a> <;> simp_all [<a>Nat.lt_succ</a>]", [{"full_name": "Nat.le_antisymm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1750, 19], "def_end_pos": [1750, 34]}, {"full_name": "Nat.lt_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [620, 9], "def_end_pos": [620, 16]}]], "state_before": "case mk.isFalse\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 f x = m' \u27e8i, h\u2082\u27e9", "state_after": "case mk.isFalse.refl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\nh\u2081 : arr.size < (arr.push x).size\nh\u2082 : arr.size < k\nh : \u00acarr.size < arr.size\n\u22a2 f x = m' \u27e8arr.size, h\u2082\u27e9"}, {"tactic": "rw [hm\u2082]", "annotated_tactic": ["rw [hm\u2082]", []], "state_before": "case mk.isFalse.refl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\nh\u2081 : arr.size < (arr.push x).size\nh\u2082 : arr.size < k\nh : \u00acarr.size < arr.size\n\u22a2 f x = m' \u27e8arr.size, h\u2082\u27e9", "state_after": "no goals"}, {"tactic": "apply Nat.le_antisymm <;> simp_all [Nat.lt_succ]", "annotated_tactic": ["apply <a>Nat.le_antisymm</a> <;> simp_all [<a>Nat.lt_succ</a>]", [{"full_name": "Nat.le_antisymm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1750, 19], "def_end_pos": [1750, 34]}, {"full_name": "Nat.lt_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [620, 9], "def_end_pos": [620, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = arr.size + 1\nhm\u2082 : \u2200 (h : arr.size < k), f x = m' \u27e8arr.size, h\u27e9\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < arr.size), m' i = f arr[\u2191i]\nthis : k = (arr.push x).size\ni : \u2115\nh\u2081 : i < (arr.push x).size\nh\u2082 : i < k\nh : \u00aci < arr.size\n\u22a2 i = arr.size", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "inv_mul_cancel_left\u2080", "start": [302, 1], "end": [305, 25], "traced_tactics": [{"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b\u271d c g h\u271d x : G\u2080\nh : a \u2260 0\nb : G\u2080\n\u22a2 a\u207b\u00b9 * a * b = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.type_le_iff", "start": [354, 1], "end": [356, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/MorphismProperty/Composition.lean", "full_name": "CategoryTheory.MorphismProperty.IsMultiplicative.of_unop", "start": [156, 1], "end": [157, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Norm.lean", "full_name": "Algebra.norm_algebraMap_of_basis", "start": [91, 1], "end": [97, 45], "traced_tactics": [{"tactic": "haveI := Classical.decEq \u03b9", "annotated_tactic": ["haveI := <a>Classical.decEq</a> \u03b9", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\n\u22a2 (norm R) ((algebraMap R S) x) = x ^ Fintype.card \u03b9", "state_after": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 (norm R) ((algebraMap R S) x) = x ^ Fintype.card \u03b9"}, {"tactic": "rw [norm_apply, \u2190 det_toMatrix b, lmul_algebraMap]", "annotated_tactic": ["rw [<a>norm_apply</a>, \u2190 <a>det_toMatrix</a> b, <a>lmul_algebraMap</a>]", [{"full_name": "Algebra.norm_apply", "def_path": "Mathlib/RingTheory/Norm.lean", "def_pos": [69, 9], "def_end_pos": [69, 19]}, {"full_name": "LinearMap.det_toMatrix", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "Algebra.lmul_algebraMap", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean", "def_pos": [44, 9], "def_end_pos": [44, 24]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 (norm R) ((algebraMap R S) x) = x ^ Fintype.card \u03b9", "state_after": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 ((toMatrix b b) ((lsmul R R S) x)).det = x ^ Fintype.card \u03b9"}, {"tactic": "convert @det_diagonal _ _ _ _ _ fun _ : \u03b9 => x", "annotated_tactic": ["convert @<a>det_diagonal</a> _ _ _ _ _ fun _ : \u03b9 => x", [{"full_name": "Matrix.det_diagonal", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [73, 9], "def_end_pos": [73, 21]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 ((toMatrix b b) ((lsmul R R S) x)).det = x ^ Fintype.card \u03b9", "state_after": "case h.e'_2.h.e'_6\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 (toMatrix b b) ((lsmul R R S) x) = diagonal fun x_1 => x\n\ncase h.e'_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 x ^ Fintype.card \u03b9 = \u220f i : \u03b9, x"}, {"tactic": "ext (i j)", "annotated_tactic": ["ext (i j)", []], "state_before": "case h.e'_2.h.e'_6\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 (toMatrix b b) ((lsmul R R S) x) = diagonal fun x_1 => x", "state_after": "case h.e'_2.h.e'_6.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 (toMatrix b b) ((lsmul R R S) x) i j = diagonal (fun x_1 => x) i j"}, {"tactic": "rw [toMatrix_lsmul]", "annotated_tactic": ["rw [<a>toMatrix_lsmul</a>]", [{"full_name": "Algebra.toMatrix_lsmul", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [890, 9], "def_end_pos": [890, 23]}]], "state_before": "case h.e'_2.h.e'_6.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 (toMatrix b b) ((lsmul R R S) x) i j = diagonal (fun x_1 => x) i j", "state_after": "no goals"}, {"tactic": "rw [Finset.prod_const, Finset.card_univ]", "annotated_tactic": ["rw [<a>Finset.prod_const</a>, <a>Finset.card_univ</a>]", [{"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1737, 9], "def_end_pos": [1737, 19]}, {"full_name": "Finset.card_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 25]}]], "state_before": "case h.e'_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 x ^ Fintype.card \u03b9 = \u220f i : \u03b9, x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "full_name": "LinearEquiv.rTensor_symm_tmul", "start": [1433, 9], "end": [1433, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Invertible/Defs.lean", "full_name": "mul_invOf_self'", "start": [109, 1], "end": [110, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "full_name": "ModularForm.mul_slash_SL2", "start": [232, 1], "end": [239, 50], "traced_tactics": [{"tactic": "apply mul_slash", "annotated_tactic": ["apply <a>mul_slash</a>", [{"full_name": "ModularForm.mul_slash", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [211, 9], "def_end_pos": [211, 18]}]], "state_before": "\u0393 : Subgroup SL(2, \u2124)\nk : \u2124\nf\u271d : \u210d \u2192 \u2102\nk1 k2 : \u2124\nA : SL(2, \u2124)\nf g : \u210d \u2192 \u2102\n\u22a2 (f * g) \u2223[k1 + k2] \u2191A = (\u2191\u2191\u2191A).det \u2022 f \u2223[k1] A * g \u2223[k2] A", "state_after": "no goals"}, {"tactic": "rw [det_coe']", "annotated_tactic": ["rw [<a>det_coe'</a>]", [{"full_name": "UpperHalfPlane.ModularGroup.det_coe'", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [319, 9], "def_end_pos": [319, 17]}]], "state_before": "\u0393 : Subgroup SL(2, \u2124)\nk : \u2124\nf\u271d : \u210d \u2192 \u2102\nk1 k2 : \u2124\nA : SL(2, \u2124)\nf g : \u210d \u2192 \u2102\n\u22a2 (\u2191\u2191\u2191A).det \u2022 f \u2223[k1] A * g \u2223[k2] A = 1 \u2022 f \u2223[k1] A * g \u2223[k2] A", "state_after": "no goals"}, {"tactic": "rw [one_smul]", "annotated_tactic": ["rw [<a>one_smul</a>]", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "\u0393 : Subgroup SL(2, \u2124)\nk : \u2124\nf\u271d : \u210d \u2192 \u2102\nk1 k2 : \u2124\nA : SL(2, \u2124)\nf g : \u210d \u2192 \u2102\n\u22a2 1 \u2022 f \u2223[k1] A * g \u2223[k2] A = f \u2223[k1] A * g \u2223[k2] A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "Right.mul_lt_one_of_le_of_lt", "start": [903, 1], "end": [906, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.Measure.restrict_restrict", "start": [183, 1], "end": [184, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean", "full_name": "UniqueMDiffWithinAt.smooth_bundle_preimage", "start": [120, 1], "end": [131, 68], "traced_tactics": [{"tactic": "set e := trivializationAt F Z p.proj", "annotated_tactic": ["set e := <a>trivializationAt</a> F Z p.proj", [{"full_name": "FiberBundle.trivializationAt", "def_path": "Mathlib/Topology/FiberBundle/Basic.lean", "def_pos": [200, 8], "def_end_pos": [200, 24]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p"}, {"tactic": "have hp : p \u2208 e.source := FiberBundle.mem_trivializationAt_proj_source", "annotated_tactic": ["have hp : p \u2208 e.source := <a>FiberBundle.mem_trivializationAt_proj_source</a>", [{"full_name": "FiberBundle.mem_trivializationAt_proj_source", "def_path": "Mathlib/Topology/FiberBundle/Basic.lean", "def_pos": [281, 9], "def_end_pos": [281, 41]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p"}, {"tactic": "have : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (e p) := by\n  rw [\u2190 Prod.mk.eta (p := e p), FiberBundle.trivializationAt_proj_fst]\n  exact hs.prod (uniqueMDiffWithinAt_univ _)", "annotated_tactic": ["have : <a>UniqueMDiffWithinAt</a> (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 <a>univ</a>) (e p) := by\n    rw [\u2190 <a>Prod.mk.eta</a> (p := e p), <a>FiberBundle.trivializationAt_proj_fst</a>]\n    exact hs.prod (<a>uniqueMDiffWithinAt_univ</a> _)", [{"full_name": "UniqueMDiffWithinAt", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Defs.lean", "def_pos": [182, 5], "def_end_pos": [182, 24]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "Prod.mk.eta", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [28, 9], "def_end_pos": [28, 15]}, {"full_name": "FiberBundle.trivializationAt_proj_fst", "def_path": "Mathlib/Topology/FiberBundle/Basic.lean", "def_pos": [287, 9], "def_end_pos": [287, 34]}, {"full_name": "uniqueMDiffWithinAt_univ", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "def_pos": [46, 9], "def_end_pos": [46, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p"}, {"tactic": "rw [\u2190 e.left_inv hp]", "annotated_tactic": ["rw [\u2190 e.left_inv hp]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) (\u2191e.symm (\u2191e.toPartialHomeomorph p))"}, {"tactic": "refine (this.preimage_partialHomeomorph e.mdifferentiable.symm (e.map_source hp)).mono ?_", "annotated_tactic": ["refine (this.preimage_partialHomeomorph e.mdifferentiable.symm (e.map_source hp)).<a>mono</a> ?_", [{"full_name": "UniqueMDiffWithinAt.mono", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "def_pos": [70, 9], "def_end_pos": [70, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) (\u2191e.symm (\u2191e.toPartialHomeomorph p))", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 e.symm.target \u2229 \u2191e.symm.symm \u207b\u00b9' s \u00d7\u02e2 univ \u2286 TotalSpace.proj \u207b\u00b9' s"}, {"tactic": "rintro y \u27e8hy, hys, -\u27e9", "annotated_tactic": ["rintro y \u27e8hy, hys, -\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 e.symm.target \u2229 \u2191e.symm.symm \u207b\u00b9' s \u00d7\u02e2 univ \u2286 TotalSpace.proj \u207b\u00b9' s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\ny : TotalSpace F Z\nhy : y \u2208 e.symm.target\nhys : (\u2191e.symm.symm y).1 \u2208 s\n\u22a2 y \u2208 TotalSpace.proj \u207b\u00b9' s"}, {"tactic": "rwa [PartialHomeomorph.symm_symm, e.coe_coe, e.coe_fst hy] at hys", "annotated_tactic": ["rwa [<a>PartialHomeomorph.symm_symm</a>, e.coe_coe, e.coe_fst hy] at hys", [{"full_name": "PartialHomeomorph.symm_symm", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [364, 29], "def_end_pos": [364, 38]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\ny : TotalSpace F Z\nhy : y \u2208 e.symm.target\nhys : (\u2191e.symm.symm y).1 \u2208 s\n\u22a2 y \u2208 TotalSpace.proj \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "rw [\u2190 Prod.mk.eta (p := e p), FiberBundle.trivializationAt_proj_fst]", "annotated_tactic": ["rw [\u2190 <a>Prod.mk.eta</a> (p := e p), <a>FiberBundle.trivializationAt_proj_fst</a>]", [{"full_name": "Prod.mk.eta", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [28, 9], "def_end_pos": [28, 15]}, {"full_name": "FiberBundle.trivializationAt_proj_fst", "def_path": "Mathlib/Topology/FiberBundle/Basic.lean", "def_pos": [287, 9], "def_end_pos": [287, 34]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (p.proj, (\u2191e p).2)"}, {"tactic": "exact hs.prod (uniqueMDiffWithinAt_univ _)", "annotated_tactic": ["exact hs.prod (<a>uniqueMDiffWithinAt_univ</a> _)", [{"full_name": "uniqueMDiffWithinAt_univ", "def_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "def_pos": [46, 9], "def_end_pos": [46, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (p.proj, (\u2191e p).2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "full_name": "List.takeD_eq_takeDTR", "start": [332, 10], "end": [333, 48], "traced_tactics": [{"tactic": "funext \u03b1 f n l", "annotated_tactic": ["funext \u03b1 f n l", []], "state_before": "\u22a2 @takeD = @takeDTR", "state_after": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : List \u03b1\nl : \u03b1\n\u22a2 takeD f n l = takeDTR f n l"}, {"tactic": "simp [takeDTR, takeDTR_go_eq]", "annotated_tactic": ["simp [<a>takeDTR</a>, <a>takeDTR_go_eq</a>]", [{"full_name": "List.takeDTR", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [320, 5], "def_end_pos": [320, 12]}, {"full_name": "List.takeDTR_go_eq", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [327, 9], "def_end_pos": [327, 22]}]], "state_before": "case h.h.h.h\n\u03b1 : Type u_1\nf : Nat\nn : List \u03b1\nl : \u03b1\n\u22a2 takeD f n l = takeDTR f n l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "full_name": "StarAlgHom.norm_apply_le", "start": [145, 1], "end": [146, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.IsRotated.perm", "start": [471, 1], "end": [472, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UnitInterval.lean", "full_name": "unitInterval.mem_iff_one_sub_mem", "start": [62, 1], "end": [64, 53], "traced_tactics": [{"tactic": "rw [mem_Icc, mem_Icc]", "annotated_tactic": ["rw [<a>mem_Icc</a>, <a>mem_Icc</a>]", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}]], "state_before": "t : \u211d\n\u22a2 t \u2208 I \u2194 1 - t \u2208 I", "state_after": "t : \u211d\n\u22a2 0 \u2264 t \u2227 t \u2264 1 \u2194 0 \u2264 1 - t \u2227 1 - t \u2264 1"}, {"tactic": "constructor <;> intro <;> constructor <;> linarith", "annotated_tactic": ["constructor <;> intro <;> constructor <;> linarith", []], "state_before": "t : \u211d\n\u22a2 0 \u2264 t \u2227 t \u2264 1 \u2194 0 \u2264 1 - t \u2227 1 - t \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.Fork.IsLimit.lift_\u03b9", "start": [431, 1], "end": [432, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order.lean", "full_name": "isOpen_sup", "start": [669, 1], "end": [671, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_tsub_one_le_encard_diff_singleton", "start": [252, 1], "end": [254, 62], "traced_tactics": [{"tactic": "rw [\u2190 encard_singleton x]", "annotated_tactic": ["rw [\u2190 <a>encard_singleton</a> x]", [{"full_name": "Set.encard_singleton", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [107, 17], "def_end_pos": [107, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 s.encard - 1 \u2264 (s \\ {x}).encard", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 s.encard - {x}.encard \u2264 (s \\ {x}).encard"}, {"tactic": "apply tsub_encard_le_encard_diff", "annotated_tactic": ["apply <a>tsub_encard_le_encard_diff</a>", [{"full_name": "Set.tsub_encard_le_encard_diff", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [225, 9], "def_end_pos": [225, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 s.encard - {x}.encard \u2264 (s \\ {x}).encard", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "AddSubgroup.toSubgroup_comap", "start": [1355, 1], "end": [1357, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "Ideal.IsPrime.mul_mem_pow", "start": [1243, 1], "end": [1252, 94], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn : \u2115\nh : a * b \u2208 I ^ n\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ n", "state_after": "case zero\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nh : a * b \u2208 I ^ 0\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ 0\n\ncase succ\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)"}, {"tactic": "by_cases hI0 : I = \u22a5", "annotated_tactic": ["by_cases hI0 : I = \u22a5", []], "state_before": "case succ\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\nhI0 : I = \u22a5\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)\n\ncase neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\nhI0 : \u00acI = \u22a5\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)"}, {"tactic": "simp only [\u2190 Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, \u2190 Ideal.dvd_iff_le, \u2190\n  Ideal.span_singleton_mul_span_singleton] at h \u22a2", "annotated_tactic": ["simp only [\u2190 <a>Submodule.span_singleton_le_iff_mem</a>, <a>Ideal.submodule_span_eq</a>, \u2190 <a>Ideal.dvd_iff_le</a>, \u2190\n    <a>Ideal.span_singleton_mul_span_singleton</a>] at h \u22a2", [{"full_name": "Submodule.span_singleton_le_iff_mem", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [624, 9], "def_end_pos": [624, 34]}, {"full_name": "Ideal.submodule_span_eq", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 26]}, {"full_name": "Ideal.dvd_iff_le", "def_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "def_pos": [636, 9], "def_end_pos": [636, 25]}, {"full_name": "Ideal.span_singleton_mul_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [514, 9], "def_end_pos": [514, 42]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\nhI0 : \u00acI = \u22a5\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)", "state_after": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}"}, {"tactic": "by_cases ha : I \u2223 span {a}", "annotated_tactic": ["by_cases ha : I \u2223 <a>span</a> {a}", [{"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\nha : I \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}\n\ncase neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\nha : \u00acI \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}"}, {"tactic": "rw [mul_comm] at h", "annotated_tactic": ["rw [<a>mul_comm</a>] at h", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\nha : \u00acI \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}", "state_after": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {b} * span {a}\nha : \u00acI \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}"}, {"tactic": "exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Prime.pow_dvd_of_dvd_mul_right</a> ((<a>Ideal.prime_iff_isPrime</a> hI0).<a>mpr</a> hI) _ ha h)", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Prime.pow_dvd_of_dvd_mul_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [148, 9], "def_end_pos": [148, 39]}, {"full_name": "Ideal.prime_iff_isPrime", "def_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "def_pos": [715, 9], "def_end_pos": [715, 32]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {b} * span {a}\nha : \u00acI \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nh : a * b \u2208 I ^ 0\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ 0", "state_after": "no goals"}, {"tactic": "simpa [pow_succ, hI0] using h", "annotated_tactic": ["simpa [<a>pow_succ</a>, hI0] using h", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nh : a * b \u2208 I ^ (n\u271d + 1)\nhI0 : I = \u22a5\n\u22a2 a \u2208 I \u2228 b \u2208 I ^ (n\u271d + 1)", "state_after": "no goals"}, {"tactic": "exact Or.inl ha", "annotated_tactic": ["exact <a>Or.inl</a> ha", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Field K\ninst\u271d : IsDedekindDomain R\nI : Ideal R\nhI : I.IsPrime\na b : R\nn\u271d : \u2115\nhI0 : \u00acI = \u22a5\nh : I ^ (n\u271d + 1) \u2223 span {a} * span {b}\nha : I \u2223 span {a}\n\u22a2 I \u2223 span {a} \u2228 I ^ (n\u271d + 1) \u2223 span {b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "full_name": "MeasureTheory.mem\u2112p_zero_iff_aestronglyMeasurable", "start": [183, 1], "end": [184, 83], "traced_tactics": [{"tactic": "simp [Mem\u2112p, snorm_exponent_zero]", "annotated_tactic": ["simp [<a>Mem\u2112p</a>, <a>snorm_exponent_zero</a>]", [{"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [113, 5], "def_end_pos": [113, 10]}, {"full_name": "MeasureTheory.snorm_exponent_zero", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\n\u22a2 Mem\u2112p f 0 \u03bc \u2194 AEStronglyMeasurable f \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectLimit.lean", "full_name": "Ring.DirectLimit.congr_symm_apply_of", "start": [950, 1], "end": [955, 82], "traced_tactics": [{"tactic": "simp only [congr, RingEquiv.ofHomInv_symm_apply, map_apply_of, RingHom.coe_coe]", "annotated_tactic": ["simp only [<a>congr</a>, <a>RingEquiv.ofHomInv_symm_apply</a>, <a>map_apply_of</a>, <a>RingHom.coe_coe</a>]", [{"full_name": "Ring.DirectLimit.congr", "def_path": "Mathlib/Algebra/DirectLimit.lean", "def_pos": [929, 5], "def_end_pos": [929, 10]}, {"full_name": "RingEquiv.ofHomInv_symm_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [815, 3], "def_end_pos": [815, 8]}, {"full_name": "Ring.DirectLimit.map_apply_of", "def_path": "Mathlib/Algebra/DirectLimit.lean", "def_pos": [902, 15], "def_end_pos": [902, 27]}, {"full_name": "RingHom.coe_coe", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [446, 9], "def_end_pos": [446, 16]}]], "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\n\u03b9 : Type v\ninst\u271d\u2076 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2075 : (i : \u03b9) \u2192 CommRing (G i)\nf\u271d : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192 G j\nf'\u271d : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192+* G j\nP : Type u\u2081\ninst\u271d\u2074 : CommRing P\ng\u271d : (i : \u03b9) \u2192 G i \u2192+* P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), (g\u271d j) (f\u271d i j hij x) = (g\u271d i) x\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u2192+* G j\nG' : \u03b9 \u2192 Type v'\ninst\u271d\u00b3 : (i : \u03b9) \u2192 CommRing (G' i)\nf' : (i j : \u03b9) \u2192 i \u2264 j \u2192 G' i \u2192+* G' j\nG'' : \u03b9 \u2192 Type v''\ninst\u271d\u00b2 : (i : \u03b9) \u2192 CommRing (G'' i)\nf'' : (i j : \u03b9) \u2192 i \u2264 j \u2192 G'' i \u2192+* G'' j\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ne : (i : \u03b9) \u2192 G i \u2243+* G' i\nhe : \u2200 (i j : \u03b9) (h : i \u2264 j), (e j).toRingHom.comp (f i j h) = (f' i j h).comp \u2191(e i)\ni : \u03b9\ng : G' i\n\u22a2 (congr e he).symm ((of G' (fun x x_1 h => \u21d1(f' x x_1 h)) i) g) = (of G (fun x x_1 h => \u21d1(f x x_1 h)) i) ((e i).symm g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.comap_le_comap_of_surjective", "start": [2992, 1], "end": [2994, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "BilinForm.toMatrix_apply", "start": [244, 1], "end": [246, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Field/Power.lean", "full_name": "zpow_lt_iff_lt", "start": [79, 1], "end": [80, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "eq_one_div_of_mul_eq_one_left", "start": [515, 1], "end": [516, 44], "traced_tactics": [{"tactic": "rw [eq_inv_of_mul_eq_one_left h, one_div]", "annotated_tactic": ["rw [<a>eq_inv_of_mul_eq_one_left</a> h, <a>one_div</a>]", [{"full_name": "eq_inv_of_mul_eq_one_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 34]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\nM : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na b c d : \u03b1\nh : b * a = 1\n\u22a2 b = 1 / a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProdFun_tsum_left", "start": [146, 1], "end": [148, 75], "traced_tactics": [{"tactic": "simp_rw [compProdFun, (measure_sum_seq \u03ba _).symm, lintegral_sum_measure]", "annotated_tactic": ["simp_rw [<a>compProdFun</a>, (<a>measure_sum_seq</a> \u03ba _).<a>symm</a>, <a>lintegral_sum_measure</a>]", [{"full_name": "ProbabilityTheory.kernel.compProdFun", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [88, 19], "def_end_pos": [88, 30]}, {"full_name": "ProbabilityTheory.kernel.measure_sum_seq", "def_path": "Mathlib/Probability/Kernel/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 24]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "MeasureTheory.lintegral_sum_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [631, 9], "def_end_pos": [631, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns\u271d : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03ba\na : \u03b1\ns : Set (\u03b2 \u00d7 \u03b3)\n\u22a2 compProdFun \u03ba \u03b7 a s = \u2211' (n : \u2115), compProdFun (seq \u03ba n) \u03b7 a s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.foldl_swap", "start": [1487, 1], "end": [1489, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Support.lean", "full_name": "MulAction.Supports.smul", "start": [58, 1], "end": [63, 48], "traced_tactics": [{"tactic": "rintro g' hg'", "annotated_tactic": ["rintro g' hg'", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\n\u22a2 Supports G (g \u2022 s) (g \u2022 b)", "state_after": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 g' \u2022 g \u2022 b = g \u2022 b"}, {"tactic": "rw [smul_comm, h]", "annotated_tactic": ["rw [<a>smul_comm</a>, h]", [{"full_name": "SMulCommClass.smul_comm", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [183, 3], "def_end_pos": [183, 12]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 g' \u2022 g \u2022 b = g \u2022 b", "state_after": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 g' \u2022 a = a"}, {"tactic": "rintro a ha", "annotated_tactic": ["rintro a ha", []], "state_before": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 g' \u2022 a = a", "state_after": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\n\u22a2 g' \u2022 a = a"}, {"tactic": "have := Set.forall_mem_image.1 hg' ha", "annotated_tactic": ["have := <a>Set.forall_mem_image</a>.1 hg' ha", [{"full_name": "Set.forall_mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [223, 9], "def_end_pos": [223, 25]}]], "state_before": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\n\u22a2 g' \u2022 a = a", "state_after": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\nthis : g' \u2022 g \u2022 a = g \u2022 a\n\u22a2 g' \u2022 a = a"}, {"tactic": "rwa [smul_comm, smul_left_cancel_iff] at this", "annotated_tactic": ["rwa [<a>smul_comm</a>, <a>smul_left_cancel_iff</a>] at this", [{"full_name": "SMulCommClass.smul_comm", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [183, 3], "def_end_pos": [183, 12]}, {"full_name": "smul_left_cancel_iff", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [162, 9], "def_end_pos": [162, 29]}]], "state_before": "case a\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\nthis : g' \u2022 g \u2022 a = g \u2022 a\n\u22a2 g' \u2022 a = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Surreal/Basic.lean", "full_name": "SetTheory.PGame.Numeric.neg", "start": [213, 1], "end": [215, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "full_name": "Set.projIci_eq_self", "start": [99, 1], "end": [99, 98], "traced_tactics": [{"tactic": "simp [projIci, Subtype.ext_iff]", "annotated_tactic": ["simp [<a>projIci</a>, <a>Subtype.ext_iff</a>]", [{"full_name": "Set.projIci", "def_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "def_pos": [40, 5], "def_end_pos": [40, 12]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\n\u22a2 projIci a x = \u27e8a, \u22ef\u27e9 \u2194 x \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Commute/Units.lean", "full_name": "Commute.units_zpow_right", "start": [104, 1], "end": [106, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Polynomial.lean", "full_name": "AnalyticOn.eval_continuousLinearMap'", "start": [61, 1], "end": [63, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "full_name": "nnnorm_mul_le", "start": [237, 1], "end": [239, 41], "traced_tactics": [{"tactic": "simpa only [\u2190 norm_toNNReal, \u2190 Real.toNNReal_mul (norm_nonneg _)] using\n  Real.toNNReal_mono (norm_mul_le _ _)", "annotated_tactic": ["simpa only [\u2190 <a>norm_toNNReal</a>, \u2190 <a>Real.toNNReal_mul</a> (<a>norm_nonneg</a> _)] using\n    <a>Real.toNNReal_mono</a> (<a>norm_mul_le</a> _ _)", [{"full_name": "norm_toNNReal", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [776, 15], "def_end_pos": [776, 28]}, {"full_name": "Real.toNNReal_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [838, 9], "def_end_pos": [838, 21]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "Real.toNNReal_mono", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [392, 19], "def_end_pos": [392, 44]}, {"full_name": "norm_mul_le", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [233, 9], "def_end_pos": [233, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : NonUnitalSeminormedRing \u03b1\na b : \u03b1\n\u22a2 \u2016a * b\u2016\u208a \u2264 \u2016a\u2016\u208a * \u2016b\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Disjoint.lean", "full_name": "IsGLB.biUnion_Ioi_eq", "start": [201, 1], "end": [205, 30], "traced_tactics": [{"tactic": "refine (iUnion\u2082_subset fun x hx => ?_).antisymm fun x hx => ?_", "annotated_tactic": ["refine (<a>iUnion\u2082_subset</a> fun x hx => ?_).<a>antisymm</a> fun x hx => ?_", [{"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [658, 7], "def_end_pos": [658, 32]}]], "state_before": "\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\n\u22a2 \u22c3 x \u2208 s, Ioi x = Ioi a", "state_after": "case refine_1\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 s\n\u22a2 Ioi x \u2286 Ioi a\n\ncase refine_2\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 Ioi a\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ioi i"}, {"tactic": "exact Ioi_subset_Ioi (h.1 hx)", "annotated_tactic": ["exact <a>Ioi_subset_Ioi</a> (h.1 hx)", [{"full_name": "Set.Ioi_subset_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 23]}]], "state_before": "case refine_1\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 s\n\u22a2 Ioi x \u2286 Ioi a", "state_after": "no goals"}, {"tactic": "rcases h.exists_between hx with \u27e8y, hys, _, hyx\u27e9", "annotated_tactic": ["rcases h.exists_between hx with \u27e8y, hys, _, hyx\u27e9", []], "state_before": "case refine_2\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 Ioi a\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ioi i", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ioi i"}, {"tactic": "exact mem_biUnion hys hyx", "annotated_tactic": ["exact <a>mem_biUnion</a> hys hyx", [{"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [818, 9], "def_end_pos": [818, 20]}]], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\nh : IsGLB s a\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ioi i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Module/Defs.lean", "full_name": "PosSMulReflectLT.toPosSMulMono", "start": [399, 1], "end": [400, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.conj_smul", "start": [370, 1], "end": [372, 56], "traced_tactics": [{"tactic": "rw [conj_eq_re_sub_im, conj_eq_re_sub_im, smul_re, smul_im, ofReal_mul, ofReal_mul,\n  real_smul_eq_coe_mul r (_ - _), mul_sub, mul_assoc]", "annotated_tactic": ["rw [<a>conj_eq_re_sub_im</a>, <a>conj_eq_re_sub_im</a>, <a>smul_re</a>, <a>smul_im</a>, <a>ofReal_mul</a>, <a>ofReal_mul</a>,\n    <a>real_smul_eq_coe_mul</a> r (_ - _), <a>mul_sub</a>, <a>mul_assoc</a>]", [{"full_name": "RCLike.conj_eq_re_sub_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [358, 9], "def_end_pos": [358, 26]}, {"full_name": "RCLike.conj_eq_re_sub_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [358, 9], "def_end_pos": [358, 26]}, {"full_name": "RCLike.smul_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [263, 9], "def_end_pos": [263, 16]}, {"full_name": "RCLike.smul_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [268, 9], "def_end_pos": [268, 16]}, {"full_name": "RCLike.ofReal_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 19]}, {"full_name": "RCLike.ofReal_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [226, 9], "def_end_pos": [226, 19]}, {"full_name": "RCLike.real_smul_eq_coe_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 29]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nr : \u211d\nz : K\n\u22a2 (starRingEnd K) (r \u2022 z) = r \u2022 (starRingEnd K) z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Monotone.lean", "full_name": "Monotone.map_sSup_of_continuousAt'", "start": [31, 1], "end": [36, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "himp_iff_imp", "start": [1131, 1], "end": [1132, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Nilpotent.lean", "full_name": "Polynomial.isNilpotent_C_mul_pow_X_of_isNilpotent", "start": [33, 1], "end": [38, 24], "traced_tactics": [{"tactic": "refine Commute.isNilpotent_mul_left (commute_X_pow _ _).symm ?_", "annotated_tactic": ["refine <a>Commute.isNilpotent_mul_left</a> (<a>commute_X_pow</a> _ _).<a>symm</a> ?_", [{"full_name": "Commute.isNilpotent_mul_left", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [229, 9], "def_end_pos": [229, 29]}, {"full_name": "Polynomial.commute_X_pow", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [632, 9], "def_end_pos": [632, 22]}, {"full_name": "Commute.symm", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [70, 19], "def_end_pos": [70, 23]}]], "state_before": "R : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (C r * X ^ n)", "state_after": "R : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (C r)"}, {"tactic": "obtain \u27e8m, hm\u27e9 := hnil", "annotated_tactic": ["obtain \u27e8m, hm\u27e9 := hnil", []], "state_before": "R : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (C r)", "state_after": "case intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 IsNilpotent (C r)"}, {"tactic": "refine \u27e8m, ?_\u27e9", "annotated_tactic": ["refine \u27e8m, ?_\u27e9", []], "state_before": "case intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 IsNilpotent (C r)", "state_after": "case intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 C r ^ m = 0"}, {"tactic": "rw [\u2190 C_pow, hm, C_0]", "annotated_tactic": ["rw [\u2190 <a>C_pow</a>, hm, <a>C_0</a>]", [{"full_name": "Polynomial.C_pow", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [541, 9], "def_end_pos": [541, 14]}, {"full_name": "Polynomial.C_0", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 12]}]], "state_before": "case intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 C r ^ m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean", "full_name": "AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ", "start": [615, 1], "end": [621, 28], "traced_tactics": [{"tactic": "ext z", "annotated_tactic": ["ext z", []], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\n\u22a2 awayToSection \ud835\udc9c f \u226b (structureSheaf \ud835\udc9c).presheaf.germ x =\n    HomogeneousLocalization.mapId \ud835\udc9c \u22ef \u226b (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv", "state_after": "case w\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nz : (forget CommRingCat).obj (CommRingCat.of (A\u2070_ f))\n\u22a2 (awayToSection \ud835\udc9c f \u226b (structureSheaf \ud835\udc9c).presheaf.germ x) z =\n    (HomogeneousLocalization.mapId \ud835\udc9c \u22ef \u226b (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv) z"}, {"tactic": "apply (Proj.stalkIso' \ud835\udc9c x).eq_symm_apply.mpr", "annotated_tactic": ["apply (<a>Proj.stalkIso'</a> \ud835\udc9c x).eq_symm_apply.mpr", [{"full_name": "AlgebraicGeometry.Proj.stalkIso'", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean", "def_pos": [373, 5], "def_end_pos": [373, 19]}]], "state_before": "case w\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nz : (forget CommRingCat).obj (CommRingCat.of (A\u2070_ f))\n\u22a2 (awayToSection \ud835\udc9c f \u226b (structureSheaf \ud835\udc9c).presheaf.germ x) z =\n    (HomogeneousLocalization.mapId \ud835\udc9c \u22ef \u226b (Proj.stalkIso' \ud835\udc9c \u2191x).toCommRingCatIso.inv) z", "state_after": "case w\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nz : (forget CommRingCat).obj (CommRingCat.of (A\u2070_ f))\n\u22a2 (Proj.stalkIso' \ud835\udc9c \u2191x).toEquiv ((awayToSection \ud835\udc9c f \u226b (structureSheaf \ud835\udc9c).presheaf.germ x) z) =\n    (HomogeneousLocalization.mapId \ud835\udc9c \u22ef) z"}, {"tactic": "apply Proj.stalkIso'_germ", "annotated_tactic": ["apply <a>Proj.stalkIso'_germ</a>", [{"full_name": "AlgebraicGeometry.Proj.stalkIso'_germ", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean", "def_pos": [389, 9], "def_end_pos": [389, 28]}]], "state_before": "case w\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nx : \u21a5(pbo f)\nz : (forget CommRingCat).obj (CommRingCat.of (A\u2070_ f))\n\u22a2 (Proj.stalkIso' \ud835\udc9c \u2191x).toEquiv ((awayToSection \ud835\udc9c f \u226b (structureSheaf \ud835\udc9c).presheaf.germ x) z) =\n    (HomogeneousLocalization.mapId \ud835\udc9c \u22ef) z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "full_name": "Measurable.nnreal_tsum", "start": [410, 1], "end": [413, 85], "traced_tactics": [{"tactic": "simp_rw [NNReal.tsum_eq_toNNReal_tsum]", "annotated_tactic": ["simp_rw [<a>NNReal.tsum_eq_toNNReal_tsum</a>]", [{"full_name": "NNReal.tsum_eq_toNNReal_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1133, 9], "def_end_pos": [1133, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun x => \u2211' (i : \u03b9), f i x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun x => (\u2211' (b : \u03b9), \u2191(f b x)).toNNReal"}, {"tactic": "exact (Measurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal", "annotated_tactic": ["exact (<a>Measurable.ennreal_tsum</a> fun i => (h i).<a>coe_nnreal_ennreal</a>).<a>ennreal_toNNReal</a>", [{"full_name": "Measurable.ennreal_tsum", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "def_pos": [395, 9], "def_end_pos": [395, 32]}, {"full_name": "Measurable.coe_nnreal_ennreal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "def_pos": [181, 9], "def_end_pos": [181, 38]}, {"full_name": "Measurable.ennreal_toNNReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "def_pos": [358, 9], "def_end_pos": [358, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Sort y\ns t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 Measurable fun x => (\u2211' (b : \u03b9), \u2191(f b x)).toNNReal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.to_nat_inj", "start": [522, 1], "end": [523, 91], "traced_tactics": [{"tactic": "rw [\u2190 PosNum.of_to_nat, \u2190 PosNum.of_to_nat, h]", "annotated_tactic": ["rw [\u2190 <a>PosNum.of_to_nat</a>, \u2190 <a>PosNum.of_to_nat</a>, h]", [{"full_name": "PosNum.of_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [517, 9], "def_end_pos": [517, 18]}, {"full_name": "PosNum.of_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [517, 9], "def_end_pos": [517, 18]}]], "state_before": "\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m = \u2191n\n\u22a2 pos m = pos n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.add_conj", "start": [752, 1], "end": [753, 42], "traced_tactics": [{"tactic": "simp [two_mul, ofReal']", "annotated_tactic": ["simp [<a>two_mul</a>, <a>ofReal'</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [179, 9], "def_end_pos": [179, 16]}, {"full_name": "Complex.ofReal'", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [90, 5], "def_end_pos": [90, 12]}]], "state_before": "z : \u2102\n\u22a2 (z + (starRingEnd \u2102) z).re = (\u2191(2 * z.re)).re \u2227 (z + (starRingEnd \u2102) z).im = (\u2191(2 * z.re)).im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/AdjoinRoot.lean", "full_name": "AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk", "start": [823, 1], "end": [833, 45], "traced_tactics": [{"tactic": "rw [quotAdjoinRootEquivQuotPolynomialQuot, RingEquiv.symm_trans_apply,\n  RingEquiv.symm_trans_apply, RingEquiv.symm_trans_apply, RingEquiv.symm_symm,\n  Polynomial.quotQuotEquivComm_mk, Ideal.quotEquivOfEq_symm, Ideal.quotEquivOfEq_mk, \u2190\n  RingHom.comp_apply, \u2190 DoubleQuot.quotQuotMk,\n  quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk,\n  quotMapOfEquivQuotMapCMapSpanMk_symm_mk]", "annotated_tactic": ["rw [<a>quotAdjoinRootEquivQuotPolynomialQuot</a>, <a>RingEquiv.symm_trans_apply</a>,\n    <a>RingEquiv.symm_trans_apply</a>, <a>RingEquiv.symm_trans_apply</a>, <a>RingEquiv.symm_symm</a>,\n    <a>Polynomial.quotQuotEquivComm_mk</a>, <a>Ideal.quotEquivOfEq_symm</a>, <a>Ideal.quotEquivOfEq_mk</a>, \u2190\n    <a>RingHom.comp_apply</a>, \u2190 <a>DoubleQuot.quotQuotMk</a>,\n    <a>quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk</a>,\n    <a>quotMapOfEquivQuotMapCMapSpanMk_symm_mk</a>]", [{"full_name": "AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [805, 5], "def_end_pos": [805, 42]}, {"full_name": "RingEquiv.symm_trans_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [333, 9], "def_end_pos": [333, 25]}, {"full_name": "RingEquiv.symm_trans_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [333, 9], "def_end_pos": [333, 25]}, {"full_name": "RingEquiv.symm_trans_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [333, 9], "def_end_pos": [333, 25]}, {"full_name": "RingEquiv.symm_symm", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [282, 9], "def_end_pos": [282, 18]}, {"full_name": "AdjoinRoot.Polynomial.quotQuotEquivComm_mk", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [786, 9], "def_end_pos": [786, 40]}, {"full_name": "Ideal.quotEquivOfEq_symm", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [316, 9], "def_end_pos": [316, 27]}, {"full_name": "Ideal.quotEquivOfEq_mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [310, 9], "def_end_pos": [310, 25]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [670, 9], "def_end_pos": [670, 19]}, {"full_name": "DoubleQuot.quotQuotMk", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [728, 5], "def_end_pos": [728, 15]}, {"full_name": "AdjoinRoot.quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [765, 9], "def_end_pos": [765, 68]}, {"full_name": "AdjoinRoot.quotMapOfEquivQuotMapCMapSpanMk_symm_mk", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [737, 9], "def_end_pos": [737, 48]}]], "state_before": "R : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf p : R[X]\n\u22a2 (quotAdjoinRootEquivQuotPolynomialQuot I f).symm\n      ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) p)) =\n    (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/NormNum/Inv.lean", "full_name": "Mathlib.Meta.NormNum.isRat_inv_pos", "start": [106, 1], "end": [110, 24], "traced_tactics": [{"tactic": "rintro \u27e8_, rfl\u27e9", "annotated_tactic": ["rintro \u27e8_, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\na : \u03b1\nn d : \u2115\n\u22a2 IsRat a (Int.ofNat n.succ) d \u2192 IsRat a\u207b\u00b9 (Int.ofNat d) n.succ", "state_after": "case mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn d : \u2115\ninv\u271d : Invertible \u2191d\n\u22a2 IsRat (\u2191(Int.ofNat n.succ) * \u215f\u2191d)\u207b\u00b9 (Int.ofNat d) n.succ"}, {"tactic": "have := invertibleOfNonzero (\u03b1 := \u03b1) (Nat.cast_ne_zero.2 (Nat.succ_ne_zero n))", "annotated_tactic": ["have := <a>invertibleOfNonzero</a> (\u03b1 := \u03b1) (<a>Nat.cast_ne_zero</a>.2 (<a>Nat.succ_ne_zero</a> n))", [{"full_name": "invertibleOfNonzero", "def_path": "Mathlib/Algebra/GroupWithZero/Invertible.lean", "def_pos": [51, 5], "def_end_pos": [51, 24]}, {"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "Nat.succ_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [708, 17], "def_end_pos": [708, 29]}]], "state_before": "case mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn d : \u2115\ninv\u271d : Invertible \u2191d\n\u22a2 IsRat (\u2191(Int.ofNat n.succ) * \u215f\u2191d)\u207b\u00b9 (Int.ofNat d) n.succ", "state_after": "case mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn d : \u2115\ninv\u271d : Invertible \u2191d\nthis : Invertible \u2191n.succ\n\u22a2 IsRat (\u2191(Int.ofNat n.succ) * \u215f\u2191d)\u207b\u00b9 (Int.ofNat d) n.succ"}, {"tactic": "exact \u27e8this, by simp\u27e9", "annotated_tactic": ["exact \u27e8this, by simp\u27e9", []], "state_before": "case mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn d : \u2115\ninv\u271d : Invertible \u2191d\nthis : Invertible \u2191n.succ\n\u22a2 IsRat (\u2191(Int.ofNat n.succ) * \u215f\u2191d)\u207b\u00b9 (Int.ofNat d) n.succ", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn d : \u2115\ninv\u271d : Invertible \u2191d\nthis : Invertible \u2191n.succ\n\u22a2 (\u2191(Int.ofNat n.succ) * \u215f\u2191d)\u207b\u00b9 = \u2191(Int.ofNat d) * \u215f\u2191n.succ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Interval.lean", "full_name": "Sum.Lex.Ioo_inr_inl", "start": [407, 1], "end": [408, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean", "full_name": "CategoryTheory.ShortComplex.cyclesMap_i", "start": [523, 1], "end": [525, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "full_name": "ProbabilityTheory.kernel.density_nonneg", "start": [497, 1], "end": [501, 75], "traced_tactics": [{"tactic": "refine le_limsup_of_frequently_le ?_ ?_", "annotated_tactic": ["refine <a>le_limsup_of_frequently_le</a> ?_ ?_", [{"full_name": "Filter.le_limsup_of_frequently_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\na : \u03b1\nx : \u03b3\ns : Set \u03b2\n\u22a2 0 \u2264 density \u03ba \u03bd a x s", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\na : \u03b1\nx : \u03b3\ns : Set \u03b2\n\u22a2 \u2203\u1da0 (x_1 : \u2115) in atTop, 0 \u2264 densityProcess \u03ba \u03bd x_1 a x s\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\na : \u03b1\nx : \u03b3\ns : Set \u03b2\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => densityProcess \u03ba \u03bd n a x s"}, {"tactic": "exact frequently_of_forall (fun n \u21a6 densityProcess_nonneg _ _ _ _ _ _)", "annotated_tactic": ["exact <a>frequently_of_forall</a> (fun n \u21a6 <a>densityProcess_nonneg</a> _ _ _ _ _ _)", [{"full_name": "Filter.frequently_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 29]}, {"full_name": "ProbabilityTheory.kernel.densityProcess_nonneg", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [160, 7], "def_end_pos": [160, 28]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\na : \u03b1\nx : \u03b3\ns : Set \u03b2\n\u22a2 \u2203\u1da0 (x_1 : \u2115) in atTop, 0 \u2264 densityProcess \u03ba \u03bd x_1 a x s", "state_after": "no goals"}, {"tactic": "exact isBoundedUnder_of \u27e81, fun n \u21a6 densityProcess_le_one h\u03ba\u03bd _ _ _ _\u27e9", "annotated_tactic": ["exact <a>isBoundedUnder_of</a> \u27e81, fun n \u21a6 <a>densityProcess_le_one</a> h\u03ba\u03bd _ _ _ _\u27e9", [{"full_name": "Filter.isBoundedUnder_of", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [70, 9], "def_end_pos": [70, 26]}, {"full_name": "ProbabilityTheory.kernel.densityProcess_le_one", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [176, 7], "def_end_pos": [176, 28]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\na : \u03b1\nx : \u03b3\ns : Set \u03b2\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => densityProcess \u03ba \u03bd n a x s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.le_inter", "start": [1840, 1], "end": [1847, 67], "traced_tactics": [{"tactic": "revert s u", "annotated_tactic": ["revert s u", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nh\u2081 : s \u2264 t\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 t \u2229 u", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u"}, {"tactic": "refine @(Multiset.induction_on t ?_ fun a t IH => ?_) <;> intros s u h\u2081 h\u2082", "annotated_tactic": ["refine @(<a>Multiset.induction_on</a> t ?_ fun a t IH => ?_) <;> intros s u h\u2081 h\u2082", [{"full_name": "Multiset.induction_on", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\ns u : Multiset \u03b1\nh\u2081 : s \u2264 0\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 0 \u2229 u\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u"}, {"tactic": "by_cases h : a \u2208 u", "annotated_tactic": ["by_cases h : a \u2208 u", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2209 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u"}, {"tactic": "simpa only [zero_inter, nonpos_iff_eq_zero] using h\u2081", "annotated_tactic": ["simpa only [<a>zero_inter</a>, <a>nonpos_iff_eq_zero</a>] using h\u2081", [{"full_name": "Multiset.zero_inter", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 19]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\ns u : Multiset \u03b1\nh\u2081 : s \u2264 0\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 0 \u2229 u", "state_after": "no goals"}, {"tactic": "rw [cons_inter_of_pos _ h, \u2190 erase_le_iff_le_cons]", "annotated_tactic": ["rw [<a>cons_inter_of_pos</a> _ h, \u2190 <a>erase_le_iff_le_cons</a>]", [{"full_name": "Multiset.cons_inter_of_pos", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1822, 9], "def_end_pos": [1822, 26]}, {"full_name": "Multiset.erase_le_iff_le_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1149, 9], "def_end_pos": [1149, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 s.erase a \u2264 t \u2229 u.erase a"}, {"tactic": "exact IH (erase_le_iff_le_cons.2 h\u2081) (erase_le_erase _ h\u2082)", "annotated_tactic": ["exact IH (<a>erase_le_iff_le_cons</a>.2 h\u2081) (<a>erase_le_erase</a> _ h\u2082)", [{"full_name": "Multiset.erase_le_iff_le_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1149, 9], "def_end_pos": [1149, 29]}, {"full_name": "Multiset.erase_le_erase", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1145, 9], "def_end_pos": [1145, 23]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 s.erase a \u2264 t \u2229 u.erase a", "state_after": "no goals"}, {"tactic": "rw [cons_inter_of_neg _ h]", "annotated_tactic": ["rw [<a>cons_inter_of_neg</a> _ h]", [{"full_name": "Multiset.cons_inter_of_neg", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1827, 9], "def_end_pos": [1827, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2209 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2209 u\n\u22a2 s \u2264 t \u2229 u"}, {"tactic": "exact IH ((le_cons_of_not_mem <| mt (mem_of_le h\u2082) h).1 h\u2081) h\u2082", "annotated_tactic": ["exact IH ((<a>le_cons_of_not_mem</a> <| <a>mt</a> (<a>mem_of_le</a> h\u2082) h).1 h\u2081) h\u2082", [{"full_name": "Multiset.le_cons_of_not_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [606, 9], "def_end_pos": [606, 27]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "Multiset.mem_of_le", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [546, 9], "def_end_pos": [546, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2209 u\n\u22a2 s \u2264 t \u2229 u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/Basic.lean", "full_name": "Bundle.smoothWithinAt_proj", "start": [248, 1], "end": [250, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/WithBot.lean", "full_name": "Nat.WithBot.add_one_le_of_lt", "start": [83, 1], "end": [89, 28], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n m : WithBot \u2115\nh : n < m\n\u22a2 n + 1 \u2264 m", "state_after": "case bot\nm : WithBot \u2115\nh : \u22a5 < m\n\u22a2 \u22a5 + 1 \u2264 m\n\ncase coe\nm : WithBot \u2115\na\u271d : \u2115\nh : \u2191a\u271d < m\n\u22a2 \u2191a\u271d + 1 \u2264 m"}, {"tactic": "cases m", "annotated_tactic": ["cases m", []], "state_before": "case coe\nm : WithBot \u2115\na\u271d : \u2115\nh : \u2191a\u271d < m\n\u22a2 \u2191a\u271d + 1 \u2264 m", "state_after": "case coe.bot\na\u271d : \u2115\nh : \u2191a\u271d < \u22a5\n\u22a2 \u2191a\u271d + 1 \u2264 \u22a5\n\ncase coe.coe\na\u271d\u00b9 a\u271d : \u2115\nh : \u2191a\u271d\u00b9 < \u2191a\u271d\n\u22a2 \u2191a\u271d\u00b9 + 1 \u2264 \u2191a\u271d"}, {"tactic": "simp only [WithBot.bot_add, bot_le]", "annotated_tactic": ["simp only [<a>WithBot.bot_add</a>, <a>bot_le</a>]", [{"full_name": "WithBot.bot_add", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [625, 9], "def_end_pos": [625, 16]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "case bot\nm : WithBot \u2115\nh : \u22a5 < m\n\u22a2 \u22a5 + 1 \u2264 m", "state_after": "no goals"}, {"tactic": "exact (not_lt_bot h).elim", "annotated_tactic": ["exact (<a>not_lt_bot</a> h).<a>elim</a>", [{"full_name": "not_lt_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [284, 9], "def_end_pos": [284, 19]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case coe.bot\na\u271d : \u2115\nh : \u2191a\u271d < \u22a5\n\u22a2 \u2191a\u271d + 1 \u2264 \u22a5", "state_after": "no goals"}, {"tactic": "rwa [WithBot.coe_lt_coe, \u2190 Nat.add_one_le_iff, \u2190 WithBot.coe_le_coe, WithBot.coe_add,\n  WithBot.coe_one] at h", "annotated_tactic": ["rwa [<a>WithBot.coe_lt_coe</a>, \u2190 <a>Nat.add_one_le_iff</a>, \u2190 <a>WithBot.coe_le_coe</a>, <a>WithBot.coe_add</a>,\n      <a>WithBot.coe_one</a>] at h", [{"full_name": "WithBot.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [299, 9], "def_end_pos": [299, 19]}, {"full_name": "Nat.add_one_le_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 23]}, {"full_name": "WithBot.coe_le_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [213, 9], "def_end_pos": [213, 19]}, {"full_name": "WithBot.coe_add", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [617, 9], "def_end_pos": [617, 16]}, {"full_name": "WithBot.coe_one", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [479, 48], "def_end_pos": [479, 55]}]], "state_before": "case coe.coe\na\u271d\u00b9 a\u271d : \u2115\nh : \u2191a\u271d\u00b9 < \u2191a\u271d\n\u22a2 \u2191a\u271d\u00b9 + 1 \u2264 \u2191a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "full_name": "Matrix.det_updateColumn_add_self", "start": [478, 1], "end": [481, 38], "traced_tactics": [{"tactic": "rw [\u2190 det_transpose, \u2190 updateRow_transpose, \u2190 det_transpose A]", "annotated_tactic": ["rw [\u2190 <a>det_transpose</a>, \u2190 <a>updateRow_transpose</a>, \u2190 <a>det_transpose</a> A]", [{"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"full_name": "Matrix.updateRow_transpose", "def_path": "Mathlib/Data/Matrix/RowCol.lean", "def_pos": [255, 9], "def_end_pos": [255, 28]}, {"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}]], "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 (A.updateColumn i fun k => A k i + A k j).det = A.det", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 (A\u1d40.updateRow i fun k => A k i + A k j).det = A\u1d40.det"}, {"tactic": "exact det_updateRow_add_self A\u1d40 hij", "annotated_tactic": ["exact <a>det_updateRow_add_self</a> A\u1d40 hij", [{"full_name": "Matrix.det_updateRow_add_self", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [472, 9], "def_end_pos": [472, 31]}]], "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 (A\u1d40.updateRow i fun k => A k i + A k j).det = A\u1d40.det", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Complement.lean", "full_name": "Subgroup.MemRightTransversals.toEquiv_apply", "start": [591, 1], "end": [595, 94], "traced_tactics": [{"tactic": "refine (Subtype.ext_iff.mp ?_).trans (Subtype.coe_mk (f q) \u27e8q, rfl\u27e9)", "annotated_tactic": ["refine (Subtype.ext_iff.mp ?_).<a>trans</a> (<a>Subtype.coe_mk</a> (f q) \u27e8q, <a>rfl</a>\u27e9)", [{"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 \u2191((toEquiv \u22ef) q) = f q", "state_after": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 (toEquiv \u22ef) q = \u27e8f q, \u22ef\u27e9"}, {"tactic": "exact (toEquiv (range_mem_rightTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm", "annotated_tactic": ["exact (<a>toEquiv</a> (<a>range_mem_rightTransversals</a> hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).<a>symm</a>", [{"full_name": "Subgroup.MemRightTransversals.toEquiv", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [577, 19], "def_end_pos": [577, 26]}, {"full_name": "Subgroup.range_mem_rightTransversals", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [310, 9], "def_end_pos": [310, 36]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 (toEquiv \u22ef) q = \u27e8f q, \u22ef\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Darboux.lean", "full_name": "Set.OrdConnected.image_deriv", "start": [102, 1], "end": [104, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.tendsto_pure_left", "start": [3259, 1], "end": [3261, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/WSeq.lean", "full_name": "Stream'.WSeq.exists_of_liftRel_right", "start": [1064, 1], "end": [1065, 88], "traced_tactics": [{"tactic": "rw [\u2190 LiftRel.swap] at H", "annotated_tactic": ["rw [\u2190 <a>LiftRel.swap</a>] at H", [{"full_name": "Stream'.WSeq.LiftRel.swap", "def_path": "Mathlib/Data/Seq/WSeq.lean", "def_pos": [559, 9], "def_end_pos": [559, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : WSeq \u03b1\nt : WSeq \u03b2\nH : LiftRel R s t\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2203 a, a \u2208 s \u2227 R a b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : WSeq \u03b1\nt : WSeq \u03b2\nH : swap (LiftRel fun x y => R y x) s t\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2203 a, a \u2208 s \u2227 R a b"}, {"tactic": "exact exists_of_liftRel_left H h", "annotated_tactic": ["exact <a>exists_of_liftRel_left</a> H h", [{"full_name": "Stream'.WSeq.exists_of_liftRel_left", "def_path": "Mathlib/Data/Seq/WSeq.lean", "def_pos": [1053, 9], "def_end_pos": [1053, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : WSeq \u03b1\nt : WSeq \u03b2\nH : swap (LiftRel fun x y => R y x) s t\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2203 a, a \u2208 s \u2227 R a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict", "start": [205, 1], "end": [212, 27], "traced_tactics": [{"tactic": "intro x _", "annotated_tactic": ["intro x _", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\n\u22a2 LocallyIntegrableOn f s \u03bc", "state_after": "X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc"}, {"tactic": "obtain \u27e8t, ht_mem, ht_int\u27e9 := hf x", "annotated_tactic": ["obtain \u27e8t, ht_mem, ht_int\u27e9 := hf x", []], "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc", "state_after": "case intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc"}, {"tactic": "obtain \u27e8u, hu_sub, hu_o, hu_mem\u27e9 := mem_nhds_iff.mp ht_mem", "annotated_tactic": ["obtain \u27e8u, hu_sub, hu_o, hu_mem\u27e9 := mem_nhds_iff.mp ht_mem", []], "state_before": "case intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc", "state_after": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc"}, {"tactic": "refine \u27e8_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), ?_\u27e9", "annotated_tactic": ["refine \u27e8_, <a>inter_mem_nhdsWithin</a> s (hu_o.mem_nhds hu_mem), ?_\u27e9", [{"full_name": "inter_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [155, 9], "def_end_pos": [155, 29]}]], "state_before": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x) \u03bc", "state_after": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableOn f (s \u2229 u) \u03bc"}, {"tactic": "simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using\n  ht_int.mono_set hu_sub", "annotated_tactic": ["simpa only [<a>IntegrableOn</a>, <a>Measure.restrict_restrict</a> hu_o.measurableSet, <a>inter_comm</a>] using\n    ht_int.mono_set hu_sub", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [183, 9], "def_end_pos": [183, 26]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}]], "state_before": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\nR : Type u_5\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\nf g : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f (\u03bc.restrict s)\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t (\u03bc.restrict s)\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableOn f (s \u2229 u) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Comp.lean", "full_name": "DifferentiableAt.comp_differentiableWithinAt", "start": [126, 1], "end": [128, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.smul_sSup", "start": [675, 1], "end": [678, 70], "traced_tactics": [{"tactic": "simp_rw [\u2190 smul_one_mul c (sSup s), ENNReal.mul_sSup, smul_one_mul]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_one_mul</a> c (<a>sSup</a> s), <a>ENNReal.mul_sSup</a>, <a>smul_one_mul</a>]", [{"full_name": "smul_one_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [650, 7], "def_end_pos": [650, 19]}, {"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [42, 3], "def_end_pos": [42, 7]}, {"full_name": "ENNReal.mul_sSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [661, 9], "def_end_pos": [661, 17]}, {"full_name": "smul_one_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [650, 7], "def_end_pos": [650, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d : Set \u211d\u22650\u221e\nR : Type u_4\ninst\u271d\u00b9 : SMul R \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nc : R\n\u22a2 c \u2022 sSup s = \u2a06 i \u2208 s, c \u2022 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/UniqueProds.lean", "full_name": "UniqueMul.of_image_filter", "start": [216, 1], "end": [224, 48], "traced_tactics": [{"tactic": "specialize huH (mem_image_of_mem _ ha) (mem_image_of_mem _ hb)", "annotated_tactic": ["specialize huH (<a>mem_image_of_mem</a> _ ha) (<a>mem_image_of_mem</a> _ hb)", [{"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}]], "state_before": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuH : UniqueMul (image (\u21d1f) A) (image (\u21d1f) B) aH bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\n\u22a2 a = aG \u2227 b = bG", "state_after": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : f a * f b = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a = aG \u2227 b = bG"}, {"tactic": "rw [\u2190 map_mul, he, map_mul, hae, hbe] at huH", "annotated_tactic": ["rw [\u2190 <a>map_mul</a>, he, <a>map_mul</a>, hae, hbe] at huH", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}]], "state_before": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : f a * f b = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a = aG \u2227 b = bG", "state_after": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a = aG \u2227 b = bG"}, {"tactic": "refine huG ?_ ?_ he <;> rw [mem_filter]", "annotated_tactic": ["refine huG ?_ ?_ he <;> rw [<a>mem_filter</a>]", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}]], "state_before": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a = aG \u2227 b = bG", "state_after": "case refine_1\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a \u2208 A \u2227 f a = aH\n\ncase refine_2\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 b \u2208 B \u2227 f b = bH"}, {"tactic": "exacts [\u27e8ha, (huH rfl).1\u27e9, \u27e8hb, (huH rfl).2\u27e9]", "annotated_tactic": ["exacts [\u27e8ha, (huH <a>rfl</a>).1\u27e9, \u27e8hb, (huH <a>rfl</a>).2\u27e9]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case refine_1\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 a \u2208 A \u2227 f a = aH\n\ncase refine_2\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b2 : Mul G\ninst\u271d\u00b9 : Mul H\nA\u271d B\u271d : Finset G\na0 b0 : G\ninst\u271d : DecidableEq H\nf : G \u2192\u2099* H\nA B : Finset G\naG bG : G\naH bH : H\nhae : f aG = aH\nhbe : f bG = bH\nhuG : UniqueMul (filter (fun x => f x = aH) A) (filter (fun x => f x = bH) B) aG bG\na b : G\nha : a \u2208 A\nhb : b \u2208 B\nhe : a * b = aG * bG\nhuH : aH * bH = aH * bH \u2192 f a = aH \u2227 f b = bH\n\u22a2 b \u2208 B \u2227 f b = bH", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.mul_refl", "start": [936, 1], "end": [937, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Nat.prod_pow_pos_of_zero_not_mem_support", "start": [694, 1], "end": [696, 98], "traced_tactics": [{"tactic": "subst H", "annotated_tactic": ["subst H", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH\u271d : Type u_13\nR : Type u_14\nS : Type u_15\nf : \u2115 \u2192\u2080 \u2115\nhf : 0 \u2209 f.support\na : \u2115\nha : a \u2208 f.support\nH : a = 0\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\nf : \u2115 \u2192\u2080 \u2115\nhf : 0 \u2209 f.support\nha : 0 \u2208 f.support\n\u22a2 False"}, {"tactic": "exact hf ha", "annotated_tactic": ["exact hf ha", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\nf : \u2115 \u2192\u2080 \u2115\nhf : 0 \u2209 f.support\nha : 0 \u2208 f.support\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "DifferentiableAt.csin", "start": [391, 1], "end": [393, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean", "full_name": "ContinuousLinearEquiv.mdifferentiable", "start": [101, 11], "end": [102, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Involutive.surjective", "start": [909, 11], "end": [909, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "full_name": "Matrix.PosSemidef.fromBlocks\u2081\u2081", "start": [532, 1], "end": [548, 16], "traced_tactics": [{"tactic": "rw [PosSemidef, IsHermitian.fromBlocks\u2081\u2081 _ _ hA.1]", "annotated_tactic": ["rw [<a>PosSemidef</a>, <a>IsHermitian.fromBlocks\u2081\u2081</a> _ _ hA.1]", [{"full_name": "Matrix.PosSemidef", "def_path": "Mathlib/LinearAlgebra/Matrix/PosDef.lean", "def_pos": [47, 5], "def_end_pos": [47, 15]}, {"full_name": "Matrix.IsHermitian.fromBlocks\u2081\u2081", "def_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "def_pos": [508, 9], "def_end_pos": [508, 33]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 (fromBlocks A B B\u1d34 D).PosSemidef \u2194 (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef", "state_after": "l : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 ((D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x) \u2194\n    (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 ((D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x) \u2194\n    (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef", "state_after": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 ((D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x) \u2192\n    (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\n\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef \u2192\n    (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x"}, {"tactic": "refine fun h => \u27e8h.1, fun x => ?_\u27e9", "annotated_tactic": ["refine fun h => \u27e8h.1, fun x => ?_\u27e9", []], "state_before": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 ((D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x) \u2192\n    (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef", "state_after": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x"}, {"tactic": "have := h.2 (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x)", "annotated_tactic": ["have := h.2 (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x)", []], "state_before": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x", "state_after": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x) \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x)\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x"}, {"tactic": "rw [dotProduct_mulVec, schur_complement_eq\u2081\u2081 B D _ _ hA.1, neg_add_self, dotProduct_zero,\n  zero_add] at this", "annotated_tactic": ["rw [<a>dotProduct_mulVec</a>, <a>schur_complement_eq\u2081\u2081</a> B D _ _ hA.1, <a>neg_add_self</a>, <a>dotProduct_zero</a>,\n      <a>zero_add</a>] at this", [{"full_name": "Matrix.dotProduct_mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1745, 9], "def_end_pos": [1745, 26]}, {"full_name": "Matrix.schur_complement_eq\u2081\u2081", "def_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "def_pos": [484, 9], "def_end_pos": [484, 30]}, {"full_name": "neg_add_self", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1227, 3], "def_end_pos": [1227, 14]}, {"full_name": "Matrix.dotProduct_zero", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [805, 9], "def_end_pos": [805, 24]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x) \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 (-((A\u207b\u00b9 * B) *\u1d65 x) \u2295\u1d65 x)\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x", "state_after": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x"}, {"tactic": "rw [dotProduct_mulVec]", "annotated_tactic": ["rw [<a>dotProduct_mulVec</a>]", [{"full_name": "Matrix.dotProduct_mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1745, 9], "def_end_pos": [1745, 26]}]], "state_before": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 star x \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x", "state_after": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "case mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 star x \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x", "state_after": "no goals"}, {"tactic": "refine fun h => \u27e8h.1, fun x => ?_\u27e9", "annotated_tactic": ["refine fun h => \u27e8h.1, fun x => ?_\u27e9", []], "state_before": "case mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\n\u22a2 (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef \u2192\n    (D - B\u1d34 * A\u207b\u00b9 * B).IsHermitian \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x", "state_after": "case mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x"}, {"tactic": "rw [dotProduct_mulVec, \u2190 Sum.elim_comp_inl_inr x, schur_complement_eq\u2081\u2081 B D _ _ hA.1]", "annotated_tactic": ["rw [<a>dotProduct_mulVec</a>, \u2190 <a>Sum.elim_comp_inl_inr</a> x, <a>schur_complement_eq\u2081\u2081</a> B D _ _ hA.1]", [{"full_name": "Matrix.dotProduct_mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1745, 9], "def_end_pos": [1745, 26]}, {"full_name": "Sum.elim_comp_inl_inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean", "def_pos": [112, 17], "def_end_pos": [112, 34]}, {"full_name": "Matrix.schur_complement_eq\u2081\u2081", "def_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "def_pos": [484, 9], "def_end_pos": [484, 30]}]], "state_before": "case mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 fromBlocks A B B\u1d34 D *\u1d65 x", "state_after": "case mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264\n    star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u1d65* A \u2b1d\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) +\n      star (x \u2218 Sum.inr) \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr"}, {"tactic": "apply le_add_of_nonneg_of_le", "annotated_tactic": ["apply <a>le_add_of_nonneg_of_le</a>", [{"full_name": "le_add_of_nonneg_of_le", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [936, 3], "def_end_pos": [936, 14]}]], "state_before": "case mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264\n    star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u1d65* A \u2b1d\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) +\n      star (x \u2218 Sum.inr) \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr", "state_after": "case mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u1d65* A \u2b1d\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr)\n\ncase mpr.hbc\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inr) \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr"}, {"tactic": "rw [\u2190 dotProduct_mulVec]", "annotated_tactic": ["rw [\u2190 <a>dotProduct_mulVec</a>]", [{"full_name": "Matrix.dotProduct_mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1745, 9], "def_end_pos": [1745, 26]}]], "state_before": "case mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u1d65* A \u2b1d\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr)", "state_after": "case mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u2b1d\u1d65 A *\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr)"}, {"tactic": "apply hA.posSemidef.2", "annotated_tactic": ["apply hA.posSemidef.2", []], "state_before": "case mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr) \u2b1d\u1d65 A *\u1d65 (x \u2218 Sum.inl + (A\u207b\u00b9 * B) *\u1d65 x \u2218 Sum.inr)", "state_after": "no goals"}, {"tactic": "rw [\u2190 dotProduct_mulVec (star (x \u2218 Sum.inr))]", "annotated_tactic": ["rw [\u2190 <a>dotProduct_mulVec</a> (<a>star</a> (x \u2218 <a>Sum.inr</a>))]", [{"full_name": "Matrix.dotProduct_mulVec", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1745, 9], "def_end_pos": [1745, 26]}, {"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [47, 3], "def_end_pos": [47, 7]}, {"full_name": "Sum.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [134, 5], "def_end_pos": [134, 8]}]], "state_before": "case mpr.hbc\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inr) \u1d65* (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr", "state_after": "case mpr.hbc\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2077 : CommRing \ud835\udd5c\ninst\u271d\u2076 : PartialOrder \ud835\udd5c\ninst\u271d\u2075 : StarRing \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : A.PosDef\ninst\u271d : Invertible A\nh : (D - B\u1d34 * A\u207b\u00b9 * B).PosSemidef\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inr) \u2b1d\u1d65 (D - B\u1d34 * A\u207b\u00b9 * B) *\u1d65 x \u2218 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: \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactification/OnePoint.lean", "full_name": "OnePoint.comap_coe_nhds", "start": [305, 1], "end": [306, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Multiset.lean", "full_name": "Multiset.Icc_eq_zero_iff", "start": [138, 1], "end": [139, 56], "traced_tactics": [{"tactic": "rw [Icc, Finset.val_eq_zero, Finset.Icc_eq_empty_iff]", "annotated_tactic": ["rw [<a>Icc</a>, <a>Finset.val_eq_zero</a>, <a>Finset.Icc_eq_empty_iff</a>]", [{"full_name": "Multiset.Icc", "def_path": "Mathlib/Order/Interval/Multiset.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "Finset.val_eq_zero", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 20]}, {"full_name": "Finset.Icc_eq_empty_iff", "def_path": "Mathlib/Order/Interval/Finset/Basic.lean", "def_pos": [78, 9], "def_end_pos": [78, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b = 0 \u2194 \u00aca \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": 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"def_pos": [391, 9], "def_end_pos": [391, 32]}, {"full_name": "Finset.affineCombination_apply", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "def_pos": [391, 9], "def_end_pos": [391, 32]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\nw : \u03b9 \u2192 k\np : \u03b9 \u2192 P\npred : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred pred\nh : \u2200 i \u2208 s, w i \u2260 0 \u2192 pred i\n\u22a2 (affineCombination k (filter pred s) p) w = (affineCombination k s p) w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.image_inj", "start": [492, 1], "end": [493, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.pi_eq_bot", "start": [181, 1], "end": [182, 67], "traced_tactics": [{"tactic": "simpa using @pi_inf_principal_univ_pi_eq_bot \u03b9 \u03b1 f fun _ => univ", "annotated_tactic": ["simpa using @<a>pi_inf_principal_univ_pi_eq_bot</a> \u03b9 \u03b1 f fun _ => <a>univ</a>", [{"full_name": "Filter.pi_inf_principal_univ_pi_eq_bot", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [142, 9], "def_end_pos": [142, 40]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\n\u22a2 pi f = \u22a5 \u2194 \u2203 i, f i = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Perfect.lean", "full_name": "iterateFrobeniusEquiv_symm_add_apply", "start": [104, 1], "end": [107, 93], "traced_tactics": [{"tactic": "rw [RingEquiv.apply_symm_apply, add_comm,\niterateFrobeniusEquiv_add_apply, RingEquiv.apply_symm_apply, RingEquiv.apply_symm_apply]", "annotated_tactic": ["rw [<a>RingEquiv.apply_symm_apply</a>, <a>add_comm</a>,\n    <a>iterateFrobeniusEquiv_add_apply</a>, <a>RingEquiv.apply_symm_apply</a>, <a>RingEquiv.apply_symm_apply</a>]", [{"full_name": "RingEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [355, 9], "def_end_pos": [355, 25]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "iterateFrobeniusEquiv_add_apply", "def_path": "Mathlib/FieldTheory/Perfect.lean", "def_pos": [96, 9], "def_end_pos": [96, 40]}, {"full_name": "RingEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [355, 9], "def_end_pos": [355, 25]}, {"full_name": "RingEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [355, 9], "def_end_pos": [355, 25]}]], "state_before": "R : Type u_1\np m n : \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : ExpChar R p\ninst\u271d : PerfectRing R p\nx : R\n\u22a2 (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p (m + n)).symm x) =\n    (iterateFrobeniusEquiv R p (m + n)) ((iterateFrobeniusEquiv R p m).symm ((iterateFrobeniusEquiv R p n).symm x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.Ioi_mul_Ici_subset'", "start": [104, 1], "end": [107, 39], "traced_tactics": [{"tactic": "haveI := covariantClass_le_of_lt", "annotated_tactic": ["haveI := <a>covariantClass_le_of_lt</a>", [{"full_name": "covariantClass_le_of_lt", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap HMul.hMul) LT.lt\na b : \u03b1\n\u22a2 Ioi a * Ici b \u2286 Ioi (a * b)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap HMul.hMul) LT.lt\na b : \u03b1\nthis :\n  \u2200 (M : Type ?u.11667) (N : Type ?u.11666) (\u03bc : M \u2192 N \u2192 N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N \u03bc fun x x_1 => x < x_1], CovariantClass M N \u03bc fun x x_1 => x \u2264 x_1\n\u22a2 Ioi a * Ici b \u2286 Ioi (a * b)"}, {"tactic": "rintro x \u27e8y, hya, z, hzb, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y, hya, z, hzb, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap HMul.hMul) LT.lt\na b : \u03b1\nthis :\n  \u2200 (M : Type ?u.11667) (N : Type ?u.11666) (\u03bc : M \u2192 N \u2192 N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N \u03bc fun x x_1 => x < x_1], CovariantClass M N \u03bc fun x x_1 => x \u2264 x_1\n\u22a2 Ioi a * Ici b \u2286 Ioi (a * b)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap HMul.hMul) LT.lt\na b : \u03b1\nthis :\n  \u2200 (M : Type ?u.11667) (N : Type ?u.11666) (\u03bc : M \u2192 N \u2192 N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N \u03bc fun x x_1 => x < x_1], CovariantClass M N \u03bc fun x x_1 => x \u2264 x_1\ny : \u03b1\nhya : y \u2208 Ioi a\nz : \u03b1\nhzb : z \u2208 Ici b\n\u22a2 (fun x x_1 => x * x_1) y z \u2208 Ioi (a * b)"}, {"tactic": "exact mul_lt_mul_of_lt_of_le hya hzb", "annotated_tactic": ["exact <a>mul_lt_mul_of_lt_of_le</a> hya hzb", [{"full_name": "mul_lt_mul_of_lt_of_le", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 31]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap HMul.hMul) LT.lt\na b : \u03b1\nthis :\n  \u2200 (M : Type ?u.11667) (N : Type ?u.11666) (\u03bc : M \u2192 N \u2192 N) [inst : PartialOrder N]\n    [inst_1 : CovariantClass M N \u03bc fun x x_1 => x < x_1], CovariantClass M N \u03bc fun x x_1 => x \u2264 x_1\ny : \u03b1\nhya : y \u2208 Ioi a\nz : \u03b1\nhzb : z \u2208 Ici b\n\u22a2 (fun x x_1 => x * x_1) y z \u2208 Ioi (a * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monad/Basic.lean", "full_name": "CategoryTheory.Monad.right_unit", "start": [155, 1], "end": [157, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.iInf_mul_of_ne", "start": [633, 1], "end": [637, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "HasStrictFDerivAt.isBigO_sub_rev", "start": [828, 1], "end": [832, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.image_coe_Iio", "start": [497, 1], "end": [501, 6], "traced_tactics": [{"tactic": "refine (image_comp WithBot.some WithTop.some _).trans ?_", "annotated_tactic": ["refine (<a>image_comp</a> <a>WithBot.some</a> <a>WithTop.some</a> _).<a>trans</a> ?_", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [263, 9], "def_end_pos": [263, 19]}, {"full_name": "WithBot.some", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [44, 27], "def_end_pos": [44, 31]}, {"full_name": "WithTop.some", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [622, 27], "def_end_pos": [622, 31]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "x : \u211d\n\u22a2 Real.toEReal '' Iio x = Ioo \u22a5 \u2191x", "state_after": "x : \u211d\n\u22a2 WithBot.some '' (WithTop.some '' Iio x) = Ioo \u22a5 \u2191x"}, {"tactic": "rw [WithTop.image_coe_Iio, WithBot.image_coe_Iio]", "annotated_tactic": ["rw [<a>WithTop.image_coe_Iio</a>, <a>WithBot.image_coe_Iio</a>]", [{"full_name": "WithTop.image_coe_Iio", "def_path": "Mathlib/Order/Interval/Set/WithBotTop.lean", "def_pos": [97, 9], "def_end_pos": [97, 22]}, {"full_name": "WithBot.image_coe_Iio", "def_path": "Mathlib/Order/Interval/Set/WithBotTop.lean", "def_pos": [197, 9], "def_end_pos": [197, 22]}]], "state_before": "x : \u211d\n\u22a2 WithBot.some '' (WithTop.some '' Iio x) = Ioo \u22a5 \u2191x", "state_after": "x : \u211d\n\u22a2 Ioo \u22a5 \u2191\u2191x = Ioo \u22a5 \u2191x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "x : \u211d\n\u22a2 Ioo \u22a5 \u2191\u2191x = Ioo \u22a5 \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Real.tendsto_exp_div_pow_atTop", "start": [254, 1], "end": [271, 85], "traced_tactics": [{"tactic": "refine (atTop_basis_Ioi.tendsto_iff (atTop_basis' 1)).2 fun C hC\u2081 => ?_", "annotated_tactic": ["refine (atTop_basis_Ioi.tendsto_iff (<a>atTop_basis'</a> 1)).2 fun C hC\u2081 => ?_", [{"full_name": "Filter.atTop_basis'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [144, 9], "def_end_pos": [144, 21]}]], "state_before": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\n\u22a2 Tendsto (fun x => rexp x / x ^ n) atTop atTop", "state_after": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "have hC\u2080 : 0 < C := zero_lt_one.trans_le hC\u2081", "annotated_tactic": ["have hC\u2080 : 0 < C := zero_lt_one.trans_le hC\u2081", []], "state_before": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C", "state_after": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "have : 0 < (exp 1 * C)\u207b\u00b9 := inv_pos.2 (mul_pos (exp_pos _) hC\u2080)", "annotated_tactic": ["have : 0 < (<a>exp</a> 1 * C)\u207b\u00b9 := <a>inv_pos</a>.2 (<a>mul_pos</a> (<a>exp_pos</a> _) hC\u2080)", [{"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [47, 15], "def_end_pos": [47, 22]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [406, 7], "def_end_pos": [406, 14]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}]], "state_before": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C", "state_after": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, \u2200 k \u2265 N, (\u2191k : \u211d) ^ n / exp 1 ^ k < (exp 1 * C)\u207b\u00b9 :=\n  eventually_atTop.1\n    ((tendsto_pow_const_div_const_pow_of_one_lt n (one_lt_exp_iff.2 zero_lt_one)).eventually\n      (gt_mem_nhds this))", "annotated_tactic": ["obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, \u2200 k \u2265 N, (\u2191k : \u211d) ^ n / <a>exp</a> 1 ^ k < (<a>exp</a> 1 * C)\u207b\u00b9 :=\n    <a>eventually_atTop</a>.1\n      ((<a>tendsto_pow_const_div_const_pow_of_one_lt</a> n (<a>one_lt_exp_iff</a>.2 <a>zero_lt_one</a>)).<a>eventually</a>\n        (<a>gt_mem_nhds</a> this))", [{"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Filter.eventually_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [182, 9], "def_end_pos": [182, 25]}, {"full_name": "tendsto_pow_const_div_const_pow_of_one_lt", "def_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "def_pos": [232, 9], "def_end_pos": [232, 50]}, {"full_name": "Real.one_lt_exp_iff", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Filter.Tendsto.eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3014, 9], "def_end_pos": [3014, 27]}, {"full_name": "gt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}]], "state_before": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n / rexp 1 ^ k < (rexp 1 * C)\u207b\u00b9\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "simp only [\u2190 exp_nat_mul, mul_one, div_lt_iff, exp_pos, \u2190 div_eq_inv_mul] at hN", "annotated_tactic": ["simp only [\u2190 <a>exp_nat_mul</a>, <a>mul_one</a>, <a>div_lt_iff</a>, <a>exp_pos</a>, \u2190 <a>div_eq_inv_mul</a>] at hN", [{"full_name": "Real.exp_nat_mul", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [854, 16], "def_end_pos": [854, 27]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 19]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n / rexp 1 ^ k < (rexp 1 * C)\u207b\u00b9\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "refine \u27e8N, trivial, fun x hx => ?_\u27e9", "annotated_tactic": ["refine \u27e8N, <a>trivial</a>, fun x hx => ?_\u27e9", [{"full_name": "trivial", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [645, 35], "def_end_pos": [645, 42]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\n\u22a2 \u2203 ia, True \u2227 \u2200 x \u2208 Set.Ioi ia, rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : x \u2208 Set.Ioi \u2191N\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "rw [Set.mem_Ioi] at hx", "annotated_tactic": ["rw [<a>Set.mem_Ioi</a>] at hx", [{"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : x \u2208 Set.Ioi \u2191N\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "have hx\u2080 : 0 < x := (Nat.cast_nonneg N).trans_lt hx", "annotated_tactic": ["have hx\u2080 : 0 < x := (<a>Nat.cast_nonneg</a> N).<a>trans_lt</a> hx", [{"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C"}, {"tactic": "rw [Set.mem_Ici, le_div_iff (pow_pos hx\u2080 _), \u2190 le_div_iff' hC\u2080]", "annotated_tactic": ["rw [<a>Set.mem_Ici</a>, <a>le_div_iff</a> (<a>pow_pos</a> hx\u2080 _), \u2190 <a>le_div_iff'</a> hC\u2080]", [{"full_name": "Set.mem_Ici", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 16]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 19]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 rexp x / x ^ n \u2208 Set.Ici C", "state_after": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 x ^ n \u2264 rexp x / C"}, {"tactic": "calc\n  x ^ n \u2264 \u2308x\u2309\u208a ^ n := mod_cast pow_le_pow_left hx\u2080.le (Nat.le_ceil _) _\n  _ \u2264 exp \u2308x\u2309\u208a / (exp 1 * C) := mod_cast (hN _ (Nat.lt_ceil.2 hx).le).le\n  _ \u2264 exp (x + 1) / (exp 1 * C) := by gcongr; exact (Nat.ceil_lt_add_one hx\u2080.le).le\n  _ = exp x / C := by rw [add_comm, exp_add, mul_div_mul_left _ _ (exp_pos _).ne']", "annotated_tactic": ["calc\n    x ^ n \u2264 \u2308x\u2309\u208a ^ n := mod_cast <a>pow_le_pow_left</a> hx\u2080.le (<a>Nat.le_ceil</a> _) _\n    _ \u2264 <a>exp</a> \u2308x\u2309\u208a / (<a>exp</a> 1 * C) := mod_cast (hN _ (<a>Nat.lt_ceil</a>.2 hx).<a>le</a>).<a>le</a>\n    _ \u2264 <a>exp</a> (x + 1) / (<a>exp</a> 1 * C) := by gcongr; exact (<a>Nat.ceil_lt_add_one</a> hx\u2080.le).<a>le</a>\n    _ = <a>exp</a> x / C := by rw [<a>add_comm</a>, <a>exp_add</a>, <a>mul_div_mul_left</a> _ _ (<a>exp_pos</a> _).<a>ne'</a>]", [{"full_name": "pow_le_pow_left", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 24]}, {"full_name": "Nat.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Nat.lt_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [290, 9], "def_end_pos": [290, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Nat.ceil_lt_add_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [518, 9], "def_end_pos": [518, 24]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Real.exp_add", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [828, 16], "def_end_pos": [828, 23]}, {"full_name": "mul_div_mul_left", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [520, 7], "def_end_pos": [520, 23]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case intro\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 x ^ n \u2264 rexp x / C", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 rexp \u2191\u2308x\u2309\u208a / (rexp 1 * C) \u2264 rexp (x + 1) / (rexp 1 * C)", "state_after": "case hab.h\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 \u2191\u2308x\u2309\u208a \u2264 x + 1"}, {"tactic": "exact (Nat.ceil_lt_add_one hx\u2080.le).le", "annotated_tactic": ["exact (<a>Nat.ceil_lt_add_one</a> hx\u2080.le).<a>le</a>", [{"full_name": "Nat.ceil_lt_add_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [518, 9], "def_end_pos": [518, 24]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case hab.h\n\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 \u2191\u2308x\u2309\u208a \u2264 x + 1", "state_after": "no goals"}, {"tactic": "rw [add_comm, exp_add, mul_div_mul_left _ _ (exp_pos _).ne']", "annotated_tactic": ["rw [<a>add_comm</a>, <a>exp_add</a>, <a>mul_div_mul_left</a> _ _ (<a>exp_pos</a> _).<a>ne'</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Real.exp_add", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [828, 16], "def_end_pos": [828, 23]}, {"full_name": "mul_div_mul_left", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [520, 7], "def_end_pos": [520, 23]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "\u03b1 : Type u_1\nx\u271d y z : \u211d\nl : Filter \u03b1\nn : \u2115\nC : \u211d\nhC\u2081 : 1 \u2264 C\nhC\u2080 : 0 < C\nthis : 0 < (rexp 1 * C)\u207b\u00b9\nN : \u2115\nhN : \u2200 k \u2265 N, \u2191k ^ n < rexp \u2191k / (rexp 1 * C)\nx : \u211d\nhx : \u2191N < x\nhx\u2080 : 0 < x\n\u22a2 rexp (x + 1) / (rexp 1 * C) = rexp x / C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "full_name": "FractionalIdeal.count_zero", "start": [282, 1], "end": [283, 93], "traced_tactics": [{"tactic": "simp only [count, dif_pos]", "annotated_tactic": ["simp only [<a>count</a>, <a>dif_pos</a>]", [{"full_name": "FractionalIdeal.count", "def_path": "Mathlib/RingTheory/DedekindDomain/Factorization.lean", "def_pos": [275, 5], "def_end_pos": [275, 10]}, {"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsDedekindDomain R\nv : HeightOneSpectrum R\n\u22a2 count K v 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EqToHom.lean", "full_name": "CategoryTheory.eqToHom_comp_iff", "start": [65, 1], "end": [68, 51], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX X' Y : C\np : X = X'\nf : X \u27f6 Y\ng : X' \u27f6 Y\nh : eqToHom p \u226b g = f\n\u22a2 g = eqToHom \u22ef \u226b eqToHom p \u226b g", "state_after": "no goals"}, {"tactic": "simp [whisker_eq _ h]", "annotated_tactic": ["simp [<a>whisker_eq</a> _ h]", [{"full_name": "CategoryTheory.whisker_eq", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 19]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX X' Y : C\np : X = X'\nf : X \u27f6 Y\ng : X' \u27f6 Y\nh : g = eqToHom \u22ef \u226b f\n\u22a2 eqToHom p \u226b eqToHom \u22ef \u226b f = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Partial.lean", "full_name": "Filter.mem_pmap", "start": [215, 1], "end": [216, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.smul_single'", "start": [1579, 1], "end": [1581, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Real.exists_rat_pow_btwn_rat_aux", "start": [927, 1], "end": [939, 26], "traced_tactics": [{"tactic": "have hn' : 0 < (n : \u211d) := mod_cast hn.bot_lt", "annotated_tactic": ["have hn' : 0 < (n : \u211d) := mod_cast hn.bot_lt", []], "state_before": "z x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "z x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "obtain \u27e8q, hxq, hqy\u27e9 :=\n  exists_rat_btwn (rpow_lt_rpow (le_max_left 0 x) (max_lt hy h) <| inv_pos.mpr hn')", "annotated_tactic": ["obtain \u27e8q, hxq, hqy\u27e9 :=\n    <a>exists_rat_btwn</a> (<a>rpow_lt_rpow</a> (<a>le_max_left</a> 0 x) (<a>max_lt</a> hy h) <| inv_pos.mpr hn')", [{"full_name": "exists_rat_btwn", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [287, 9], "def_end_pos": [287, 24]}, {"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "max_lt", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [166, 9], "def_end_pos": [166, 15]}]], "state_before": "z x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "have := rpow_nonneg (le_max_left 0 x) n\u207b\u00b9", "annotated_tactic": ["have := <a>rpow_nonneg</a> (<a>le_max_left</a> 0 x) n\u207b\u00b9", [{"full_name": "Real.rpow_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [169, 9], "def_end_pos": [169, 20]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "have hq := this.trans_lt hxq", "annotated_tactic": ["have hq := this.trans_lt hxq", []], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "replace hxq := rpow_lt_rpow this hxq hn'", "annotated_tactic": ["replace hxq := <a>rpow_lt_rpow</a> this hxq hn'", [{"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}]], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "replace hqy := rpow_lt_rpow hq.le hqy hn'", "annotated_tactic": ["replace hqy := <a>rpow_lt_rpow</a> hq.le hqy hn'", [{"full_name": "Real.rpow_lt_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [543, 9], "def_end_pos": [543, 21]}]], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\nhqy : \u2191q ^ \u2191n < (y ^ (\u2191n)\u207b\u00b9) ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y"}, {"tactic": "rw [rpow_natCast, rpow_natCast, rpow_inv_natCast_pow _ hn] at hxq hqy", "annotated_tactic": ["rw [<a>rpow_natCast</a>, <a>rpow_natCast</a>, <a>rpow_inv_natCast_pow</a> _ hn] at hxq hqy", [{"full_name": "Real.rpow_natCast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [73, 9], "def_end_pos": [73, 21]}, {"full_name": "Real.rpow_natCast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [73, 9], "def_end_pos": [73, 21]}, {"full_name": "Real.rpow_inv_natCast_pow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [517, 9], "def_end_pos": [517, 29]}]], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\nhqy : \u2191q ^ \u2191n < (y ^ (\u2191n)\u207b\u00b9) ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 y\n\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ n < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 max 0 x"}, {"tactic": "exact \u27e8q, mod_cast hq, (le_max_right _ _).trans_lt hxq, hqy\u27e9", "annotated_tactic": ["exact \u27e8q, mod_cast hq, (<a>le_max_right</a> _ _).<a>trans_lt</a> hxq, hqy\u27e9", [{"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}]], "state_before": "case intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y", "state_after": "no goals"}, {"tactic": "exact hy.le", "annotated_tactic": ["exact hy.le", []], "state_before": "z x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 y", "state_after": "no goals"}, {"tactic": "exact le_max_left _ _", "annotated_tactic": ["exact <a>le_max_left</a> _ _", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "z x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ n < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 max 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "start": [628, 1], "end": [634, 23], "traced_tactics": [{"tactic": "rw [normalizedFactors, factors]", "annotated_tactic": ["rw [<a>normalizedFactors</a>, <a>factors</a>]", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "UniqueFactorizationMonoid.factors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [470, 19], "def_end_pos": [470, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 x \u2208 normalizedFactors a, normalize x = x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 x \u2208 Multiset.map (\u21d1normalize) (if h : a = 0 then 0 else Classical.choose \u22ef), normalize x = x"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 x \u2208 Multiset.map (\u21d1normalize) (if h : a = 0 then 0 else Classical.choose \u22ef), normalize x = x", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : a = 0\n\u22a2 \u2200 x \u2208 Multiset.map (fun x => normalize x) 0, normalize x = x\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\n\u22a2 \u2200 x \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef), normalize x = x"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\n\u22a2 \u2200 x \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef), normalize x = x", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\nx : \u03b1\nhx : x \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef)\n\u22a2 normalize x = x"}, {"tactic": "obtain \u27e8y, _, rfl\u27e9 := Multiset.mem_map.1 hx", "annotated_tactic": ["obtain \u27e8y, _, rfl\u27e9 := <a>Multiset.mem_map</a>.1 hx", [{"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1277, 9], "def_end_pos": [1277, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\nx : \u03b1\nhx : x \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef)\n\u22a2 normalize x = x", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\ny : \u03b1\nleft\u271d : y \u2208 Classical.choose \u22ef\nhx : normalize y \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef)\n\u22a2 normalize (normalize y) = normalize y"}, {"tactic": "apply normalize_idem", "annotated_tactic": ["apply <a>normalize_idem</a>", [{"full_name": "normalize_idem", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 23]}]], "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\ny : \u03b1\nleft\u271d : y \u2208 Classical.choose \u22ef\nhx : normalize y \u2208 Multiset.map (fun x => normalize x) (Classical.choose \u22ef)\n\u22a2 normalize (normalize y) = normalize y", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : a = 0\n\u22a2 \u2200 x \u2208 Multiset.map (fun x => normalize x) 0, normalize x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Small/Defs.lean", "full_name": "Small.mk'", "start": [38, 1], "end": [39, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean", "full_name": "Real.mul_le_sin", "start": [290, 1], "end": [294, 59], "traced_tactics": [{"tactic": "rw [\u2190 inv_div]", "annotated_tactic": ["rw [\u2190 <a>inv_div</a>]", [{"full_name": "inv_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}]], "state_before": "x : \u211d\nhx : 0 \u2264 x\nhx' : x \u2264 \u03c0 / 2\n\u22a2 2 / \u03c0 * x \u2264 sin x", "state_after": "x : \u211d\nhx : 0 \u2264 x\nhx' : x \u2264 \u03c0 / 2\n\u22a2 (\u03c0 / 2)\u207b\u00b9 * x \u2264 sin x"}, {"tactic": "simpa [-inv_div, mul_inv_cancel_left\u2080 pi_div_two_pos.ne'] using @le_sin_mul ((\u03c0 / 2)\u207b\u00b9 * x)\n  (mul_nonneg (inv_nonneg.2 pi_div_two_pos.le) hx)\n  (by rwa [\u2190 div_eq_inv_mul, div_le_one pi_div_two_pos])", "annotated_tactic": ["simpa [-<a>inv_div</a>, <a>mul_inv_cancel_left\u2080</a> pi_div_two_pos.ne'] using @<a>le_sin_mul</a> ((\u03c0 / 2)\u207b\u00b9 * x)\n    (<a>mul_nonneg</a> (<a>inv_nonneg</a>.2 pi_div_two_pos.le) hx)\n    (by rwa [\u2190 <a>div_eq_inv_mul</a>, <a>div_le_one</a> <a>pi_div_two_pos</a>])", [{"full_name": "inv_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}, {"full_name": "mul_inv_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [259, 9], "def_end_pos": [259, 29]}, {"full_name": "Real.le_sin_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean", "def_pos": [276, 9], "def_end_pos": [276, 19]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [55, 15], "def_end_pos": [55, 25]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}, {"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 19]}, {"full_name": "Real.pi_div_two_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 23]}]], "state_before": "x : \u211d\nhx : 0 \u2264 x\nhx' : x \u2264 \u03c0 / 2\n\u22a2 (\u03c0 / 2)\u207b\u00b9 * x \u2264 sin x", "state_after": "no goals"}, {"tactic": "rwa [\u2190 div_eq_inv_mul, div_le_one pi_div_two_pos]", "annotated_tactic": ["rwa [\u2190 <a>div_eq_inv_mul</a>, <a>div_le_one</a> <a>pi_div_two_pos</a>]", [{"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}, {"full_name": "div_le_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 19]}, {"full_name": "Real.pi_div_two_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 23]}]], "state_before": "x : \u211d\nhx : 0 \u2264 x\nhx' : x \u2264 \u03c0 / 2\n\u22a2 (\u03c0 / 2)\u207b\u00b9 * x \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean", "full_name": "CategoryTheory.Monoidal.leftUnitor_inv_app", "start": [135, 1], "end": [137, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/PathCategory.lean", "full_name": "CategoryTheory.composePath_toPath", "start": [172, 1], "end": [172, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Filter.EventuallyEq.isBigO", "start": [602, 1], "end": [603, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "full_name": "Bornology.isVonNBounded_insert", "start": [263, 1], "end": [265, 88], "traced_tactics": [{"tactic": "simp only [\u2190 singleton_union, isVonNBounded_union, isVonNBounded_singleton, true_and]", "annotated_tactic": ["simp only [\u2190 <a>singleton_union</a>, <a>isVonNBounded_union</a>, <a>isVonNBounded_singleton</a>, <a>true_and</a>]", [{"full_name": "Set.singleton_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1299, 9], "def_end_pos": [1299, 24]}, {"full_name": "Bornology.isVonNBounded_union", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [96, 9], "def_end_pos": [96, 28]}, {"full_name": "Bornology.isVonNBounded_singleton", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [258, 9], "def_end_pos": [258, 32]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nx : E\ns : Set E\n\u22a2 IsVonNBounded \ud835\udd5c (insert x s) \u2194 IsVonNBounded \ud835\udd5c s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "full_name": "CategoryTheory.Sheaf.Hom.add_app", "start": [511, 1], "end": [512, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monovary.lean", "full_name": "monovary_inv\u2080", "start": [274, 1], "end": [275, 53], "traced_tactics": [{"tactic": "rw [monovary_inv_left\u2080 hf, antivary_inv_right\u2080 hg]", "annotated_tactic": ["rw [<a>monovary_inv_left\u2080</a> hf, <a>antivary_inv_right\u2080</a> hg]", [{"full_name": "monovary_inv_left\u2080", "def_path": "Mathlib/Algebra/Order/Monovary.lean", "def_pos": [262, 15], "def_end_pos": [262, 33]}, {"full_name": "antivary_inv_right\u2080", "def_path": "Mathlib/Algebra/Order/Monovary.lean", "def_pos": [271, 15], "def_end_pos": [271, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : LinearOrderedSemifield \u03b2\ns : Set \u03b9\nf f\u2081 f\u2082 : \u03b9 \u2192 \u03b1\ng g\u2081 g\u2082 : \u03b9 \u2192 \u03b2\nhf : StrongLT 0 f\nhg : StrongLT 0 g\n\u22a2 Monovary f\u207b\u00b9 g\u207b\u00b9 \u2194 Monovary f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean", "full_name": "smul_balancedCore_subset", "start": [85, 1], "end": [90, 57], "traced_tactics": [{"tactic": "rintro x \u27e8y, hy, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y, hy, rfl\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 balancedCore \ud835\udd5c s \u2286 balancedCore \ud835\udd5c s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : y \u2208 balancedCore \ud835\udd5c s\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s"}, {"tactic": "rw [mem_balancedCore_iff] at hy", "annotated_tactic": ["rw [<a>mem_balancedCore_iff</a>] at hy", [{"full_name": "mem_balancedCore_iff", "def_path": "Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean", "def_pos": [81, 9], "def_end_pos": [81, 29]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : y \u2208 balancedCore \ud835\udd5c s\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2203 t, Balanced \ud835\udd5c t \u2227 t \u2286 s \u2227 y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s"}, {"tactic": "rcases hy with \u27e8t, ht1, ht2, hy\u27e9", "annotated_tactic": ["rcases hy with \u27e8t, ht1, ht2, hy\u27e9", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2203 t, Balanced \ud835\udd5c t \u2227 t \u2286 s \u2227 y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s", "state_after": "case intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t\u271d : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nt : Set E\nht1 : Balanced \ud835\udd5c t\nht2 : t \u2286 s\nhy : y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s"}, {"tactic": "exact \u27e8t, \u27e8ht1, ht2\u27e9, ht1 a ha (smul_mem_smul_set hy)\u27e9", "annotated_tactic": ["exact \u27e8t, \u27e8ht1, ht2\u27e9, ht1 a ha (<a>smul_mem_smul_set</a> hy)\u27e9", [{"full_name": "Set.smul_mem_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [322, 9], "def_end_pos": [322, 26]}]], "state_before": "case intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t\u271d : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nt : Set E\nht1 : Balanced \ud835\udd5c t\nht2 : t \u2286 s\nhy : y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Enum.lean", "full_name": "List.enumFrom_eq_nil", "start": [178, 1], "end": [179, 19], "traced_tactics": [{"tactic": "cases l <;> simp", "annotated_tactic": ["cases l <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nl : List \u03b1\n\u22a2 enumFrom n l = [] \u2194 l = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictMonoOn.monotoneOn", "start": [504, 11], "end": [505, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.or_exists_succ", "start": [214, 1], "end": [217, 40], "traced_tactics": [{"tactic": "rintro \u27e8_ | n, hn\u27e9", "annotated_tactic": ["rintro \u27e8_ | n, hn\u27e9", []], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\n\u22a2 (\u2203 n, p n) \u2192 p 0 \u2228 \u2203 n, p (n + 1)", "state_after": "case intro.zero\na b c d m n k : \u2115\np q : \u2115 \u2192 Prop\nhn : p 0\n\u22a2 p 0 \u2228 \u2203 n, p (n + 1)\n\ncase intro.succ\na b c d m n\u271d k : \u2115\np q : \u2115 \u2192 Prop\nn : \u2115\nhn : p (n + 1)\n\u22a2 p 0 \u2228 \u2203 n, p (n + 1)"}, {"tactic": "exacts [Or.inl hn, Or.inr \u27e8n, hn\u27e9]", "annotated_tactic": ["exacts [<a>Or.inl</a> hn, <a>Or.inr</a> \u27e8n, hn\u27e9]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro.zero\na b c d m n k : \u2115\np q : \u2115 \u2192 Prop\nhn : p 0\n\u22a2 p 0 \u2228 \u2203 n, p (n + 1)\n\ncase intro.succ\na b c d m n\u271d k : \u2115\np q : \u2115 \u2192 Prop\nn : \u2115\nhn : p (n + 1)\n\u22a2 p 0 \u2228 \u2203 n, p (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.induction_on'", "start": [1289, 1], "end": [1295, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean", "full_name": "TensorProduct.gradedComm_algebraMap_tmul", "start": [161, 1], "end": [163, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.append_mem_mul", "start": [119, 1], "end": [120, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "full_name": "IsUnit.smul_sub_iff_sub_inv_smul", "start": [423, 1], "end": [425, 66], "traced_tactics": [{"tactic": "rw [\u2190 isUnit_smul_iff r (1 - r\u207b\u00b9 \u2022 a), smul_sub, smul_inv_smul]", "annotated_tactic": ["rw [\u2190 <a>isUnit_smul_iff</a> r (1 - r\u207b\u00b9 \u2022 a), <a>smul_sub</a>, <a>smul_inv_smul</a>]", [{"full_name": "isUnit_smul_iff", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [418, 9], "def_end_pos": [418, 24]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}, {"full_name": "smul_inv_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [35, 9], "def_end_pos": [35, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : AddGroup \u03b2\ninst\u271d\u00b2 : DistribMulAction \u03b1 \u03b2\ninst\u271d\u00b9 : IsScalarTower \u03b1 \u03b2 \u03b2\ninst\u271d : SMulCommClass \u03b1 \u03b2 \u03b2\nr : \u03b1\na : \u03b2\n\u22a2 IsUnit (r \u2022 1 - a) \u2194 IsUnit (1 - r\u207b\u00b9 \u2022 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/CPolynomial.lean", "full_name": "HasFiniteFPowerSeriesOnBall.changeOrigin", "start": [487, 1], "end": [501, 97], "traced_tactics": [{"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\n\u22a2 0 < r - \u2191\u2016y\u2016\u208a", "state_after": "no goals"}, {"tactic": "have : f (x + y + z) =\n    FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by\n  rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz\n  rw [p.changeOrigin_eval_of_finite hf.finite, add_assoc, hf.sum]\n  refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)\n  exact mod_cast nnnorm_add_le y z", "annotated_tactic": ["have : f (x + y + z) =\n        <a>FormalMultilinearSeries.sum</a> (<a>FormalMultilinearSeries.changeOrigin</a> p y) z := by\n      rw [<a>mem_emetric_ball_zero_iff</a>, <a>lt_tsub_iff_right</a>, <a>add_comm</a>] at hz\n      rw [p.changeOrigin_eval_of_finite hf.finite, <a>add_assoc</a>, hf.sum]\n      refine <a>mem_emetric_ball_zero_iff</a>.2 (<a>lt_of_le_of_lt</a> ?_ hz)\n      exact mod_cast <a>nnnorm_add_le</a> y z", [{"full_name": "FormalMultilinearSeries.sum", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [92, 15], "def_end_pos": [92, 18]}, {"full_name": "FormalMultilinearSeries.changeOrigin", "def_path": "Mathlib/Analysis/Analytic/Basic.lean", "def_pos": [1178, 5], "def_end_pos": [1178, 17]}, {"full_name": "mem_emetric_ball_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [888, 3], "def_end_pos": [888, 14]}, {"full_name": "lt_tsub_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [417, 9], "def_end_pos": [417, 26]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "mem_emetric_ball_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [888, 3], "def_end_pos": [888, 14]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [805, 15], "def_end_pos": [805, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\n\u22a2 HasSum (fun n => (p.changeOrigin y n) fun x => z) (f (x + y + z))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\nthis : f (x + y + z) = (p.changeOrigin y).sum z\n\u22a2 HasSum (fun n => (p.changeOrigin y n) fun x => z) (f (x + y + z))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\nthis : f (x + y + z) = (p.changeOrigin y).sum z\n\u22a2 HasSum (fun n => (p.changeOrigin y n) fun x => z) (f (x + y + z))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\nthis : f (x + y + z) = (p.changeOrigin y).sum z\n\u22a2 HasSum (fun n => (p.changeOrigin y n) fun x => z) ((p.changeOrigin y).sum z)"}, {"tactic": "apply (p.changeOrigin y).hasSum_of_finite fun _ => p.changeOrigin_finite_of_finite hf.finite", "annotated_tactic": ["apply (p.changeOrigin y).<a>hasSum_of_finite</a> fun _ => p.changeOrigin_finite_of_finite hf.finite", [{"full_name": "FormalMultilinearSeries.hasSum_of_finite", "def_path": "Mathlib/Analysis/Analytic/CPolynomial.lean", "def_pos": [347, 19], "def_end_pos": [347, 59]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\nthis : f (x + y + z) = (p.changeOrigin y).sum z\n\u22a2 HasSum (fun n => (p.changeOrigin y n) fun x => z) ((p.changeOrigin y).sum z)", "state_after": "no goals"}, {"tactic": "rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz", "annotated_tactic": ["rw [<a>mem_emetric_ball_zero_iff</a>, <a>lt_tsub_iff_right</a>, <a>add_comm</a>] at hz", [{"full_name": "mem_emetric_ball_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [888, 3], "def_end_pos": [888, 14]}, {"full_name": "lt_tsub_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [417, 9], "def_end_pos": [417, 26]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : z \u2208 EMetric.ball 0 (r - \u2191\u2016y\u2016\u208a)\n\u22a2 f (x + y + z) = (p.changeOrigin y).sum z", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 f (x + y + z) = (p.changeOrigin y).sum z"}, {"tactic": "rw [p.changeOrigin_eval_of_finite hf.finite, add_assoc, hf.sum]", "annotated_tactic": ["rw [p.changeOrigin_eval_of_finite hf.finite, <a>add_assoc</a>, hf.sum]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 f (x + y + z) = (p.changeOrigin y).sum z", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 y + z \u2208 EMetric.ball 0 r"}, {"tactic": "refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)", "annotated_tactic": ["refine <a>mem_emetric_ball_zero_iff</a>.2 (<a>lt_of_le_of_lt</a> ?_ hz)", [{"full_name": "mem_emetric_ball_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [888, 3], "def_end_pos": [888, 14]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 y + z \u2208 EMetric.ball 0 r", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 \u2191\u2016y + z\u2016\u208a \u2264 \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a"}, {"tactic": "exact mod_cast nnnorm_add_le y z", "annotated_tactic": ["exact mod_cast <a>nnnorm_add_le</a> y z", [{"full_name": "nnnorm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [805, 15], "def_end_pos": [805, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\nx y : E\nhf : HasFiniteFPowerSeriesOnBall f p x n r\nh : \u2191\u2016y\u2016\u208a < r\nz : E\nhz : \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a < r\n\u22a2 \u2191\u2016y + z\u2016\u208a \u2264 \u2191\u2016y\u2016\u208a + \u2191\u2016z\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.of_diff", "start": [784, 1], "end": [785, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_mono", "start": [259, 1], "end": [261, 57], "traced_tactics": [{"tactic": "rintro - \u27e8a, ha, rfl\u27e9", "annotated_tactic": ["rintro - \u27e8a, ha, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh : s \u2286 t\n\u22a2 f '' s \u2286 f '' t", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh : s \u2286 t\na : \u03b1\nha : a \u2208 s\n\u22a2 f a \u2208 f '' t"}, {"tactic": "exact mem_image_of_mem f (h ha)", "annotated_tactic": ["exact <a>mem_image_of_mem</a> f (h ha)", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh : s \u2286 t\na : \u03b1\nha : a \u2208 s\n\u22a2 f a \u2208 f '' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffAt \u211d n f x\nhg : ContDiffAt \u211d n g x\nhne : f x \u2260 g x\n\u22a2 ContDiffAt \u211d n (fun y => Dist.dist (f y) (g y)) x", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffAt \u211d n f x\nhg : ContDiffAt \u211d n g x\nhne : f x \u2260 g x\n\u22a2 ContDiffAt \u211d n (fun y => \u2016f y - g y\u2016) x"}, {"tactic": "exact (hf.sub hg).norm \ud835\udd5c (sub_ne_zero.2 hne)", "annotated_tactic": ["exact (hf.sub hg).<a>norm</a> \ud835\udd5c (<a>sub_ne_zero</a>.2 hne)", [{"full_name": "ContDiffAt.norm", "def_path": "Mathlib/Analysis/InnerProductSpace/Calculus.lean", "def_pos": [174, 9], "def_end_pos": [174, 24]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffAt \u211d n f x\nhg : ContDiffAt \u211d n g x\nhne : f x \u2260 g x\n\u22a2 ContDiffAt \u211d n (fun y => \u2016f y - g y\u2016) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "full_name": "hasFTaylorSeriesUpToOn_top_iff'", "start": [255, 1], "end": [263, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_const_sub_uIcc", "start": [548, 1], "end": [550, 40], "traced_tactics": [{"tactic": "have := image_comp (fun x => a + x) fun x => -x", "annotated_tactic": ["have := <a>image_comp</a> (fun x => a + x) fun x => -x", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [263, 9], "def_end_pos": [263, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nthis : \u2200 (a_1 : Set \u03b1), ((fun x => a + x) \u2218 fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n\u22a2 (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]"}, {"tactic": "dsimp [Function.comp_def] at this", "annotated_tactic": ["dsimp [<a>Function.comp_def</a>] at this", [{"full_name": "Function.comp_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [37, 9], "def_end_pos": [37, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nthis : \u2200 (a_1 : Set \u03b1), ((fun x => a + x) \u2218 fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n\u22a2 (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nthis : \u2200 (a_1 : Set \u03b1), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n\u22a2 (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]"}, {"tactic": "simp [sub_eq_add_neg, this, add_comm]", "annotated_tactic": ["simp [<a>sub_eq_add_neg</a>, this, <a>add_comm</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nthis : \u2200 (a_1 : Set \u03b1), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n\u22a2 (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.div_ne_zero_iff_of_dvd", "start": [1145, 1], "end": [1146, 95], "traced_tactics": [{"tactic": "obtain rfl | hb := eq_or_ne b 0 <;> simp [Nat.div_ne_zero_iff, Nat.le_iff_ne_zero_of_dvd, *]", "annotated_tactic": ["obtain rfl | hb := <a>eq_or_ne</a> b 0 <;> simp [<a>Nat.div_ne_zero_iff</a>, <a>Nat.le_iff_ne_zero_of_dvd</a>, *]", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}, {"full_name": "Nat.div_ne_zero_iff", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [1135, 17], "def_end_pos": [1135, 32]}, {"full_name": "Nat.le_iff_ne_zero_of_dvd", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [1141, 7], "def_end_pos": [1141, 28]}]], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\nhba : b \u2223 a\n\u22a2 a / b \u2260 0 \u2194 a \u2260 0 \u2227 b \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.results_thinkN", "start": [580, 1], "end": [583, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "full_name": "CategoryTheory.MonoidalOpposite.unmop_hom_braiding", "start": [732, 1], "end": [733, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/QuasiSeparated.lean", "full_name": "isQuasiSeparated_univ_iff", "start": [53, 1], "end": [56, 26], "traced_tactics": [{"tactic": "rw [quasiSeparatedSpace_iff]", "annotated_tactic": ["rw [<a>quasiSeparatedSpace_iff</a>]", [{"full_name": "quasiSeparatedSpace_iff", "def_path": "Mathlib/Topology/QuasiSeparated.lean", "def_pos": [46, 3], "def_end_pos": [46, 9]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 QuasiSeparatedSpace \u03b1", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 \u2200 (U V : Set \u03b1), IsOpen U \u2192 IsCompact U \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)"}, {"tactic": "simp [IsQuasiSeparated]", "annotated_tactic": ["simp [<a>IsQuasiSeparated</a>]", [{"full_name": "IsQuasiSeparated", "def_path": "Mathlib/Topology/QuasiSeparated.lean", "def_pos": [40, 5], "def_end_pos": [40, 21]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 \u2200 (U V : Set \u03b1), IsOpen U \u2192 IsCompact U \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.measure_le_average_pos", "start": [565, 1], "end": [568, 20], "traced_tactics": [{"tactic": "simpa using measure_le_setAverage_pos (Measure.measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _)\n  hf.integrableOn", "annotated_tactic": ["simpa using <a>measure_le_setAverage_pos</a> (<a>Measure.measure_univ_ne_zero</a>.2 h\u03bc) (<a>measure_ne_top</a> _ _)\n    hf.integrableOn", [{"full_name": "MeasureTheory.measure_le_setAverage_pos", "def_path": "Mathlib/MeasureTheory/Integral/Average.lean", "def_pos": [520, 9], "def_end_pos": [520, 34]}, {"full_name": "MeasureTheory.Measure.measure_univ_ne_zero", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1150, 9], "def_end_pos": [1150, 29]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f \u03bc\n\u22a2 0 < \u03bc {x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bitwise_bit", "start": [390, 1], "end": [398, 71], "traced_tactics": [{"tactic": "cases' m with m m <;> cases' n with n n <;>\nsimp [bitwise, ofNat_eq_coe, bit_coe_nat, natBitwise, Bool.not_false, Bool.not_eq_false',\n  bit_negSucc]", "annotated_tactic": ["cases' m with m m <;> cases' n with n n <;>\n  simp [<a>bitwise</a>, <a>ofNat_eq_coe</a>, <a>bit_coe_nat</a>, <a>natBitwise</a>, <a>Bool.not_false</a>, <a>Bool.not_eq_false'</a>,\n    <a>bit_negSucc</a>]", [{"full_name": "Int.bitwise", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [58, 5], "def_end_pos": [58, 12]}, {"full_name": "Int.ofNat_eq_coe", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [71, 17], "def_end_pos": [71, 29]}, {"full_name": "Int.bit_coe_nat", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [251, 9], "def_end_pos": [251, 20]}, {"full_name": "Int.natBitwise", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [52, 5], "def_end_pos": [52, 15]}, {"full_name": "Bool.not_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [230, 17], "def_end_pos": [230, 31]}, {"full_name": "Bool.not_eq_false'", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [234, 17], "def_end_pos": [234, 35]}, {"full_name": "Int.bit_negSucc", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [257, 9], "def_end_pos": [257, 20]}]], "state_before": "f : Bool \u2192 Bool \u2192 Bool\na : Bool\nm : \u2124\nb : Bool\nn : \u2124\n\u22a2 bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n)", "state_after": "case ofNat.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false false then -[Nat.bitwise (fun x y => !f x y) (Nat.bit a m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise f (Nat.bit a m) (Nat.bit b n))) =\n    bit (f a b) (bif f false false then -[Nat.bitwise (fun x y => !f x y) m n+1] else \u2191(Nat.bitwise f m n))\n\ncase ofNat.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false true then -[Nat.bitwise (fun x y => !f x !y) (Nat.bit a m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f x !y) (Nat.bit a m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f false true then -[Nat.bitwise (fun x y => !f x !y) m n+1] else \u2191(Nat.bitwise (fun x y => f x !y) m n))\n\ncase negSucc.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) (Nat.bit (!a) m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) y) (Nat.bit (!a) m) (Nat.bit b n))) =\n    bit (f a b)\n      (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) y) m n))\n\ncase negSucc.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) !y) m n))"}, {"tactic": "by_cases h : f false false <;> simp (config := {decide := true}) [h]", "annotated_tactic": ["by_cases h : f <a>false</a> <a>false</a> <;> simp (config := {decide := <a>true</a>}) [h]", [{"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case ofNat.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false false then -[Nat.bitwise (fun x y => !f x y) (Nat.bit a m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise f (Nat.bit a m) (Nat.bit b n))) =\n    bit (f a b) (bif f false false then -[Nat.bitwise (fun x y => !f x y) m n+1] else \u2191(Nat.bitwise f m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f false true <;> simp (config := {decide := true}) [h]", "annotated_tactic": ["by_cases h : f <a>false</a> <a>true</a> <;> simp (config := {decide := <a>true</a>}) [h]", [{"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case ofNat.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f false true then -[Nat.bitwise (fun x y => !f x !y) (Nat.bit a m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f x !y) (Nat.bit a m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f false true then -[Nat.bitwise (fun x y => !f x !y) m n+1] else \u2191(Nat.bitwise (fun x y => f x !y) m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f true false <;> simp (config := {decide := true}) [h]", "annotated_tactic": ["by_cases h : f <a>true</a> <a>false</a> <;> simp (config := {decide := <a>true</a>}) [h]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case negSucc.ofNat\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) (Nat.bit (!a) m) (Nat.bit b n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) y) (Nat.bit (!a) m) (Nat.bit b n))) =\n    bit (f a b)\n      (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) y) m n))", "state_after": "no goals"}, {"tactic": "by_cases h : f true true <;> simp (config := {decide := true}) [h]", "annotated_tactic": ["by_cases h : f <a>true</a> <a>true</a> <;> simp (config := {decide := <a>true</a>}) [h]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case negSucc.negSucc\nf : Bool \u2192 Bool \u2192 Bool\na b : Bool\nm n : \u2115\n\u22a2 (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]\n    else \u2191(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =\n    bit (f a b)\n      (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) !y) m n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.MapsTo.comp_right", "start": [536, 1], "end": [537, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.norm_int", "start": [176, 1], "end": [177, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WellFoundedSet.lean", "full_name": "Set.isPWO_iff_exists_monotone_subseq", "start": [444, 1], "end": [446, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/NormNum/Prime.lean", "full_name": "Mathlib.Meta.NormNum.isNat_prime_1", "start": [170, 1], "end": [171, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "full_name": "Real.log_sqrt", "start": [315, 1], "end": [317, 20], "traced_tactics": [{"tactic": "rw [eq_div_iff, mul_comm, \u2190 Nat.cast_two, \u2190 log_pow, sq_sqrt hx]", "annotated_tactic": ["rw [<a>eq_div_iff</a>, <a>mul_comm</a>, \u2190 <a>Nat.cast_two</a>, \u2190 <a>log_pow</a>, <a>sq_sqrt</a> hx]", [{"full_name": "eq_div_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [357, 22], "def_end_pos": [357, 32]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [208, 9], "def_end_pos": [208, 17]}, {"full_name": "Real.log_pow", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 16]}, {"full_name": "Real.sq_sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [201, 9], "def_end_pos": [201, 16]}]], "state_before": "x\u271d y x : \u211d\nhx : 0 \u2264 x\n\u22a2 log \u221ax = log x / 2", "state_after": "x\u271d y x : \u211d\nhx : 0 \u2264 x\n\u22a2 2 \u2260 0"}, {"tactic": "exact two_ne_zero", "annotated_tactic": ["exact <a>two_ne_zero</a>", [{"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "x\u271d y x : \u211d\nhx : 0 \u2264 x\n\u22a2 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "full_name": "Batteries.TransCmp.cmp_congr_left'", "start": [102, 1], "end": [103, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometryEquiv.symm_trans_self", "start": [576, 1], "end": [577, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalHom.lean", "full_name": "NonUnitalAlgHom.toMulHom_eq_coe", "start": [243, 1], "end": [244, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.nodupKeys_of_nodupKeys_cons", "start": [111, 1], "end": [113, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "openEmbedding_sigma_map", "start": [1727, 1], "end": [1730, 16], "traced_tactics": [{"tactic": "simp only [openEmbedding_iff_embedding_open, isOpenMap_sigma_map, embedding_sigma_map h,\n  forall_and]", "annotated_tactic": ["simp only [<a>openEmbedding_iff_embedding_open</a>, <a>isOpenMap_sigma_map</a>, <a>embedding_sigma_map</a> h,\n    <a>forall_and</a>]", [{"full_name": "openEmbedding_iff_embedding_open", "def_path": "Mathlib/Topology/Maps.lean", "def_pos": [597, 9], "def_end_pos": [597, 41]}, {"full_name": "isOpenMap_sigma_map", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1710, 9], "def_end_pos": [1710, 28]}, {"full_name": "embedding_sigma_map", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1722, 9], "def_end_pos": [1722, 28]}, {"full_name": "forall_and", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [256, 9], "def_end_pos": [256, 19]}]], "state_before": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\n\u03c3 : \u03b9 \u2192 Type u_7\n\u03c4 : \u03ba \u2192 Type u_8\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c3 i)\ninst\u271d\u00b9 : (k : \u03ba) \u2192 TopologicalSpace (\u03c4 k)\ninst\u271d : TopologicalSpace X\nf\u2081 : \u03b9 \u2192 \u03ba\nf\u2082 : (i : \u03b9) \u2192 \u03c3 i \u2192 \u03c4 (f\u2081 i)\nh : Injective f\u2081\n\u22a2 OpenEmbedding (Sigma.map f\u2081 f\u2082) \u2194 \u2200 (i : \u03b9), OpenEmbedding (f\u2082 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.map\u2082_coe_right", "start": [815, 1], "end": [816, 18], "traced_tactics": [{"tactic": "cases a <;> rfl", "annotated_tactic": ["cases a <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\na\u271d b\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : WithTop \u03b1\nb : \u03b2\n\u22a2 map\u2082 f a \u2191b = map (fun x => f x b) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "full_name": "Matrix.isUnit_fromBlocks_zero\u2081\u2082", "start": [194, 1], "end": [197, 60], "traced_tactics": [{"tactic": "simp only [\u2190 nonempty_invertible_iff_isUnit, \u2190 nonempty_prod,\n  (fromBlocksZero\u2081\u2082InvertibleEquiv _ _ _).nonempty_congr]", "annotated_tactic": ["simp only [\u2190 <a>nonempty_invertible_iff_isUnit</a>, \u2190 <a>nonempty_prod</a>,\n    (<a>fromBlocksZero\u2081\u2082InvertibleEquiv</a> _ _ _).<a>nonempty_congr</a>]", [{"full_name": "nonempty_invertible_iff_isUnit", "def_path": "Mathlib/Algebra/Group/Invertible/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 39]}, {"full_name": "nonempty_prod", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [175, 9], "def_end_pos": [175, 22]}, {"full_name": "Matrix.fromBlocksZero\u2081\u2082InvertibleEquiv", "def_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "def_pos": [169, 5], "def_end_pos": [169, 36]}, {"full_name": "Equiv.nonempty_congr", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [235, 9], "def_end_pos": [235, 23]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsUnit (fromBlocks A 0 C D) \u2194 IsUnit A \u2227 IsUnit D", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.DominatedFinMeasAdditive.of_smul_measure", "start": [251, 1], "end": [263, 7], "traced_tactics": [{"tactic": "have h : \u2200 s, MeasurableSet s \u2192 c \u2022 \u03bc s = \u221e \u2192 \u03bc s = \u221e := by\n  intro s _ hc\u03bcs\n  simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false_iff, Ne,\n    false_and_iff] at hc\u03bcs\n  exact hc\u03bcs.2", "annotated_tactic": ["have h : \u2200 s, <a>MeasurableSet</a> s \u2192 c \u2022 \u03bc s = \u221e \u2192 \u03bc s = \u221e := by\n    intro s _ hc\u03bcs\n    simp only [hc_ne_top, <a>Algebra.id.smul_eq_mul</a>, <a>ENNReal.mul_eq_top</a>, <a>or_false_iff</a>, <a>Ne</a>,\n      <a>false_and_iff</a>] at hc\u03bcs\n    exact hc\u03bcs.2", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [225, 9], "def_end_pos": [225, 19]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 21]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\n\u22a2 DominatedFinMeasAdditive \u03bc T (c.toReal * C)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\n\u22a2 DominatedFinMeasAdditive \u03bc T (c.toReal * C)"}, {"tactic": "refine \u27e8hT.1.of_eq_top_imp_eq_top (\u03bc := c \u2022 \u03bc) h, fun s hs h\u03bcs => ?_\u27e9", "annotated_tactic": ["refine \u27e8hT.1.<a>of_eq_top_imp_eq_top</a> (\u03bc := c \u2022 \u03bc) h, fun s hs h\u03bcs => ?_\u27e9", [{"full_name": "MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [117, 9], "def_end_pos": [117, 29]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\n\u22a2 DominatedFinMeasAdditive \u03bc T (c.toReal * C)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal"}, {"tactic": "have hc\u03bcs : c \u2022 \u03bc s \u2260 \u221e := mt (h s hs) h\u03bcs.ne", "annotated_tactic": ["have hc\u03bcs : c \u2022 \u03bc s \u2260 \u221e := <a>mt</a> (h s hs) h\u03bcs.ne", [{"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c \u2022 \u03bc s \u2260 \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal"}, {"tactic": "rw [smul_eq_mul] at hc\u03bcs", "annotated_tactic": ["rw [<a>smul_eq_mul</a>] at hc\u03bcs", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c \u2022 \u03bc s \u2260 \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal"}, {"tactic": "simp_rw [DominatedFinMeasAdditive, Measure.smul_apply, smul_eq_mul, toReal_mul] at hT", "annotated_tactic": ["simp_rw [<a>DominatedFinMeasAdditive</a>, <a>Measure.smul_apply</a>, <a>smul_eq_mul</a>, <a>toReal_mul</a>] at hT", [{"full_name": "MeasureTheory.DominatedFinMeasAdditive", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [186, 5], "def_end_pos": [186, 29]}, {"full_name": "MeasureTheory.Measure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [893, 9], "def_end_pos": [893, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\nhT : FinMeasAdditive (c \u2022 \u03bc) T \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c * \u03bc s < \u22a4 \u2192 \u2016T s\u2016 \u2264 C * (c.toReal * (\u03bc s).toReal)\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal"}, {"tactic": "refine (hT.2 s hs hc\u03bcs.lt_top).trans (le_of_eq ?_)", "annotated_tactic": ["refine (hT.2 s hs hc\u03bcs.lt_top).<a>trans</a> (<a>le_of_eq</a> ?_)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\nhT : FinMeasAdditive (c \u2022 \u03bc) T \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c * \u03bc s < \u22a4 \u2192 \u2016T s\u2016 \u2264 C * (c.toReal * (\u03bc s).toReal)\n\u22a2 \u2016T s\u2016 \u2264 c.toReal * C * (\u03bc s).toReal", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\nhT : FinMeasAdditive (c \u2022 \u03bc) T \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c * \u03bc s < \u22a4 \u2192 \u2016T s\u2016 \u2264 C * (c.toReal * (\u03bc s).toReal)\n\u22a2 C * (c.toReal * (\u03bc s).toReal) = c.toReal * C * (\u03bc s).toReal"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s < \u22a4\nhc\u03bcs : c * \u03bc s \u2260 \u22a4\nhT : FinMeasAdditive (c \u2022 \u03bc) T \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c * \u03bc s < \u22a4 \u2192 \u2016T s\u2016 \u2264 C * (c.toReal * (\u03bc s).toReal)\n\u22a2 C * (c.toReal * (\u03bc s).toReal) = c.toReal * C * (\u03bc s).toReal", "state_after": "no goals"}, {"tactic": "intro s _ hc\u03bcs", "annotated_tactic": ["intro s _ hc\u03bcs", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 c \u2022 \u03bc s = \u22a4 \u2192 \u03bc s = \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\ns : Set \u03b1\na\u271d : MeasurableSet s\nhc\u03bcs : c \u2022 \u03bc s = \u22a4\n\u22a2 \u03bc s = \u22a4"}, {"tactic": "simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false_iff, Ne,\n  false_and_iff] at hc\u03bcs", "annotated_tactic": ["simp only [hc_ne_top, <a>Algebra.id.smul_eq_mul</a>, <a>ENNReal.mul_eq_top</a>, <a>or_false_iff</a>, <a>Ne</a>,\n      <a>false_and_iff</a>] at hc\u03bcs", [{"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [225, 9], "def_end_pos": [225, 19]}, {"full_name": "or_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 21]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "false_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\ns : Set \u03b1\na\u271d : MeasurableSet s\nhc\u03bcs : c \u2022 \u03bc s = \u22a4\n\u22a2 \u03bc s = \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\ns : Set \u03b1\na\u271d : MeasurableSet s\nhc\u03bcs : \u00acc = 0 \u2227 \u03bc s = \u22a4\n\u22a2 \u03bc s = \u22a4"}, {"tactic": "exact hc\u03bcs.2", "annotated_tactic": ["exact hc\u03bcs.2", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_7\ninst\u271d : SeminormedAddCommGroup \u03b2\nT T' : Set \u03b1 \u2192 \u03b2\nC C' : \u211d\nc : \u211d\u22650\u221e\nhc_ne_top : c \u2260 \u22a4\nhT : DominatedFinMeasAdditive (c \u2022 \u03bc) T C\ns : Set \u03b1\na\u271d : MeasurableSet s\nhc\u03bcs : \u00acc = 0 \u2227 \u03bc s = \u22a4\n\u22a2 \u03bc s = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Integral.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.ae_integrable_condKernel_iff", "start": [285, 1], "end": [291, 42], "traced_tactics": [{"tactic": "rw [\u2190 \u03c1.compProd_fst_condKernel] at hf", "annotated_tactic": ["rw [\u2190 \u03c1.compProd_fst_condKernel] at hf", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : StandardBorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03c1\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03c1.fst, Integrable (fun \u03c9 => f (a, \u03c9)) (\u03c1.condKernel a)) \u2227\n      Integrable (fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u03c1.condKernel a) \u03c1.fst \u2194\n    Integrable f \u03c1", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : StandardBorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03c1.fst, Integrable (fun \u03c9 => f (a, \u03c9)) (\u03c1.condKernel a)) \u2227\n      Integrable (fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u03c1.condKernel a) \u03c1.fst \u2194\n    Integrable f \u03c1"}, {"tactic": "conv_rhs => rw [\u2190 \u03c1.compProd_fst_condKernel]", "annotated_tactic": ["conv_rhs => rw [\u2190 \u03c1.compProd_fst_condKernel]", []], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : StandardBorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03c1.fst, Integrable (fun \u03c9 => f (a, \u03c9)) (\u03c1.condKernel a)) \u2227\n      Integrable (fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u03c1.condKernel a) \u03c1.fst \u2194\n    Integrable f \u03c1", "state_after": "\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : StandardBorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03c1.fst, Integrable (fun \u03c9 => f (a, \u03c9)) (\u03c1.condKernel a)) \u2227\n      Integrable (fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u03c1.condKernel a) \u03c1.fst \u2194\n    Integrable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)"}, {"tactic": "rw [Measure.integrable_compProd_iff hf]", "annotated_tactic": ["rw [<a>Measure.integrable_compProd_iff</a> hf]", [{"full_name": "MeasureTheory.Measure.integrable_compProd_iff", "def_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "def_pos": [110, 7], "def_end_pos": [110, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03a9\ninst\u271d\u2075 : StandardBorelSpace \u03a9\ninst\u271d\u2074 : Nonempty \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03c1.fst, Integrable (fun \u03c9 => f (a, \u03c9)) (\u03c1.condKernel a)) \u2227\n      Integrable (fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u03c1.condKernel a) \u03c1.fst \u2194\n    Integrable f (\u03c1.fst \u2297\u2098 \u03c1.condKernel)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "full_name": "smul_inv_smul", "start": [35, 1], "end": [36, 42], "traced_tactics": [{"tactic": "rw [smul_smul, mul_right_inv, one_smul]", "annotated_tactic": ["rw [<a>smul_smul</a>, <a>mul_right_inv</a>, <a>one_smul</a>]", [{"full_name": "smul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [446, 7], "def_end_pos": [446, 16]}, {"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1238, 9], "def_end_pos": [1238, 22]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nc : \u03b1\nx : \u03b2\n\u22a2 c \u2022 c\u207b\u00b9 \u2022 x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/IsPerfectClosure.lean", "full_name": "RingHom.pNilradical_le_ker_of_perfectRing", "start": [196, 1], "end": [201, 29], "traced_tactics": [{"tactic": "obtain \u27e8n, h\u27e9 := mem_pNilradical.1 h", "annotated_tactic": ["obtain \u27e8n, h\u27e9 := <a>mem_pNilradical</a>.1 h", [{"full_name": "mem_pNilradical", "def_path": "Mathlib/FieldTheory/IsPerfectClosure.lean", "def_pos": [96, 9], "def_end_pos": [96, 24]}]], "state_before": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2076 : CommSemiring K\ninst\u271d\u2075 : CommSemiring L\ninst\u271d\u2074 : CommSemiring M\ni : K \u2192+* L\nj : K \u2192+* M\nf : L \u2192+* M\np : \u2115\ninst\u271d\u00b3 : ExpChar K p\ninst\u271d\u00b2 : ExpChar L p\ninst\u271d\u00b9 : ExpChar M p\ninst\u271d : PerfectRing L p\nx : K\nh : x \u2208 pNilradical K p\n\u22a2 x \u2208 ker i", "state_after": "case intro\nK : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2076 : CommSemiring K\ninst\u271d\u2075 : CommSemiring L\ninst\u271d\u2074 : CommSemiring M\ni : K \u2192+* L\nj : K \u2192+* M\nf : L \u2192+* M\np : \u2115\ninst\u271d\u00b3 : ExpChar K p\ninst\u271d\u00b2 : ExpChar L p\ninst\u271d\u00b9 : ExpChar M p\ninst\u271d : PerfectRing L p\nx : K\nh\u271d : x \u2208 pNilradical K p\nn : \u2115\nh : x ^ p ^ n = 0\n\u22a2 x \u2208 ker i"}, {"tactic": "replace h := congr((iterateFrobeniusEquiv L p n).symm (i $h))", "annotated_tactic": ["replace h := congr((<a>iterateFrobeniusEquiv</a> L p n).<a>symm</a> (i $h))", [{"full_name": "iterateFrobeniusEquiv", "def_path": "Mathlib/FieldTheory/Perfect.lean", "def_pos": [88, 19], "def_end_pos": [88, 40]}, {"full_name": "RingEquiv.symm", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [265, 15], "def_end_pos": [265, 19]}]], "state_before": "case intro\nK : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2076 : CommSemiring K\ninst\u271d\u2075 : CommSemiring L\ninst\u271d\u2074 : CommSemiring M\ni : K \u2192+* L\nj : K \u2192+* M\nf : L \u2192+* M\np : \u2115\ninst\u271d\u00b3 : ExpChar K p\ninst\u271d\u00b2 : ExpChar L p\ninst\u271d\u00b9 : ExpChar M p\ninst\u271d : PerfectRing L p\nx : K\nh\u271d : x \u2208 pNilradical K p\nn : \u2115\nh : x ^ p ^ n = 0\n\u22a2 x \u2208 ker i", "state_after": "case intro\nK : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2076 : CommSemiring K\ninst\u271d\u2075 : CommSemiring L\ninst\u271d\u2074 : CommSemiring M\ni : K \u2192+* L\nj : K \u2192+* M\nf : L \u2192+* M\np : \u2115\ninst\u271d\u00b3 : ExpChar K p\ninst\u271d\u00b2 : ExpChar L p\ninst\u271d\u00b9 : ExpChar M p\ninst\u271d : PerfectRing L p\nx : K\nh\u271d : x \u2208 pNilradical K p\nn : \u2115\nh : (iterateFrobeniusEquiv L p n).symm (i (x ^ p ^ n)) = (iterateFrobeniusEquiv L p n).symm (i 0)\n\u22a2 x \u2208 ker i"}, {"tactic": "rwa [map_pow, \u2190 iterateFrobenius_def, \u2190 iterateFrobeniusEquiv_apply, RingEquiv.symm_apply_apply,\n  map_zero, map_zero] at h", "annotated_tactic": ["rwa [<a>map_pow</a>, \u2190 <a>iterateFrobenius_def</a>, \u2190 <a>iterateFrobeniusEquiv_apply</a>, <a>RingEquiv.symm_apply_apply</a>,\n    <a>map_zero</a>, <a>map_zero</a>] at h", [{"full_name": "map_pow", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [473, 9], "def_end_pos": [473, 16]}, {"full_name": "iterateFrobenius_def", "def_path": "Mathlib/Algebra/CharP/ExpChar.lean", "def_pos": [296, 9], "def_end_pos": [296, 29]}, {"full_name": "iterateFrobeniusEquiv_apply", "def_path": "Mathlib/FieldTheory/Perfect.lean", "def_pos": [87, 10], "def_end_pos": [87, 15]}, {"full_name": "RingEquiv.symm_apply_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [360, 9], "def_end_pos": [360, 25]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "case intro\nK : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2076 : CommSemiring K\ninst\u271d\u2075 : CommSemiring L\ninst\u271d\u2074 : CommSemiring M\ni : K \u2192+* L\nj : K \u2192+* M\nf : L \u2192+* M\np : \u2115\ninst\u271d\u00b3 : ExpChar K p\ninst\u271d\u00b2 : ExpChar L p\ninst\u271d\u00b9 : ExpChar M p\ninst\u271d : PerfectRing L p\nx : K\nh\u271d : x \u2208 pNilradical K p\nn : \u2115\nh : (iterateFrobeniusEquiv L p n).symm (i (x ^ p ^ n)) = (iterateFrobeniusEquiv L p n).symm (i 0)\n\u22a2 x \u2208 ker i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Irreducible.lean", "full_name": "infPrime_iff_infIrred", "start": [313, 1], "end": [315, 87], "traced_tactics": [{"tactic": "simp_rw [\u2190 sup_eq_left, sup_inf_left]", "annotated_tactic": ["simp_rw [\u2190 <a>sup_eq_left</a>, <a>sup_inf_left</a>]", [{"full_name": "sup_eq_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [154, 9], "def_end_pos": [154, 20]}, {"full_name": "sup_inf_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [718, 9], "def_end_pos": [718, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 b \u2293 c \u2264 a \u2192 b \u2264 a \u2228 c \u2264 a", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) = a \u2192 a \u2294 b = a \u2228 a \u2294 c = a"}, {"tactic": "exact @h _ _", "annotated_tactic": ["exact @h _ _", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) = a \u2192 a \u2294 b = a \u2228 a \u2294 c = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "edist_pi_const_le", "start": [506, 1], "end": [507, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier", "start": [625, 1], "end": [627, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Count.lean", "full_name": "Nat.count_one", "start": [91, 1], "end": [91, 77], "traced_tactics": [{"tactic": "simp [count_succ]", "annotated_tactic": ["simp [<a>count_succ</a>]", [{"full_name": "Nat.count_succ", "def_path": "Mathlib/Data/Nat/Count.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}]], "state_before": "p : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 count p 1 = if p 0 then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/MonCat/Basic.lean", "full_name": "MonCat.one_of", "start": [161, 1], "end": [162, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "full_name": "TensorProduct.liftAux.smul", "start": [538, 1], "end": [541, 86], "traced_tactics": [{"tactic": "simp_rw [\u2190 tmul_smul, liftAux_tmul, (f p).map_smul]", "annotated_tactic": ["simp_rw [\u2190 <a>tmul_smul</a>, <a>liftAux_tmul</a>, (f p).<a>map_smul</a>]", [{"full_name": "TensorProduct.tmul_smul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [373, 9], "def_end_pos": [373, 18]}, {"full_name": "TensorProduct.liftAux_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 21]}, {"full_name": "LinearMap.map_smul", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [376, 19], "def_end_pos": [376, 27]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u2076 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2075 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2074 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\ninst\u271d\u00b9\u00b9 : AddCommMonoid P\ninst\u271d\u00b9\u2070 : AddCommMonoid Q\ninst\u271d\u2079 : AddCommMonoid S\ninst\u271d\u2078 : AddCommMonoid T\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module R P\ninst\u271d\u2074 : Module R Q\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module R T\ninst\u271d\u00b9 : DistribMulAction R' M\ninst\u271d : Module R'' M\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nr : R\nx : M \u2297[R] N\np : M\nq : N\n\u22a2 (liftAux f) (r \u2022 p \u2297\u209c[R] q) = r \u2022 (liftAux f) (p \u2297\u209c[R] q)", "state_after": "no goals"}, {"tactic": "simp_rw [smul_add, (liftAux f).map_add, ih1, ih2, smul_add]", "annotated_tactic": ["simp_rw [<a>smul_add</a>, (<a>liftAux</a> f).<a>map_add</a>, ih1, ih2, <a>smul_add</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "TensorProduct.liftAux", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [526, 5], "def_end_pos": [526, 12]}, {"full_name": "AddMonoidHom.map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u2076 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2075 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2074 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\ninst\u271d\u00b9\u00b9 : AddCommMonoid P\ninst\u271d\u00b9\u2070 : AddCommMonoid Q\ninst\u271d\u2079 : AddCommMonoid S\ninst\u271d\u2078 : AddCommMonoid T\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module R P\ninst\u271d\u2074 : Module R Q\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module R T\ninst\u271d\u00b9 : DistribMulAction R' M\ninst\u271d : Module R'' M\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nr : R\nx p q : M \u2297[R] N\nih1 : (liftAux f) (r \u2022 p) = r \u2022 (liftAux f) p\nih2 : (liftAux f) (r \u2022 q) = r \u2022 (liftAux f) q\n\u22a2 (liftAux f) (r \u2022 (p + q)) = r \u2022 (liftAux f) (p + q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "full_name": "Int.cast_comm", "start": [116, 1], "end": [116, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Basic.lean", "full_name": "NumberField.mem_span_integralBasis", "start": [288, 1], "end": [291, 54], "traced_tactics": [{"tactic": "rw [integralBasis, Basis.localizationLocalization_span, LinearMap.mem_range,\n    IsScalarTower.coe_toAlgHom', RingHom.mem_range]", "annotated_tactic": ["rw [<a>integralBasis</a>, <a>Basis.localizationLocalization_span</a>, <a>LinearMap.mem_range</a>,\n      <a>IsScalarTower.coe_toAlgHom'</a>, <a>RingHom.mem_range</a>]", [{"full_name": "NumberField.integralBasis", "def_path": "Mathlib/NumberTheory/NumberField/Basic.lean", "def_pos": [272, 19], "def_end_pos": [272, 32]}, {"full_name": "Basis.localizationLocalization_span", "def_path": "Mathlib/RingTheory/Localization/Module.lean", "def_pos": [164, 9], "def_end_pos": [164, 44]}, {"full_name": "LinearMap.mem_range", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}, {"full_name": "IsScalarTower.coe_toAlgHom'", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [156, 9], "def_end_pos": [156, 22]}, {"full_name": "RingHom.mem_range", "def_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "def_pos": [601, 9], "def_end_pos": [601, 18]}]], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Field L\nnf : NumberField K\nx : K\n\u22a2 x \u2208 Submodule.span \u2124 (Set.range \u21d1(integralBasis K)) \u2194 x \u2208 (algebraMap (\ud835\udcde K) K).range", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.weightedHomogeneousComponent_eq_zero", "start": [385, 1], "end": [394, 57], "traced_tactics": [{"tactic": "classical\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\nintro d hd\nrw [Finset.mem_filter] at hd\nexfalso\napply lt_irrefl n\nnth_rw 1 [\u2190 hd.2]\nexact lt_of_le_of_lt (le_weightedTotalDegree w hd.1) h", "annotated_tactic": ["classical\n  rw [<a>weightedHomogeneousComponent_apply</a>, <a>sum_eq_zero</a>]\n  intro d hd\n  rw [<a>Finset.mem_filter</a>] at hd\n  exfalso\n  apply <a>lt_irrefl</a> n\n  nth_rw 1 [\u2190 hd.2]\n  exact <a>lt_of_le_of_lt</a> (<a>le_weightedTotalDegree</a> w hd.1) h", [{"full_name": "MvPolynomial.weightedHomogeneousComponent_apply", "def_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "def_pos": [347, 9], "def_end_pos": [347, 43]}, {"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [426, 3], "def_end_pos": [426, 14]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MvPolynomial.le_weightedTotalDegree", "def_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "def_pos": [125, 9], "def_end_pos": [125, 31]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 (weightedHomogeneousComponent w n) \u03c6 = 0", "state_after": "no goals"}, {"tactic": "rw [weightedHomogeneousComponent_apply, sum_eq_zero]", "annotated_tactic": ["rw [<a>weightedHomogeneousComponent_apply</a>, <a>sum_eq_zero</a>]", [{"full_name": "MvPolynomial.weightedHomogeneousComponent_apply", "def_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "def_pos": [347, 9], "def_end_pos": [347, 43]}, {"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [426, 3], "def_end_pos": [426, 14]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 (weightedHomogeneousComponent w n) \u03c6 = 0", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 \u2200 x \u2208 Finset.filter (fun d => (weightedDegree w) d = n) \u03c6.support, (monomial x) (coeff x \u03c6) = 0"}, {"tactic": "intro d hd", "annotated_tactic": ["intro d hd", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 \u2200 x \u2208 Finset.filter (fun d => (weightedDegree w) d = n) \u03c6.support, (monomial x) (coeff x \u03c6) = 0", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 Finset.filter (fun d => (weightedDegree w) d = n) \u03c6.support\n\u22a2 (monomial d) (coeff d \u03c6) = 0"}, {"tactic": "rw [Finset.mem_filter] at hd", "annotated_tactic": ["rw [<a>Finset.mem_filter</a>] at hd", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 Finset.filter (fun d => (weightedDegree w) d = n) \u03c6.support\n\u22a2 (monomial d) (coeff d \u03c6) = 0", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 (monomial d) (coeff d \u03c6) = 0"}, {"tactic": "exfalso", "annotated_tactic": ["exfalso", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 (monomial d) (coeff d \u03c6) = 0", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 False"}, {"tactic": "apply lt_irrefl n", "annotated_tactic": ["apply <a>lt_irrefl</a> n", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 False", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 n < n"}, {"tactic": "nth_rw 1 [\u2190 hd.2]", "annotated_tactic": ["nth_rw 1 [\u2190 hd.2]", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 n < n", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 (weightedDegree w) d < n"}, {"tactic": "exact lt_of_le_of_lt (le_weightedTotalDegree w hd.1) h", "annotated_tactic": ["exact <a>lt_of_le_of_lt</a> (<a>le_weightedTotalDegree</a> w hd.1) h", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "MvPolynomial.le_weightedTotalDegree", "def_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "def_pos": [125, 9], "def_end_pos": [125, 31]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support \u2227 (weightedDegree w) d = n\n\u22a2 (weightedDegree w) d < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "full_name": "Real.image_tan_Ioo", "start": [99, 1], "end": [100, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.map_comap_le", "start": [486, 1], "end": [487, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "full_name": "IsPurelyInseparable.tower_bot", "start": [406, 1], "end": [411, 97], "traced_tactics": [{"tactic": "refine \u27e8\u27e8fun x \u21a6 (isIntegral' F (algebraMap E K x)).tower_bot_of_field\u27e9, fun x h \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8fun x \u21a6 (<a>isIntegral'</a> F (<a>algebraMap</a> E K x)).<a>tower_bot_of_field</a>\u27e9, fun x h \u21a6 ?_\u27e9", [{"full_name": "IsPurelyInseparable.isIntegral'", "def_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "def_pos": [150, 9], "def_end_pos": [150, 40]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IsIntegral.tower_bot_of_field", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [937, 9], "def_end_pos": [937, 38]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\n\u22a2 IsPurelyInseparable F E", "state_after": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh : (minpoly F x).Separable\n\u22a2 x \u2208 (algebraMap F E).range"}, {"tactic": "rw [\u2190 minpoly.algebraMap_eq (algebraMap E K).injective] at h", "annotated_tactic": ["rw [\u2190 <a>minpoly.algebraMap_eq</a> (<a>algebraMap</a> E K).<a>injective</a>] at h", [{"full_name": "minpoly.algebraMap_eq", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [73, 9], "def_end_pos": [73, 22]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "RingHom.injective", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 28]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh : (minpoly F x).Separable\n\u22a2 x \u2208 (algebraMap F E).range", "state_after": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh : (minpoly F ((algebraMap E K) x)).Separable\n\u22a2 x \u2208 (algebraMap F E).range"}, {"tactic": "obtain \u27e8y, h\u27e9 := inseparable F _ h", "annotated_tactic": ["obtain \u27e8y, h\u27e9 := <a>inseparable</a> F _ h", [{"full_name": "IsPurelyInseparable.inseparable", "def_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "def_pos": [158, 9], "def_end_pos": [158, 40]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh : (minpoly F ((algebraMap E K) x)).Separable\n\u22a2 x \u2208 (algebraMap F E).range", "state_after": "case intro\nF : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh\u271d : (minpoly F ((algebraMap E K) x)).Separable\ny : F\nh : (algebraMap F K) y = (algebraMap E K) x\n\u22a2 x \u2208 (algebraMap F E).range"}, {"tactic": "exact \u27e8y, (algebraMap E K).injective (h.symm \u25b8 (IsScalarTower.algebraMap_apply F E K y).symm)\u27e9", "annotated_tactic": ["exact \u27e8y, (<a>algebraMap</a> E K).<a>injective</a> (h.symm \u25b8 (<a>IsScalarTower.algebraMap_apply</a> F E K y).<a>symm</a>)\u27e9", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "RingHom.injective", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [259, 19], "def_end_pos": [259, 28]}, {"full_name": "IsScalarTower.algebraMap_apply", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [130, 9], "def_end_pos": [130, 25]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case intro\nF : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsPurelyInseparable F K\nx : E\nh\u271d : (minpoly F ((algebraMap E K) x)).Separable\ny : F\nh : (algebraMap F K) y = (algebraMap E K) x\n\u22a2 x \u2208 (algebraMap F E).range", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_union_eq_card_add_card", "start": [544, 1], "end": [545, 71], "traced_tactics": [{"tactic": "rw [\u2190 card_union_add_card_inter]", "annotated_tactic": ["rw [\u2190 <a>card_union_add_card_inter</a>]", [{"full_name": "Finset.card_union_add_card_inter", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [524, 9], "def_end_pos": [524, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (s \u222a t).card = s.card + t.card \u2194 _root_.Disjoint s t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (s \u222a t).card = (s \u222a t).card + (s \u2229 t).card \u2194 _root_.Disjoint s t"}, {"tactic": "simp [disjoint_iff_inter_eq_empty]", "annotated_tactic": ["simp [<a>disjoint_iff_inter_eq_empty</a>]", [{"full_name": "Finset.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1383, 9], "def_end_pos": [1383, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (s \u222a t).card = (s \u222a t).card + (s \u2229 t).card \u2194 _root_.Disjoint s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.comap_op_pow", "start": [569, 1], "end": [572, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BooleanAlgebra.lean", "full_name": "sdiff_lt", "start": [319, 1], "end": [322, 32], "traced_tactics": [{"tactic": "refine sdiff_le.lt_of_ne fun h => hy ?_", "annotated_tactic": ["refine sdiff_le.lt_of_ne fun h => hy ?_", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\n\u22a2 x \\ y < x", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \\ y = x\n\u22a2 y = \u22a5"}, {"tactic": "rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h", "annotated_tactic": ["rw [<a>sdiff_eq_self_iff_disjoint'</a>, <a>disjoint_iff</a>] at h", [{"full_name": "sdiff_eq_self_iff_disjoint'", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [315, 9], "def_end_pos": [315, 36]}, {"full_name": "disjoint_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [135, 9], "def_end_pos": [135, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \\ y = x\n\u22a2 y = \u22a5", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \u2293 y = \u22a5\n\u22a2 y = \u22a5"}, {"tactic": "rw [\u2190 h, inf_eq_right.mpr hx]", "annotated_tactic": ["rw [\u2190 h, inf_eq_right.mpr hx]", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \u2293 y = \u22a5\n\u22a2 y = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "map_fst_nhds", "start": [773, 1], "end": [774, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.HasAntitoneBasis.eventually_subset", "start": [1901, 1], "end": [1904, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Dioph.lean", "full_name": "Dioph.dioph_comp", "start": [551, 1], "end": [553, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Basic.lean", "full_name": "ENNReal.zero_eq_coe", "start": [393, 20], "end": [393, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Finset.prod_mulIndicator_subset", "start": [1407, 1], "end": [1413, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompletePartialOrder.lean", "full_name": "Directed.le_iSup", "start": [55, 1], "end": [56, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Instances.lean", "full_name": "nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts", "start": [312, 1], "end": [317, 54], "traced_tactics": [{"tactic": "refine \u27e8fun ha \u21a6 \u27e8.of_nonneg ha, .nnreal_of_nonneg ha\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun ha \u21a6 \u27e8.of_nonneg ha, .nnreal_of_nonneg ha\u27e9, ?_\u27e9", []], "state_before": "A : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\na : A\n\u22a2 0 \u2264 a \u2194 IsSelfAdjoint a \u2227 QuasispectrumRestricts a \u21d1ContinuousMap.realToNNReal", "state_after": "A : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\na : A\n\u22a2 IsSelfAdjoint a \u2227 QuasispectrumRestricts a \u21d1ContinuousMap.realToNNReal \u2192 0 \u2264 a"}, {"tactic": "rintro \u27e8ha\u2081, ha\u2082\u27e9", "annotated_tactic": ["rintro \u27e8ha\u2081, ha\u2082\u27e9", []], "state_before": "A : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\na : A\n\u22a2 IsSelfAdjoint a \u2227 QuasispectrumRestricts a \u21d1ContinuousMap.realToNNReal \u2192 0 \u2264 a", "state_after": "case intro\nA : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\na : A\nha\u2081 : IsSelfAdjoint a\nha\u2082 : QuasispectrumRestricts a \u21d1ContinuousMap.realToNNReal\n\u22a2 0 \u2264 a"}, {"tactic": "obtain \u27e8x, hx, -, rfl\u27e9 := CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts ha\u2081 ha\u2082", "annotated_tactic": ["obtain \u27e8x, hx, -, rfl\u27e9 := <a>CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts</a> ha\u2081 ha\u2082", [{"full_name": "CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts", "def_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Instances.lean", "def_pos": [290, 7], "def_end_pos": [290, 65]}]], "state_before": "case intro\nA : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\na : A\nha\u2081 : IsSelfAdjoint a\nha\u2082 : QuasispectrumRestricts a \u21d1ContinuousMap.realToNNReal\n\u22a2 0 \u2264 a", "state_after": "case intro.intro.intro.intro\nA : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\nx : A\nhx : IsSelfAdjoint x\nha\u2081 : IsSelfAdjoint (x * x)\nha\u2082 : QuasispectrumRestricts (x * x) \u21d1ContinuousMap.realToNNReal\n\u22a2 0 \u2264 x * x"}, {"tactic": "simpa [sq, hx.star_eq] using star_mul_self_nonneg x", "annotated_tactic": ["simpa [<a>sq</a>, hx.star_eq] using <a>star_mul_self_nonneg</a> x", [{"full_name": "sq", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [684, 41], "def_end_pos": [684, 43]}, {"full_name": "star_mul_self_nonneg", "def_path": "Mathlib/Algebra/Star/Order.lean", "def_pos": [149, 9], "def_end_pos": [149, 29]}]], "state_before": "case intro.intro.intro.intro\nA : Type u_1\ninst\u271d\u00b9\u2070 : NonUnitalRing A\ninst\u271d\u2079 : PartialOrder A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarOrderedRing A\ninst\u271d\u2076 : TopologicalSpace A\ninst\u271d\u2075 : Module \u211d A\ninst\u271d\u2074 : IsScalarTower \u211d A A\ninst\u271d\u00b3 : SMulCommClass \u211d A A\ninst\u271d\u00b2 : NonUnitalContinuousFunctionalCalculus \u211d IsSelfAdjoint\ninst\u271d\u00b9 : NonnegSpectrumClass \u211d A\ninst\u271d : UniqueNonUnitalContinuousFunctionalCalculus \u211d A\nx : A\nhx : IsSelfAdjoint x\nha\u2081 : IsSelfAdjoint (x * x)\nha\u2082 : QuasispectrumRestricts (x * x) \u21d1ContinuousMap.realToNNReal\n\u22a2 0 \u2264 x * x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "full_name": "Complex.range_arg", "start": [156, 1], "end": [157, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Basic.lean", "full_name": "Finset.Icc_subset_Icc_left", "start": [188, 1], "end": [189, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "MeasureTheory.Integrable.integral_norm_condDistrib", "start": [163, 1], "end": [166, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_subset", "start": [282, 1], "end": [283, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.norm_const", "start": [920, 1], "end": [924, 81], "traced_tactics": [{"tactic": "have := NeZero.ne \u03bc", "annotated_tactic": ["have := <a>NeZero.ne</a> \u03bc", [{"full_name": "NeZero.ne", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [30, 9], "def_end_pos": [30, 18]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\n\u22a2 \u2016(Lp.const p \u03bc) c\u2016 = \u2016c\u2016 * (\u03bc Set.univ).toReal ^ (1 / p.toReal)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\nthis : \u03bc \u2260 0\n\u22a2 \u2016(Lp.const p \u03bc) c\u2016 = \u2016c\u2016 * (\u03bc Set.univ).toReal ^ (1 / p.toReal)"}, {"tactic": "rw [\u2190 Mem\u2112p.toLp_const, Lp.norm_toLp, snorm_const] <;> try assumption", "annotated_tactic": ["rw [\u2190 <a>Mem\u2112p.toLp_const</a>, <a>Lp.norm_toLp</a>, <a>snorm_const</a>] <;> try assumption", [{"full_name": "MeasureTheory.Mem\u2112p.toLp_const", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [912, 7], "def_end_pos": [912, 23]}, {"full_name": "MeasureTheory.Lp.norm_toLp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [283, 9], "def_end_pos": [283, 18]}, {"full_name": "MeasureTheory.snorm_const", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 20]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\nthis : \u03bc \u2260 0\n\u22a2 \u2016(Lp.const p \u03bc) c\u2016 = \u2016c\u2016 * (\u03bc Set.univ).toReal ^ (1 / p.toReal)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\nthis : \u03bc \u2260 0\n\u22a2 (\u2191\u2016c\u2016\u208a * \u03bc Set.univ ^ (1 / p.toReal)).toReal = \u2016c\u2016 * (\u03bc Set.univ).toReal ^ (1 / p.toReal)"}, {"tactic": "rw [ENNReal.toReal_mul, ENNReal.coe_toReal, \u2190 ENNReal.toReal_rpow, coe_nnnorm]", "annotated_tactic": ["rw [<a>ENNReal.toReal_mul</a>, <a>ENNReal.coe_toReal</a>, \u2190 <a>ENNReal.toReal_rpow</a>, <a>coe_nnnorm</a>]", [{"full_name": "ENNReal.toReal_mul", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [262, 17], "def_end_pos": [262, 27]}, {"full_name": "ENNReal.toReal_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [885, 9], "def_end_pos": [885, 20]}, {"full_name": "coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [764, 41], "def_end_pos": [764, 51]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\nthis : \u03bc \u2260 0\n\u22a2 (\u2191\u2016c\u2016\u208a * \u03bc Set.univ ^ (1 / p.toReal)).toReal = \u2016c\u2016 * (\u03bc Set.univ).toReal ^ (1 / p.toReal)", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "case h\u03bc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\nc : E\ninst\u271d : NeZero \u03bc\nhp_zero : p \u2260 0\nthis : \u03bc \u2260 0\n\u22a2 \u03bc \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "himp_le_himp_left", "start": [353, 1], "end": [354, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/IntermediateValue.lean", "full_name": "Continuous.surjective'", "start": [628, 1], "end": [630, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto", "start": [777, 1], "end": [794, 45], "traced_tactics": [{"tactic": "refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi a f'int tendsto_id) ?_", "annotated_tactic": ["refine <a>tendsto_nhds_unique</a> (<a>intervalIntegral_tendsto_integral_Ioi</a> a f'int <a>tendsto_id</a>) ?_", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 28]}, {"full_name": "MeasureTheory.intervalIntegral_tendsto_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "def_pos": [676, 9], "def_end_pos": [676, 46]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\n\u22a2 \u222b (x : \u211d) in Ioi a, f' x = m - f a", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\n\u22a2 Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x) atTop (\ud835\udcdd (m - f a))"}, {"tactic": "apply Tendsto.congr' _ (hf.sub_const _)", "annotated_tactic": ["apply <a>Tendsto.congr'</a> _ (hf.sub_const _)", [{"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3075, 9], "def_end_pos": [3075, 23]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\n\u22a2 Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x) atTop (\ud835\udcdd (m - f a))", "state_after": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\n\u22a2 (fun x => f x - f a) =\u1da0[atTop] fun i => \u222b (x : \u211d) in a..id i, f' x"}, {"tactic": "filter_upwards [Ioi_mem_atTop a] with x hx", "annotated_tactic": ["filter_upwards [<a>Ioi_mem_atTop</a> a] with x hx", [{"full_name": "Filter.Ioi_mem_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}]], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\n\u22a2 (fun x => f x - f a) =\u1da0[atTop] fun i => \u222b (x : \u211d) in a..id i, f' x", "state_after": "case h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont\u271d : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nhcont : ContinuousOn f (Ici a)\nx : \u211d\nhx : x \u2208 Ioi a\n\u22a2 f x - f a = \u222b (x : \u211d) in a..id x, f' x"}, {"tactic": "have h'x : a \u2264 id x := le_of_lt hx", "annotated_tactic": ["have h'x : a \u2264 <a>id</a> x := <a>le_of_lt</a> hx", [{"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : 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a\n\u22a2 ContinuousWithinAt f (Ici a) x"}, {"tactic": "rcases hx.out.eq_or_lt with rfl|hx", "annotated_tactic": ["rcases hx.out.eq_or_lt with rfl|hx", []], "state_before": "E : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nx : \u211d\nhx : x \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) x", "state_after": "case inl\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f 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Tendsto f atTop (\ud835\udcdd m)\nhx : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a", "state_after": "no goals"}, {"tactic": "exact (hderiv x hx).continuousAt.continuousWithinAt", "annotated_tactic": ["exact (hderiv x hx).continuousAt.continuousWithinAt", []], "state_before": "case inr\nE : Type u_1\nf f' : \u211d \u2192 E\ng g' : \u211d \u2192 \u211d\na b l : \u211d\nm : E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (\ud835\udcdd m)\nx : \u211d\nhx\u271d : x \u2208 Ici a\nhx : a < x\n\u22a2 ContinuousWithinAt f (Ici a) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": 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<a>measurableSet_Ioc</a>", [{"full_name": "MeasurableSet.univ_pi", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [988, 19], "def_end_pos": [988, 40]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}]], "state_before": "\u03b9 : Type u_1\nI : Box \u03b9\ninst\u271d : Countable \u03b9\n\u22a2 MeasurableSet (univ.pi fun i => Ioc (I.lower i) (I.upper i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "full_name": "LinearMap.coe_copy", "start": [257, 1], "end": [258, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Deriv.lean", "full_name": "ConvexOn.deriv_le_slope", "start": [453, 1], "end": [457, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "Matrix.toBilin_symm", "start": [267, 1], "end": [268, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Bits.lean", "full_name": "Nat.shiftLeft'_add", "start": [265, 1], "end": [267, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "full_name": "IsPurelyInseparable.surjective_algebraMap_of_isSeparable", "start": [191, 1], "end": [193, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "full_name": "LinearMap.coe_addHom_mk", "start": [275, 1], "end": [277, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean", "full_name": "Sum.lex_acc_inr", "start": [227, 1], "end": [235, 25], "traced_tactics": [{"tactic": "induction acb with\n| intro _ _ IH =>\n  constructor\n  intro y h\n  cases h with\n  | inr h' => exact IH _ h'\n  | sep => exact aca _", "annotated_tactic": ["induction acb with\n  | <a>intro</a> _ _ IH =>\n    constructor\n    intro y h\n    cases h with\n    | <a>inr</a> h' => exact IH _ h'\n    | <a>sep</a> => exact aca _", [{"full_name": "Acc.intro", "def_path": ".lake/packages/lean4/src/lean/Init/WF.lean", "def_pos": [24, 5], "def_end_pos": [24, 10]}, {"full_name": "Sum.Lex.inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [144, 15], "def_end_pos": [144, 18]}, {"full_name": "Sum.Lex.sep", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [146, 5], "def_end_pos": [146, 8]}]], "state_before": "\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb : \u03b2\u271d\nacb : Acc s b\n\u22a2 Acc (Lex r s) (inr b)", "state_after": "no goals"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\n\u22a2 Acc (Lex r s) (inr x\u271d)", "state_after": "case intro.h\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\n\u22a2 \u2200 (y : \u03b1\u271d \u2295 \u03b2\u271d), Lex r s y (inr x\u271d) \u2192 Acc (Lex r s) y"}, {"tactic": "intro y h", "annotated_tactic": ["intro y h", []], "state_before": "case intro.h\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\n\u22a2 \u2200 (y : \u03b1\u271d \u2295 \u03b2\u271d), Lex r s y (inr x\u271d) \u2192 Acc (Lex r s) y", "state_after": "case intro.h\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\ny : \u03b1\u271d \u2295 \u03b2\u271d\nh : Lex r s y (inr x\u271d)\n\u22a2 Acc (Lex r s) y"}, {"tactic": "cases h with\n| inr h' => exact IH _ h'\n| sep => exact aca _", "annotated_tactic": ["cases h with\n    | <a>inr</a> h' => exact IH _ h'\n    | <a>sep</a> => exact aca _", [{"full_name": "Sum.Lex.inr", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [144, 15], "def_end_pos": [144, 18]}, {"full_name": "Sum.Lex.sep", "def_path": ".lake/packages/batteries/Batteries/Data/Sum/Basic.lean", "def_pos": [146, 5], "def_end_pos": [146, 8]}]], "state_before": "case intro.h\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\ny : \u03b1\u271d \u2295 \u03b2\u271d\nh : Lex r s y (inr x\u271d)\n\u22a2 Acc (Lex r s) y", "state_after": "no goals"}, {"tactic": "exact IH _ h'", "annotated_tactic": ["exact IH _ h'", []], "state_before": "case intro.h.inr\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\nb\u2081\u271d : \u03b2\u271d\nh' : s b\u2081\u271d x\u271d\n\u22a2 Acc (Lex r s) (inr b\u2081\u271d)", "state_after": "no goals"}, {"tactic": "exact aca _", "annotated_tactic": ["exact aca _", []], "state_before": "case intro.h.sep\n\u03b1\u271d : Type u_1\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b2\u271d : Type u_2\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\naca : \u2200 (a : \u03b1\u271d), Acc (Lex r s) (inl a)\nb x\u271d : \u03b2\u271d\nh\u271d : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc s y\nIH : \u2200 (y : \u03b2\u271d), s y x\u271d \u2192 Acc (Lex r s) (inr y)\na\u271d : \u03b1\u271d\n\u22a2 Acc (Lex r s) (inl a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/Additive.lean", "full_name": "HomologicalComplex.neg_f_apply", "start": [69, 1], "end": [70, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Bits.lean", "full_name": "Nat.bit0_val", "start": [141, 1], "end": [145, 34], "traced_tactics": [{"tactic": "rw [Nat.zero_add]", "annotated_tactic": ["rw [<a>Nat.zero_add</a>]", [{"full_name": "Nat.zero_add", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [137, 27], "def_end_pos": [137, 35]}]], "state_before": "m n\u271d n : \u2115\n\u22a2 n + n = 0 + n + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Properties.lean", "full_name": "LinearMap.BilinForm.nonDegenerateFlip_iff", "start": [454, 1], "end": [455, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "full_name": "ProbabilityTheory.kernel.densityProcess_fst_univ_ae", "start": [709, 1], "end": [735, 15], "traced_tactics": [{"tactic": "rw [ae_iff]", "annotated_tactic": ["rw [<a>ae_iff</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\n\u22a2 \u2200\u1d50 (x : \u03b3) \u2202(fst \u03ba) a, densityProcess \u03ba (fst \u03ba) n a x univ = 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\n\u22a2 ((fst \u03ba) a) {a_1 | \u00acdensityProcess \u03ba (fst \u03ba) n a a_1 univ = 1} = 0"}, {"tactic": "have : {x | \u00ac densityProcess \u03ba (fst \u03ba) n a x univ = 1}\n    \u2286 {x | fst \u03ba a (countablePartitionSet n x) = 0} := by\n  intro x hx\n  simp only [mem_setOf_eq] at hx \u22a2\n  rw [densityProcess_fst_univ] at hx\n  simpa using hx", "annotated_tactic": ["have : {x | \u00ac <a>densityProcess</a> \u03ba (<a>fst</a> \u03ba) n a x <a>univ</a> = 1}\n      \u2286 {x | <a>fst</a> \u03ba a (<a>countablePartitionSet</a> n x) = 0} := by\n    intro x hx\n    simp only [<a>mem_setOf_eq</a>] at hx \u22a2\n    rw [<a>densityProcess_fst_univ</a>] at hx\n    simpa using hx", [{"full_name": "ProbabilityTheory.kernel.densityProcess", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [93, 5], "def_end_pos": [93, 19]}, {"full_name": "ProbabilityTheory.kernel.fst", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [905, 19], "def_end_pos": [905, 22]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "ProbabilityTheory.kernel.fst", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [905, 19], "def_end_pos": [905, 22]}, {"full_name": "MeasurableSpace.countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [481, 5], "def_end_pos": [481, 26]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "ProbabilityTheory.kernel.densityProcess_fst_univ", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [689, 7], "def_end_pos": [689, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\n\u22a2 ((fst \u03ba) a) {a_1 | \u00acdensityProcess \u03ba (fst \u03ba) n a a_1 univ = 1} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 ((fst \u03ba) a) {a_1 | \u00acdensityProcess \u03ba (fst \u03ba) n a a_1 univ = 1} = 0"}, {"tactic": "refine measure_mono_null this ?_", "annotated_tactic": ["refine <a>measure_mono_null</a> this ?_", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 ((fst \u03ba) a) {a_1 | \u00acdensityProcess \u03ba (fst \u03ba) n a a_1 univ = 1} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 ((fst \u03ba) a) {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} = 0"}, {"tactic": "have : {x | fst \u03ba a (countablePartitionSet n x) = 0}\n    \u2286 \u22c3 (u) (_ : u \u2208 countablePartition \u03b3 n) (_ : fst \u03ba a u = 0), u := by\n  intro t ht\n  simp only [mem_setOf_eq, mem_iUnion, exists_prop] at ht \u22a2\n  exact \u27e8countablePartitionSet n t, countablePartitionSet_mem _ _, ht,\n    mem_countablePartitionSet _ _\u27e9", "annotated_tactic": ["have : {x | <a>fst</a> \u03ba a (<a>countablePartitionSet</a> n x) = 0}\n      \u2286 \u22c3 (u) (_ : u \u2208 <a>countablePartition</a> \u03b3 n) (_ : <a>fst</a> \u03ba a u = 0), u := by\n    intro t ht\n    simp only [<a>mem_setOf_eq</a>, <a>mem_iUnion</a>, <a>exists_prop</a>] at ht \u22a2\n    exact \u27e8<a>countablePartitionSet</a> n t, <a>countablePartitionSet_mem</a> _ _, ht,\n      <a>mem_countablePartitionSet</a> _ _\u27e9", [{"full_name": "ProbabilityTheory.kernel.fst", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [905, 19], "def_end_pos": [905, 22]}, {"full_name": "MeasurableSpace.countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [481, 5], "def_end_pos": [481, 26]}, {"full_name": "MeasurableSpace.countablePartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [425, 5], "def_end_pos": [425, 23]}, {"full_name": "ProbabilityTheory.kernel.fst", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [905, 19], "def_end_pos": [905, 22]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "MeasurableSpace.countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [481, 5], "def_end_pos": [481, 26]}, {"full_name": "MeasurableSpace.countablePartitionSet_mem", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [484, 7], "def_end_pos": [484, 32]}, {"full_name": "MeasurableSpace.mem_countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [488, 7], "def_end_pos": [488, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 ((fst \u03ba) a) {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 ((fst \u03ba) a) {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} = 0"}, {"tactic": "refine measure_mono_null this ?_", "annotated_tactic": ["refine <a>measure_mono_null</a> this ?_", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 ((fst \u03ba) a) {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 ((fst \u03ba) a) (\u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u) = 0"}, {"tactic": "rw [measure_biUnion]", "annotated_tactic": ["rw [<a>measure_biUnion</a>]", [{"full_name": "MeasureTheory.measure_biUnion", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [160, 9], "def_end_pos": [160, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 ((fst \u03ba) a) (\u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 \u2211' (p : \u2191(countablePartition \u03b3 n)), ((fst \u03ba) a) (\u22c3 (_ : ((fst \u03ba) a) \u2191p = 0), \u2191p) = 0\n\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 (countablePartition \u03b3 n).Countable\n\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 (countablePartition \u03b3 n).PairwiseDisjoint fun u => \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 \u2200 b \u2208 countablePartition \u03b3 n, MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) b = 0), b)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\n\u22a2 {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : x \u2208 {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1}\n\u22a2 x \u2208 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}"}, {"tactic": "simp only [mem_setOf_eq] at hx \u22a2", "annotated_tactic": ["simp only [<a>mem_setOf_eq</a>] at hx \u22a2", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : x \u2208 {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1}\n\u22a2 x \u2208 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1\n\u22a2 ((fst \u03ba) a) (countablePartitionSet n x) = 0"}, {"tactic": "rw [densityProcess_fst_univ] at hx", "annotated_tactic": ["rw [<a>densityProcess_fst_univ</a>] at hx", [{"full_name": "ProbabilityTheory.kernel.densityProcess_fst_univ", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [689, 7], "def_end_pos": [689, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1\n\u22a2 ((fst \u03ba) a) (countablePartitionSet n x) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : \u00ac(if ((fst \u03ba) a) (countablePartitionSet n x) = 0 then 0 else 1) = 1\n\u22a2 ((fst \u03ba) a) (countablePartitionSet n x) = 0"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nx : \u03b3\nhx : \u00ac(if ((fst \u03ba) a) (countablePartitionSet n x) = 0 then 0 else 1) = 1\n\u22a2 ((fst \u03ba) a) (countablePartitionSet n x) = 0", "state_after": "no goals"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nt : \u03b3\nht : t \u2208 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 t \u2208 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u"}, {"tactic": "simp only [mem_setOf_eq, mem_iUnion, exists_prop] at ht \u22a2", "annotated_tactic": ["simp only [<a>mem_setOf_eq</a>, <a>mem_iUnion</a>, <a>exists_prop</a>] at ht \u22a2", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nt : \u03b3\nht : t \u2208 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\n\u22a2 t \u2208 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nt : \u03b3\nht : ((fst \u03ba) a) (countablePartitionSet n t) = 0\n\u22a2 \u2203 i \u2208 countablePartition \u03b3 n, ((fst \u03ba) a) i = 0 \u2227 t \u2208 i"}, {"tactic": "exact \u27e8countablePartitionSet n t, countablePartitionSet_mem _ _, ht,\n  mem_countablePartitionSet _ _\u27e9", "annotated_tactic": ["exact \u27e8<a>countablePartitionSet</a> n t, <a>countablePartitionSet_mem</a> _ _, ht,\n      <a>mem_countablePartitionSet</a> _ _\u27e9", [{"full_name": "MeasurableSpace.countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [481, 5], "def_end_pos": [481, 26]}, {"full_name": "MeasurableSpace.countablePartitionSet_mem", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [484, 7], "def_end_pos": [484, 32]}, {"full_name": "MeasurableSpace.mem_countablePartitionSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [488, 7], "def_end_pos": [488, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nt : \u03b3\nht : ((fst \u03ba) a) (countablePartitionSet n t) = 0\n\u22a2 \u2203 i \u2208 countablePartition \u03b3 n, ((fst \u03ba) a) i = 0 \u2227 t \u2208 i", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 \u2211' (p : \u2191(countablePartition \u03b3 n)), ((fst \u03ba) a) (\u22c3 (_ : ((fst \u03ba) a) \u2191p = 0), \u2191p) = 0", "state_after": "no goals"}, {"tactic": "exact (finite_countablePartition _ _).countable", "annotated_tactic": ["exact (<a>finite_countablePartition</a> _ _).<a>countable</a>", [{"full_name": "MeasurableSpace.finite_countablePartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [434, 7], "def_end_pos": [434, 32]}, {"full_name": "Set.Finite.countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "case hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 (countablePartition \u03b3 n).Countable", "state_after": "no goals"}, {"tactic": "intro s hs t ht hst", "annotated_tactic": ["intro s hs t ht hst", []], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 (countablePartition \u03b3 n).PairwiseDisjoint fun u => \u22c3 (_ : ((fst \u03ba) a) u = 0), u", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nt : Set \u03b3\nht : t \u2208 countablePartition \u03b3 n\nhst : s \u2260 t\n\u22a2 (Disjoint on fun u => \u22c3 (_ : ((fst \u03ba) a) u = 0), u) s t"}, {"tactic": "simp only [disjoint_iUnion_right, disjoint_iUnion_left]", "annotated_tactic": ["simp only [<a>disjoint_iUnion_right</a>, <a>disjoint_iUnion_left</a>]", [{"full_name": "Set.disjoint_iUnion_right", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2064, 9], "def_end_pos": [2064, 30]}, {"full_name": "Set.disjoint_iUnion_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2058, 9], "def_end_pos": [2058, 29]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nt : Set \u03b3\nht : t \u2208 countablePartition \u03b3 n\nhst : s \u2260 t\n\u22a2 (Disjoint on fun u => \u22c3 (_ : ((fst \u03ba) a) u = 0), u) s t", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nt : Set \u03b3\nht : t \u2208 countablePartition \u03b3 n\nhst : s \u2260 t\n\u22a2 ((fst \u03ba) a) t = 0 \u2192 ((fst \u03ba) a) s = 0 \u2192 Disjoint s t"}, {"tactic": "exact fun _ _ \u21a6 disjoint_countablePartition hs ht hst", "annotated_tactic": ["exact fun _ _ \u21a6 <a>disjoint_countablePartition</a> hs ht hst", [{"full_name": "MeasurableSpace.disjoint_countablePartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [441, 7], "def_end_pos": [441, 34]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nt : Set \u03b3\nht : t \u2208 countablePartition \u03b3 n\nhst : s \u2260 t\n\u22a2 ((fst \u03ba) a) t = 0 \u2192 ((fst \u03ba) a) s = 0 \u2192 Disjoint s t", "state_after": "no goals"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\n\u22a2 \u2200 b \u2208 countablePartition \u03b3 n, MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) b = 0), b)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)"}, {"tactic": "by_cases h : fst \u03ba a s = 0", "annotated_tactic": ["by_cases h : <a>fst</a> \u03ba a s = 0", [{"full_name": "ProbabilityTheory.kernel.fst", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [905, 19], "def_end_pos": [905, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nh : ((fst \u03ba) a) s = 0\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nh : \u00ac((fst \u03ba) a) s = 0\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)"}, {"tactic": "simp [h, measurableSet_countablePartition n hs]", "annotated_tactic": ["simp [h, <a>measurableSet_countablePartition</a> n hs]", [{"full_name": "MeasurableSpace.measurableSet_countablePartition", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "def_pos": [476, 7], "def_end_pos": [476, 39]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nh : ((fst \u03ba) a) s = 0\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba\u271d : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\ninst\u271d : IsFiniteKernel \u03ba\nn : \u2115\na : \u03b1\nthis\u271d : {x | \u00acdensityProcess \u03ba (fst \u03ba) n a x univ = 1} \u2286 {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0}\nthis : {x | ((fst \u03ba) a) (countablePartitionSet n x) = 0} \u2286 \u22c3 u \u2208 countablePartition \u03b3 n, \u22c3 (_ : ((fst \u03ba) a) u = 0), u\ns : Set \u03b3\nhs : s \u2208 countablePartition \u03b3 n\nh : \u00ac((fst \u03ba) a) s = 0\n\u22a2 MeasurableSet (\u22c3 (_ : ((fst \u03ba) a) s = 0), s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/TensorProduct.lean", "full_name": "TensorProduct.LieModule.liftLie_apply", "start": [125, 1], "end": [127, 80], "traced_tactics": [{"tactic": "simp only [coe_liftLie_eq_lift_coe, LieModuleHom.coe_toLinearMap, lift_apply]", "annotated_tactic": ["simp only [<a>coe_liftLie_eq_lift_coe</a>, <a>LieModuleHom.coe_toLinearMap</a>, <a>lift_apply</a>]", [{"full_name": "TensorProduct.LieModule.coe_liftLie_eq_lift_coe", "def_path": "Mathlib/Algebra/Lie/TensorProduct.lean", "def_pos": [115, 9], "def_end_pos": [115, 32]}, {"full_name": "LieModuleHom.coe_toLinearMap", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [731, 9], "def_end_pos": [731, 24]}, {"full_name": "TensorProduct.LieModule.lift_apply", "def_path": "Mathlib/Algebra/Lie/TensorProduct.lean", "def_pos": [101, 9], "def_end_pos": [101, 19]}]], "state_before": "R : Type u\ninst\u271d\u00b9\u2078 : CommRing R\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\nQ : Type w\u2083\ninst\u271d\u00b9\u2077 : LieRing L\ninst\u271d\u00b9\u2076 : LieAlgebra R L\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : LieRingModule L N\ninst\u271d\u2078 : LieModule R L N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L P\ninst\u271d\u2074 : LieModule R L P\ninst\u271d\u00b3 : AddCommGroup Q\ninst\u271d\u00b2 : Module R Q\ninst\u271d\u00b9 : LieRingModule L Q\ninst\u271d : LieModule R L Q\nf : M \u2192\u2097\u2045R,L\u2046 N \u2192\u2097[R] P\nm : M\nn : N\n\u22a2 ((liftLie R L M N P) f) (m \u2297\u209c[R] n) = (f m) n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Rat.lean", "full_name": "Rat.divInt_nonneg", "start": [41, 1], "end": [45, 42], "traced_tactics": [{"tactic": "obtain rfl | hb := hb.eq_or_lt", "annotated_tactic": ["obtain rfl | hb := hb.eq_or_lt", []], "state_before": "a\u271d b\u271d c p q : \u211a\na b : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a /. b", "state_after": "case inl\na\u271d b c p q : \u211a\na : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 0\n\u22a2 0 \u2264 a /. 0\n\ncase inr\na\u271d b\u271d c p q : \u211a\na b : \u2124\nha : 0 \u2264 a\nhb\u271d : 0 \u2264 b\nhb : 0 < b\n\u22a2 0 \u2264 a /. b"}, {"tactic": "rwa [divInt_nonneg_iff_of_pos_right hb]", "annotated_tactic": ["rwa [<a>divInt_nonneg_iff_of_pos_right</a> hb]", [{"full_name": "Rat.divInt_nonneg_iff_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Rat.lean", "def_pos": [34, 15], "def_end_pos": [34, 45]}]], "state_before": "case inr\na\u271d b\u271d c p q : \u211a\na b : \u2124\nha : 0 \u2264 a\nhb\u271d : 0 \u2264 b\nhb : 0 < b\n\u22a2 0 \u2264 a /. b", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\na\u271d b c p q : \u211a\na : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 0\n\u22a2 0 \u2264 a /. 0", "state_after": "case inl\na\u271d b c p q : \u211a\na : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 0\n\u22a2 0 \u2264 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case inl\na\u271d b c p q : \u211a\na : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 0\n\u22a2 0 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Metrizable/Uniformity.lean", "full_name": "TotallyBounded.isSeparable", "start": [282, 1], "end": [295, 22], "traced_tactics": [{"tactic": "letI := (UniformSpace.pseudoMetricSpace (X := X)).toPseudoEMetricSpace", "annotated_tactic": ["letI := (<a>UniformSpace.pseudoMetricSpace</a> (X := X)).<a>toPseudoEMetricSpace</a>", [{"full_name": "UniformSpace.pseudoMetricSpace", "def_path": "Mathlib/Topology/Metrizable/Uniformity.lean", "def_pos": [255, 29], "def_end_pos": [255, 59]}, {"full_name": "PseudoMetricSpace.toPseudoEMetricSpace", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1187, 28], "def_end_pos": [1187, 66]}]], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nh : TotallyBounded s\n\u22a2 TopologicalSpace.IsSeparable s", "state_after": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nh : TotallyBounded s\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\n\u22a2 TopologicalSpace.IsSeparable s"}, {"tactic": "rw [EMetric.totallyBounded_iff] at h", "annotated_tactic": ["rw [<a>EMetric.totallyBounded_iff</a>] at h", [{"full_name": "EMetric.totallyBounded_iff", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [775, 9], "def_end_pos": [775, 27]}]], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nh : TotallyBounded s\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\n\u22a2 TopologicalSpace.IsSeparable s", "state_after": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 TopologicalSpace.IsSeparable s"}, {"tactic": "have h' : \u2200 \u03b5 > 0, \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5 := by\n  intro \u03b5 h\u03b5\n  obtain \u27e8t, ht\u27e9 := h \u03b5 h\u03b5\n  refine \u27e8t, ht.1.countable, subset_trans ht.2 ?_\u27e9\n  gcongr\n  exact EMetric.ball_subset_closedBall", "annotated_tactic": ["have h' : \u2200 \u03b5 > 0, \u2203 t, <a>Set.Countable</a> t \u2227 s \u2286 \u22c3 y \u2208 t, <a>EMetric.closedBall</a> y \u03b5 := by\n    intro \u03b5 h\u03b5\n    obtain \u27e8t, ht\u27e9 := h \u03b5 h\u03b5\n    refine \u27e8t, ht.1.<a>countable</a>, <a>subset_trans</a> ht.2 ?_\u27e9\n    gcongr\n    exact <a>EMetric.ball_subset_closedBall</a>", [{"full_name": "Set.Countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [45, 15], "def_end_pos": [45, 24]}, {"full_name": "EMetric.closedBall", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [552, 5], "def_end_pos": [552, 15]}, {"full_name": "Set.Finite.countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [634, 7], "def_end_pos": [634, 19]}, {"full_name": "EMetric.ball_subset_closedBall", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 31]}]], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 TopologicalSpace.IsSeparable s", "state_after": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\nh' : \u2200 \u03b5 > 0, \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5\n\u22a2 TopologicalSpace.IsSeparable s"}, {"tactic": "obtain \u27e8t, _, htc, hts\u27e9 := EMetric.subset_countable_closure_of_almost_dense_set s h'", "annotated_tactic": ["obtain \u27e8t, _, htc, hts\u27e9 := <a>EMetric.subset_countable_closure_of_almost_dense_set</a> s h'", [{"full_name": "EMetric.subset_countable_closure_of_almost_dense_set", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [796, 9], "def_end_pos": [796, 53]}]], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\nh' : \u2200 \u03b5 > 0, \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5\n\u22a2 TopologicalSpace.IsSeparable s", "state_after": "case intro.intro.intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\nh' : \u2200 \u03b5 > 0, \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5\nt : Set X\nleft\u271d : t \u2286 s\nhtc : t.Countable\nhts : s \u2286 _root_.closure t\n\u22a2 TopologicalSpace.IsSeparable s"}, {"tactic": "exact \u27e8t, htc, hts\u27e9", "annotated_tactic": ["exact \u27e8t, htc, hts\u27e9", []], "state_before": "case intro.intro.intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\nh' : \u2200 \u03b5 > 0, \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5\nt : Set X\nleft\u271d : t \u2286 s\nhtc : t.Countable\nhts : s \u2286 _root_.closure t\n\u22a2 TopologicalSpace.IsSeparable s", "state_after": "no goals"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 \u2200 \u03b5 > 0, \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5", "state_after": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5"}, {"tactic": "obtain \u27e8t, ht\u27e9 := h \u03b5 h\u03b5", "annotated_tactic": ["obtain \u27e8t, ht\u27e9 := h \u03b5 h\u03b5", []], "state_before": "X : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5", "state_after": "case intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5"}, {"tactic": "refine \u27e8t, ht.1.countable, subset_trans ht.2 ?_\u27e9", "annotated_tactic": ["refine \u27e8t, ht.1.<a>countable</a>, <a>subset_trans</a> ht.2 ?_\u27e9", [{"full_name": "Set.Finite.countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "subset_trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [634, 7], "def_end_pos": [634, 19]}]], "state_before": "case intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 \u2203 t, t.Countable \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5", "state_after": "case intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 \u22c3 y \u2208 t, EMetric.ball y \u03b5 \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "case intro\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u22a2 \u22c3 y \u2208 t, EMetric.ball y \u03b5 \u2286 \u22c3 y \u2208 t, EMetric.closedBall y \u03b5", "state_after": "case intro.h.h\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\ni\u271d\u00b9 : X\ni\u271d : i\u271d\u00b9 \u2208 t\n\u22a2 EMetric.ball i\u271d\u00b9 \u03b5 \u2286 EMetric.closedBall i\u271d\u00b9 \u03b5"}, {"tactic": "exact EMetric.ball_subset_closedBall", "annotated_tactic": ["exact <a>EMetric.ball_subset_closedBall</a>", [{"full_name": "EMetric.ball_subset_closedBall", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [567, 9], "def_end_pos": [567, 31]}]], "state_before": "case intro.h.h\nX : Type u_1\ninst\u271d : UniformSpace X\ni : (\ud835\udce4 X).IsCountablyGenerated\ns : Set X\nthis : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace\nh : \u2200 \u03b5 > 0, \u2203 t, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\n\u03b5 : ENNReal\nh\u03b5 : \u03b5 > 0\nt : Set X\nht : t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, EMetric.ball y \u03b5\ni\u271d\u00b9 : X\ni\u271d : i\u271d\u00b9 \u2208 t\n\u22a2 EMetric.ball i\u271d\u00b9 \u03b5 \u2286 EMetric.closedBall i\u271d\u00b9 \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "ne_of_ssubset", "start": [689, 1], "end": [689, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean", "full_name": "CategoryTheory.ShortComplex.leftHomologyMap'_nullHomotopic", "start": [596, 1], "end": [602, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Different.lean", "full_name": "FractionalIdeal.dual_inv", "start": [328, 1], "end": [329, 70], "traced_tactics": [{"tactic": "rw [dual_eq_mul_inv, inv_inv]", "annotated_tactic": ["rw [<a>dual_eq_mul_inv</a>, <a>inv_inv</a>]", [{"full_name": "FractionalIdeal.dual_eq_mul_inv", "def_path": "Mathlib/RingTheory/DedekindDomain/Different.lean", "def_pos": [303, 7], "def_end_pos": [303, 22]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [870, 9], "def_end_pos": [870, 16]}]], "state_before": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst\u271d\u00b9\u2079 : CommRing A\ninst\u271d\u00b9\u2078 : Field K\ninst\u271d\u00b9\u2077 : CommRing B\ninst\u271d\u00b9\u2076 : Field L\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : Algebra B L\ninst\u271d\u00b9\u00b3 : Algebra A B\ninst\u271d\u00b9\u00b2 : Algebra K L\ninst\u271d\u00b9\u00b9 : Algebra A L\ninst\u271d\u00b9\u2070 : IsScalarTower A K L\ninst\u271d\u2079 : IsScalarTower A B L\ninst\u271d\u2078 : IsDomain A\ninst\u271d\u2077 : IsDomain B\ninst\u271d\u2076 : IsFractionRing A K\ninst\u271d\u2075 : IsIntegralClosure B A L\ninst\u271d\u2074 : IsFractionRing B L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsDedekindDomain B\nI J : FractionalIdeal B\u2070 L\nhI : I \u2260 0\nhJ : J \u2260 0\n\u22a2 dual A K I\u207b\u00b9 = dual A K 1 * I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BoundedOrder.lean", "full_name": "top_eq_true", "start": [905, 1], "end": [906, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/EllipticDivisibilitySequence.lean", "full_name": "normEDS_three", "start": [235, 1], "end": [237, 71], "traced_tactics": [{"tactic": "erw [normEDS_ofNat, preNormEDS'_three, if_neg <| by decide, mul_one]", "annotated_tactic": ["erw [<a>normEDS_ofNat</a>, <a>preNormEDS'_three</a>, <a>if_neg</a> <| by decide, <a>mul_one</a>]", [{"full_name": "normEDS_ofNat", "def_path": "Mathlib/NumberTheory/EllipticDivisibilitySequence.lean", "def_pos": [219, 7], "def_end_pos": [219, 20]}, {"full_name": "preNormEDS'_three", "def_path": "Mathlib/NumberTheory/EllipticDivisibilitySequence.lean", "def_pos": [142, 7], "def_end_pos": [142, 24]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nW : \u2124 \u2192 R\nf : R \u2192+* S\nb c d : R\n\u22a2 normEDS b c d 3 = c", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nW : \u2124 \u2192 R\nf : R \u2192+* S\nb c d : R\n\u22a2 \u00acEven 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean", "full_name": "CategoryTheory.monoidalOfHasFiniteProducts.associator_hom_fst", "start": [136, 12], "end": [137, 80], "traced_tactics": [{"tactic": "simp [associator_hom]", "annotated_tactic": ["simp [<a>associator_hom</a>]", [{"full_name": "CategoryTheory.monoidalOfHasFiniteProducts.associator_hom", "def_path": "Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean", "def_pos": [124, 9], "def_end_pos": [124, 23]}]], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasBinaryProducts C\nX Y Z : C\n\u22a2 (\u03b1_ X Y Z).hom \u226b prod.fst = prod.fst \u226b prod.fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Tower.lean", "full_name": "TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul", "start": [451, 1], "end": [453, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.PushoutCocone.flip_inl", "start": [965, 1], "end": [965, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "MonoidHom.eqOn_closure", "start": [2852, 1], "end": [2853, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.cantor", "start": [736, 1], "end": [741, 30], "traced_tactics": [{"tactic": "induction' a using Cardinal.inductionOn with \u03b1", "annotated_tactic": ["induction' a using <a>Cardinal.inductionOn</a> with \u03b1", [{"full_name": "Cardinal.inductionOn", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 20]}]], "state_before": "\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 a < 2 ^ a", "state_after": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < 2 ^ #\u03b1"}, {"tactic": "rw [\u2190 mk_set]", "annotated_tactic": ["rw [\u2190 <a>mk_set</a>]", [{"full_name": "Cardinal.mk_set", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [642, 9], "def_end_pos": [642, 15]}]], "state_before": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < 2 ^ #\u03b1", "state_after": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < #(Set \u03b1)"}, {"tactic": "refine \u27e8\u27e8\u27e8singleton, fun a b => singleton_eq_singleton_iff.1\u27e9\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8\u27e8<a>singleton</a>, fun a b => <a>singleton_eq_singleton_iff</a>.1\u27e9\u27e9, ?_\u27e9", [{"full_name": "Singleton.singleton", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [467, 3], "def_end_pos": [467, 12]}, {"full_name": "Set.singleton_eq_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 35]}]], "state_before": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < #(Set \u03b1)", "state_after": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 \u00acQuotient.liftOn\u2082 (#(Set \u03b1)) (#\u03b1) (fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) instLE.proof_1"}, {"tactic": "rintro \u27e8\u27e8f, hf\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8f, hf\u27e9\u27e9", []], "state_before": "case h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 \u00acQuotient.liftOn\u2082 (#(Set \u03b1)) (#\u03b1) (fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) instLE.proof_1", "state_after": "case h.intro.mk\n\u03b1\u271d \u03b2 \u03b1 : Type u\nf : Set \u03b1 \u2192 \u03b1\nhf : Injective f\n\u22a2 False"}, {"tactic": "exact cantor_injective f hf", "annotated_tactic": ["exact <a>cantor_injective</a> f hf", [{"full_name": "Function.cantor_injective", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 25]}]], "state_before": "case h.intro.mk\n\u03b1\u271d \u03b2 \u03b1 : Type u\nf : Set \u03b1 \u2192 \u03b1\nhf : Injective f\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/MinimalPrime.lean", "full_name": "Ideal.exists_comap_eq_of_mem_minimalPrimes", "start": [134, 1], "end": [164, 64], "traced_tactics": [{"tactic": "have := H.1.1", "annotated_tactic": ["have := H.1.1", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p"}, {"tactic": "let f' := (Ideal.Quotient.mk I).comp f", "annotated_tactic": ["let f' := (<a>Ideal.Quotient.mk</a> I).<a>comp</a> f", [{"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.comp", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [655, 5], "def_end_pos": [655, 9]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p"}, {"tactic": "have e : RingHom.ker f' = I.comap f := by\n  ext1\n  exact Submodule.Quotient.mk_eq_zero _", "annotated_tactic": ["have e : <a>RingHom.ker</a> f' = I.comap f := by\n    ext1\n    exact <a>Submodule.Quotient.mk_eq_zero</a> _", [{"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}, {"full_name": "Submodule.Quotient.mk_eq_zero", "def_path": "Mathlib/LinearAlgebra/Quotient.lean", "def_pos": [100, 9], "def_end_pos": [100, 19]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p"}, {"tactic": "have : RingHom.ker (Ideal.Quotient.mk <| RingHom.ker f') \u2264 p := by\n  rw [Ideal.mk_ker, e]\n  exact H.1.2", "annotated_tactic": ["have : <a>RingHom.ker</a> (<a>Ideal.Quotient.mk</a> <| <a>RingHom.ker</a> f') \u2264 p := by\n    rw [<a>Ideal.mk_ker</a>, e]\n    exact H.1.2", [{"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}, {"full_name": "Ideal.mk_ker", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [131, 9], "def_end_pos": [131, 15]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p"}, {"tactic": "refine \u27e8\u27e8?_, bot_le\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8?_, <a>bot_le</a>\u27e9, ?_\u27e9", [{"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')", "state_after": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 (map (Quotient.mk (RingHom.ker f')) p).IsPrime\n\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 \u2200 \u2983b : Ideal (R \u29f8 RingHom.ker f')\u2984,\n    b \u2208 {p | p.IsPrime \u2227 \u22a5 \u2264 p} \u2192\n      (fun x x_1 => x \u2264 x_1) b (map (Quotient.mk (RingHom.ker f')) p) \u2192\n        (fun x x_1 => x \u2264 x_1) (map (Quotient.mk (RingHom.ker f')) p) b"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\n\u22a2 RingHom.ker f' = comap f I", "state_after": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\nx\u271d : R\n\u22a2 x\u271d \u2208 RingHom.ker f' \u2194 x\u271d \u2208 comap f I"}, {"tactic": "exact Submodule.Quotient.mk_eq_zero _", "annotated_tactic": ["exact <a>Submodule.Quotient.mk_eq_zero</a> _", [{"full_name": "Submodule.Quotient.mk_eq_zero", "def_path": "Mathlib/LinearAlgebra/Quotient.lean", "def_pos": [100, 9], "def_end_pos": [100, 19]}]], "state_before": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\nx\u271d : R\n\u22a2 x\u271d \u2208 RingHom.ker f' \u2194 x\u271d \u2208 comap f I", "state_after": "no goals"}, {"tactic": "rw [Ideal.mk_ker, e]", "annotated_tactic": ["rw [<a>Ideal.mk_ker</a>, e]", [{"full_name": "Ideal.mk_ker", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [131, 9], "def_end_pos": [131, 15]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\n\u22a2 RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\n\u22a2 comap f I \u2264 p"}, {"tactic": "exact H.1.2", "annotated_tactic": ["exact H.1.2", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\n\u22a2 comap f I \u2264 p", "state_after": "no goals"}, {"tactic": "have \u27e8p', hp\u2081, hp\u2082\u27e9 := Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective\n  (RingHom.kerLift_injective f') (p.map <| Ideal.Quotient.mk <| RingHom.ker f') this", "annotated_tactic": ["have \u27e8p', hp\u2081, hp\u2082\u27e9 := <a>Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective</a>\n      (<a>RingHom.kerLift_injective</a> f') (p.map <| <a>Ideal.Quotient.mk</a> <| <a>RingHom.ker</a> f') this", [{"full_name": "Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective", "def_path": "Mathlib/RingTheory/Ideal/MinimalPrime.lean", "def_pos": [104, 9], "def_end_pos": [104, 64]}, {"full_name": "RingHom.kerLift_injective", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [60, 9], "def_end_pos": [60, 26]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : ?m.29985\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p"}, {"tactic": "refine \u27e8p'.comap <| Ideal.Quotient.mk I, Ideal.IsPrime.comap _, ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8p'.comap <| <a>Ideal.Quotient.mk</a> I, <a>Ideal.IsPrime.comap</a> _, ?_, ?_\u27e9", [{"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "Ideal.IsPrime.comap", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [114, 10], "def_end_pos": [114, 23]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 \u2203 p', p'.IsPrime \u2227 I \u2264 p' \u2227 comap f p' = p", "state_after": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 I \u2264 comap (Quotient.mk I) p'\n\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap f (comap (Quotient.mk I) p') = p"}, {"tactic": "exact Ideal.mk_ker.symm.trans_le (Ideal.comap_mono bot_le)", "annotated_tactic": ["exact Ideal.mk_ker.symm.trans_le (<a>Ideal.comap_mono</a> <a>bot_le</a>)", [{"full_name": "Ideal.comap_mono", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [78, 9], "def_end_pos": [78, 19]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 I \u2264 comap (Quotient.mk I) p'", "state_after": "no goals"}, {"tactic": "convert congr_arg (Ideal.comap <| Ideal.Quotient.mk <| RingHom.ker f') hp\u2082", "annotated_tactic": ["convert <a>congr_arg</a> (<a>Ideal.comap</a> <| <a>Ideal.Quotient.mk</a> <| <a>RingHom.ker</a> f') hp\u2082", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Ideal.comap", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}]], "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap f (comap (Quotient.mk I) p') = p", "state_after": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p = comap (Quotient.mk (RingHom.ker f')) (map (Quotient.mk (RingHom.ker f')) p)"}, {"tactic": "rwa [Ideal.comap_map_of_surjective (Ideal.Quotient.mk <| RingHom.ker f')\n  Ideal.Quotient.mk_surjective, eq_comm, sup_eq_left]", "annotated_tactic": ["rwa [<a>Ideal.comap_map_of_surjective</a> (<a>Ideal.Quotient.mk</a> <| <a>RingHom.ker</a> f')\n        <a>Ideal.Quotient.mk_surjective</a>, <a>eq_comm</a>, <a>sup_eq_left</a>]", [{"full_name": "Ideal.comap_map_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [416, 9], "def_end_pos": [416, 32]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}, {"full_name": "Ideal.Quotient.mk_surjective", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "sup_eq_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [154, 9], "def_end_pos": [154, 20]}]], "state_before": "case h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d\u00b9 : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : p'.IsPrime\nhp\u2082 : comap f'.kerLift p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p = comap (Quotient.mk (RingHom.ker f')) (map (Quotient.mk (RingHom.ker f')) p)", "state_after": "no goals"}, {"tactic": "apply Ideal.map_isPrime_of_surjective _ this", "annotated_tactic": ["apply <a>Ideal.map_isPrime_of_surjective</a> _ this", [{"full_name": "Ideal.map_isPrime_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [757, 9], "def_end_pos": [757, 34]}]], "state_before": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 (map (Quotient.mk (RingHom.ker f')) p).IsPrime", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 Function.Surjective \u21d1(Quotient.mk (RingHom.ker f'))"}, {"tactic": "exact Ideal.Quotient.mk_surjective", "annotated_tactic": ["exact <a>Ideal.Quotient.mk_surjective</a>", [{"full_name": "Ideal.Quotient.mk_surjective", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 Function.Surjective \u21d1(Quotient.mk (RingHom.ker f'))", "state_after": "no goals"}, {"tactic": "rintro q \u27e8hq, -\u27e9 hq'", "annotated_tactic": ["rintro q \u27e8hq, -\u27e9 hq'", []], "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 \u2200 \u2983b : Ideal (R \u29f8 RingHom.ker f')\u2984,\n    b \u2208 {p | p.IsPrime \u2227 \u22a5 \u2264 p} \u2192\n      (fun x x_1 => x \u2264 x_1) b (map (Quotient.mk (RingHom.ker f')) p) \u2192\n        (fun x x_1 => x \u2264 x_1) (map (Quotient.mk (RingHom.ker f')) p) b", "state_after": "case refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264 q"}, {"tactic": "rw [\u2190 Ideal.map_comap_of_surjective\n    (Ideal.Quotient.mk (RingHom.ker ((Ideal.Quotient.mk I).comp f)))\n    Ideal.Quotient.mk_surjective q]", "annotated_tactic": ["rw [\u2190 <a>Ideal.map_comap_of_surjective</a>\n        (<a>Ideal.Quotient.mk</a> (<a>RingHom.ker</a> ((<a>Ideal.Quotient.mk</a> I).<a>comp</a> f)))\n        <a>Ideal.Quotient.mk_surjective</a> q]", [{"full_name": "Ideal.map_comap_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [292, 9], "def_end_pos": [292, 32]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "RingHom.comp", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [655, 5], "def_end_pos": [655, 9]}, {"full_name": "Ideal.Quotient.mk_surjective", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}]], "state_before": "case refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264 q", "state_after": "case refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264\n    map (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f)))\n      (comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q)"}, {"tactic": "apply Ideal.map_mono", "annotated_tactic": ["apply <a>Ideal.map_mono</a>", [{"full_name": "Ideal.map_mono", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [57, 9], "def_end_pos": [57, 17]}]], "state_before": "case refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264\n    map (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f)))\n      (comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q)", "state_after": "case refine_2.intro.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2264 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q"}, {"tactic": "apply H.2", "annotated_tactic": ["apply H.2", []], "state_before": "case refine_2.intro.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2264 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q", "state_after": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q \u2208 {p | p.IsPrime \u2227 comap f I \u2264 p}\n\ncase refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q \u2264 p"}, {"tactic": "refine \u27e8inferInstance, (Ideal.mk_ker.trans e).symm.trans_le (Ideal.comap_mono bot_le)\u27e9", "annotated_tactic": ["refine \u27e8<a>inferInstance</a>, (Ideal.mk_ker.trans e).symm.trans_le (<a>Ideal.comap_mono</a> <a>bot_le</a>)\u27e9", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}, {"full_name": "Ideal.comap_mono", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [78, 9], "def_end_pos": [78, 19]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q \u2208 {p | p.IsPrime \u2227 comap f I \u2264 p}", "state_after": "no goals"}, {"tactic": "refine (Ideal.comap_mono hq').trans ?_", "annotated_tactic": ["refine (<a>Ideal.comap_mono</a> hq').<a>trans</a> ?_", [{"full_name": "Ideal.comap_mono", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [78, 9], "def_end_pos": [78, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) q \u2264 p", "state_after": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) (map (Quotient.mk (RingHom.ker f')) p) \u2264 p"}, {"tactic": "rw [Ideal.comap_map_of_surjective]", "annotated_tactic": ["rw [<a>Ideal.comap_map_of_surjective</a>]", [{"full_name": "Ideal.comap_map_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [416, 9], "def_end_pos": [416, 32]}]], "state_before": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) (map (Quotient.mk (RingHom.ker f')) p) \u2264 p", "state_after": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2294 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) \u22a5 \u2264 p\n\ncase refine_2.intro.h.a.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 Function.Surjective \u21d1(Quotient.mk (RingHom.ker ((Quotient.mk I).comp f)))"}, {"tactic": "exacts [sup_le rfl.le this, Ideal.Quotient.mk_surjective]", "annotated_tactic": ["exacts [<a>sup_le</a> rfl.le this, <a>Ideal.Quotient.mk_surjective</a>]", [{"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [143, 9], "def_end_pos": [143, 15]}, {"full_name": "Ideal.Quotient.mk_surjective", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}]], "state_before": "case refine_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2294 comap (Quotient.mk (RingHom.ker ((Quotient.mk I).comp f))) \u22a5 \u2264 p\n\ncase refine_2.intro.h.a.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 (comap f I).minimalPrimes\nthis\u271d : p.IsPrime\nf' : R \u2192+* S \u29f8 I := (Quotient.mk I).comp f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : q.IsPrime\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 Function.Surjective \u21d1(Quotient.mk (RingHom.ker ((Quotient.mk I).comp f)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.dvd_term_of_isRoot_of_dvd_terms", "start": [502, 1], "end": [504, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.leadingCoeff_prod", "start": [370, 1], "end": [371, 53], "traced_tactics": [{"tactic": "simpa using leadingCoeff_multiset_prod (s.1.map f)", "annotated_tactic": ["simpa using <a>leadingCoeff_multiset_prod</a> (s.1.<a>map</a> f)", [{"full_name": "Polynomial.leadingCoeff_multiset_prod", "def_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "def_pos": [358, 9], "def_end_pos": [358, 35]}, {"full_name": "Multiset.map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1193, 5], "def_end_pos": [1193, 8]}]], "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 (\u220f i \u2208 s, f i).leadingCoeff = \u220f i \u2208 s, (f i).leadingCoeff", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.Lex.inl_mono", "start": [402, 1], "end": [403, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean", "full_name": "CochainComplex.mappingCone.ext_cochain_to_iff", "start": [188, 1], "end": [202, 19], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081 = \u03b3\u2082 \u2194 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef", "state_after": "case mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081 = \u03b3\u2082 \u2192 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\n\ncase mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef \u2192 \u03b3\u2081 = \u03b3\u2082"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081 = \u03b3\u2082 \u2192 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef", "state_after": "case mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2081.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2081.comp (snd \u03c6) \u22ef"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2081.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2081.comp (snd \u03c6) \u22ef", "state_after": "no goals"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "case mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\n\u22a2 \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij \u2227 \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef \u2192 \u03b3\u2081 = \u03b3\u2082", "state_after": "case mpr.intro\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\n\u22a2 \u03b3\u2081 = \u03b3\u2082"}, {"tactic": "ext p q hpq", "annotated_tactic": ["ext p q hpq", []], "state_before": "case mpr.intro\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\n\u22a2 \u03b3\u2081 = \u03b3\u2082", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\n\u22a2 \u03b3\u2081.v p q hpq = \u03b3\u2082.v p q hpq"}, {"tactic": "rw [ext_to_iff \u03c6 q (q + 1) rfl]", "annotated_tactic": ["rw [<a>ext_to_iff</a> \u03c6 q (q + 1) <a>rfl</a>]", [{"full_name": "CochainComplex.mappingCone.ext_to_iff", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean", "def_pos": [153, 7], "def_end_pos": [153, 17]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\n\u22a2 \u03b3\u2081.v p q hpq = \u03b3\u2082.v p q hpq", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef"}, {"tactic": "replace h\u2081 := Cochain.congr_v h\u2081 p (q + 1) (by omega)", "annotated_tactic": ["replace h\u2081 := <a>Cochain.congr_v</a> h\u2081 p (q + 1) (by omega)", [{"full_name": "CochainComplex.HomComplex.Cochain.congr_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [88, 7], "def_end_pos": [88, 14]}]], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\nh\u2081 : (\u03b3\u2081.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef = (\u03b3\u2082.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef"}, {"tactic": "replace h\u2082 := Cochain.congr_v h\u2082 p q hpq", "annotated_tactic": ["replace h\u2082 := <a>Cochain.congr_v</a> h\u2082 p q hpq", [{"full_name": "CochainComplex.HomComplex.Cochain.congr_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [88, 7], "def_end_pos": [88, 14]}]], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\nh\u2081 : (\u03b3\u2081.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef = (\u03b3\u2082.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2081 : (\u03b3\u2081.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef = (\u03b3\u2082.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef\nh\u2082 : (\u03b3\u2081.comp (snd \u03c6) \u22ef).v p q hpq = (\u03b3\u2082.comp (snd \u03c6) \u22ef).v p q hpq\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef"}, {"tactic": "simp only [Cochain.comp_v _ _ _ p q (q + 1) hpq rfl] at h\u2081", "annotated_tactic": ["simp only [<a>Cochain.comp_v</a> _ _ _ p q (q + 1) hpq <a>rfl</a>] at h\u2081", [{"full_name": "CochainComplex.HomComplex.Cochain.comp_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [236, 7], "def_end_pos": [236, 13]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2081 : (\u03b3\u2081.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef = (\u03b3\u2082.comp (\u2191(fst \u03c6)) hij).v p (q + 1) \u22ef\nh\u2082 : (\u03b3\u2081.comp (snd \u03c6) \u22ef).v p q hpq = (\u03b3\u2082.comp (snd \u03c6) \u22ef).v p q hpq\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2082 : (\u03b3\u2081.comp (snd \u03c6) \u22ef).v p q hpq = (\u03b3\u2082.comp (snd \u03c6) \u22ef).v p q hpq\nh\u2081 : \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef"}, {"tactic": "simp only [Cochain.comp_zero_cochain_v] at h\u2082", "annotated_tactic": ["simp only [<a>Cochain.comp_zero_cochain_v</a>] at h\u2082", [{"full_name": "CochainComplex.HomComplex.Cochain.comp_zero_cochain_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [243, 7], "def_end_pos": [243, 26]}]], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2082 : (\u03b3\u2081.comp (snd \u03c6) \u22ef).v p q hpq = (\u03b3\u2082.comp (snd \u03c6) \u22ef).v p q hpq\nh\u2081 : \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef", "state_after": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2081 : \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef\nh\u2082 : \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef"}, {"tactic": "exact \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["exact \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "case mpr.intro.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\np q : \u2124\nhpq : p + i = q\nh\u2081 : \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef\nh\u2082 : \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef\n\u22a2 \u03b3\u2081.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef = \u03b3\u2082.v p q hpq \u226b (\u2191(fst \u03c6)).v q (q + 1) \u22ef \u2227\n    \u03b3\u2081.v p q hpq \u226b (snd \u03c6).v q q \u22ef = \u03b3\u2082.v p q hpq \u226b (snd \u03c6).v q q \u22ef", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\nF G : CochainComplex C \u2124\n\u03c6 : F \u27f6 G\ninst\u271d : HasHomotopyCofiber \u03c6\ni j : \u2124\nhij : i + 1 = j\nK : CochainComplex C \u2124\n\u03b3\u2081 \u03b3\u2082 : Cochain K (mappingCone \u03c6) i\nh\u2081 : \u03b3\u2081.comp (\u2191(fst \u03c6)) hij = \u03b3\u2082.comp (\u2191(fst \u03c6)) hij\nh\u2082 : \u03b3\u2081.comp (snd \u03c6) \u22ef = \u03b3\u2082.comp (snd \u03c6) \u22ef\np q : \u2124\nhpq : p + i = q\n\u22a2 p + j = q + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "full_name": "Batteries.RBSet.toStream_eq", "start": [858, 1], "end": [858, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/HashMap/WF.lean", "full_name": "Batteries.HashMap.Imp.Buckets.WF.update", "start": [48, 1], "end": [65, 26], "traced_tactics": [{"tactic": "refine \u27e8fun l hl => ?_, fun i hi p hp => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun l hl => ?_, fun i hi p hp => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : i.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) buckets.val[i] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) d\n\u22a2 (buckets.update i d h).WF", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : i.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) buckets.val[i] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) d\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nhl : l \u2208 (buckets.update i d h).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l.toList\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nhp : p \u2208 (buckets.update i\u271d d h).val[i].toList\n\u22a2 (fun k x => ((hash k).toUSize % (buckets.update i\u271d d h).val.size).toNat = i) p.fst p.snd"}, {"tactic": "exact match List.mem_or_eq_of_mem_set hl with\n| .inl hl => H.1 _ hl\n| .inr rfl => h\u2081 (H.1 _ (Array.getElem_mem_data ..))", "annotated_tactic": ["exact match <a>List.mem_or_eq_of_mem_set</a> hl with\n    | .inl hl => H.1 _ hl\n    | .inr <a>rfl</a> => h\u2081 (H.1 _ (<a>Array.getElem_mem_data</a> ..))", [{"full_name": "List.mem_or_eq_of_mem_set", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [513, 9], "def_end_pos": [513, 29]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Array.getElem_mem_data", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [354, 9], "def_end_pos": [354, 25]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni : USize\nd : AssocList \u03b1 \u03b2\nh : i.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) buckets.val[i] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i.toNat) d\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nl : AssocList \u03b1 \u03b2\nhl : l \u2208 (buckets.update i d h).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l.toList", "state_after": "no goals"}, {"tactic": "revert hp", "annotated_tactic": ["revert hp", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nhp : p \u2208 (buckets.update i\u271d d h).val[i].toList\n\u22a2 (fun k x => ((hash k).toUSize % (buckets.update i\u271d d h).val.size).toNat = i) p.fst p.snd", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 (buckets.update i\u271d d h).val[i].toList \u2192\n    (fun k x => ((hash k).toUSize % (buckets.update i\u271d d h).val.size).toNat = i) p.fst p.snd"}, {"tactic": "simp only [Array.getElem_eq_data_getElem, update_data, List.getElem_set, Array.data_length,\n  update_size]", "annotated_tactic": ["simp only [<a>Array.getElem_eq_data_getElem</a>, <a>update_data</a>, <a>List.getElem_set</a>, <a>Array.data_length</a>,\n      <a>update_size</a>]", [{"full_name": "Array.getElem_eq_data_getElem", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 32]}, {"full_name": "Batteries.HashMap.Imp.Buckets.update_data", "def_path": ".lake/packages/batteries/Batteries/Data/HashMap/WF.lean", "def_pos": [20, 9], "def_end_pos": [20, 20]}, {"full_name": "List.getElem_set", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [457, 9], "def_end_pos": [457, 20]}, {"full_name": "Array.data_length", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [29, 17], "def_end_pos": [29, 28]}, {"full_name": "Batteries.HashMap.Imp.Buckets.update_size", "def_path": ".lake/packages/batteries/Batteries/Data/HashMap/Basic.lean", "def_pos": [43, 17], "def_end_pos": [43, 28]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 (buckets.update i\u271d d h).val[i].toList \u2192\n    (fun k x => ((hash k).toUSize % (buckets.update i\u271d d h).val.size).toNat = i) p.fst p.snd", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 (if i\u271d.toNat = i then d else buckets.val.data[i]).toList \u2192 ((hash p.fst).toUSize % buckets.val.size).toNat = i"}, {"tactic": "split <;> intro hp", "annotated_tactic": ["split <;> intro hp", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\n\u22a2 p \u2208 (if i\u271d.toNat = i then d else buckets.val.data[i]).toList \u2192 ((hash p.fst).toUSize % buckets.val.size).toNat = i", "state_after": "case refine_2.isTrue\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nh\u271d : i\u271d.toNat = i\nhp : p \u2208 d.toList\n\u22a2 ((hash p.fst).toUSize % buckets.val.size).toNat = i\n\ncase refine_2.isFalse\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00aci\u271d.toNat = i\nhp : p \u2208 buckets.val.data[i].toList\n\u22a2 ((hash p.fst).toUSize % buckets.val.size).toNat = i"}, {"tactic": "next eq => exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", "annotated_tactic": ["next eq => exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", []], "state_before": "case refine_2.isTrue\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nh\u271d : i\u271d.toNat = i\nhp : p \u2208 d.toList\n\u22a2 ((hash p.fst).toUSize % buckets.val.size).toNat = i", "state_after": "no goals"}, {"tactic": "exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", "annotated_tactic": ["exact eq \u25b8 h\u2082 (H.2 _ _) _ hp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\neq : i\u271d.toNat = i\nhp : p \u2208 d.toList\n\u22a2 ((hash p.fst).toUSize % buckets.val.size).toNat = i", "state_after": "no goals"}, {"tactic": "simp only [update_size, Array.data_length] at hi", "annotated_tactic": ["simp only [<a>update_size</a>, <a>Array.data_length</a>] at hi", [{"full_name": "Batteries.HashMap.Imp.Buckets.update_size", "def_path": ".lake/packages/batteries/Batteries/Data/HashMap/Basic.lean", "def_pos": [43, 17], "def_end_pos": [43, 28]}, {"full_name": "Array.data_length", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [29, 17], "def_end_pos": [29, 28]}]], "state_before": "case refine_2.isFalse\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) d\ni : Nat\nhi : i < (buckets.update i\u271d d h).val.size\np : \u03b1 \u00d7 \u03b2\nh\u271d : \u00aci\u271d.toNat = i\nhp : p \u2208 buckets.val.data[i].toList\n\u22a2 ((hash p.fst).toUSize % buckets.val.size).toNat = i", "state_after": "case refine_2.isFalse\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nbuckets : Buckets \u03b1 \u03b2\ni\u271d : USize\nd : AssocList \u03b1 \u03b2\nh : i\u271d.toNat < buckets.val.size\nH : buckets.WF\nh\u2081 :\n  \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],\n    List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) buckets.val[i\u271d].toList \u2192\n      List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) d.toList\nh\u2082 :\n  AssocList.All (fun k x => ((hash k).toUSize % buckets.val.size).toNat = i\u271d.toNat) buckets.val[i\u271d] \u2192\n    AssocList.All (fun k x => 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u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd5c\n\u22a2 HasDerivAt (fun x => x * c) c x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "OrderIso.tendsto_atTop", "start": [483, 1], "end": [484, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "full_name": "Complex.arg_eq_nhds_of_re_neg_of_im_neg", "start": [608, 1], "end": [614, 99], "traced_tactics": [{"tactic": "suffices h_forall_nhds : \u2200\u1da0 y : \u2102 in \ud835\udcdd x, y.re < 0 \u2227 y.im < 0 from\n  h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2", "annotated_tactic": ["suffices h_forall_nhds : \u2200\u1da0 y : \u2102 in \ud835\udcdd x, y.re < 0 \u2227 y.im < 0 from\n    h_forall_nhds.mono fun y hy => <a>arg_of_re_neg_of_im_neg</a> hy.1 hy.2", [{"full_name": "Complex.arg_of_re_neg_of_im_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [313, 9], "def_end_pos": [313, 32]}]], "state_before": "a x z : \u2102\nhx_re : x.re < 0\nhx_im : x.im < 0\n\u22a2 arg =\u1da0[\ud835\udcdd x] fun x => Real.arcsin ((-x).im / abs x) - \u03c0", "state_after": "a x z : \u2102\nhx_re : x.re < 0\nhx_im : x.im < 0\n\u22a2 \u2200\u1da0 (y : \u2102) in \ud835\udcdd x, y.re < 0 \u2227 y.im < 0"}, {"tactic": "refine IsOpen.eventually_mem ?_ (\u27e8hx_re, hx_im\u27e9 : x.re < 0 \u2227 x.im < 0)", "annotated_tactic": ["refine <a>IsOpen.eventually_mem</a> ?_ (\u27e8hx_re, hx_im\u27e9 : x.re < 0 \u2227 x.im < 0)", [{"full_name": "IsOpen.eventually_mem", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [882, 9], "def_end_pos": [882, 30]}]], "state_before": "a x z : \u2102\nhx_re : x.re < 0\nhx_im : x.im < 0\n\u22a2 \u2200\u1da0 (y : \u2102) in \ud835\udcdd x, y.re < 0 \u2227 y.im < 0", "state_after": "a x z : \u2102\nhx_re : x.re < 0\nhx_im : x.im < 0\n\u22a2 IsOpen fun y => y.re < 0 \u2227 y.im < 0"}, {"tactic": "exact\n  IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero)", "annotated_tactic": ["exact\n    <a>IsOpen.and</a> (<a>isOpen_lt</a> <a>continuous_re</a> <a>continuous_zero</a>) (<a>isOpen_lt</a> <a>continuous_im</a> <a>continuous_zero</a>)", [{"full_name": "IsOpen.and", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 19]}, {"full_name": "isOpen_lt", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [735, 9], "def_end_pos": [735, 18]}, {"full_name": "Complex.continuous_re", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 22]}, {"full_name": "continuous_zero", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [34, 3], "def_end_pos": [34, 14]}, {"full_name": "isOpen_lt", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [735, 9], "def_end_pos": [735, 18]}, {"full_name": "Complex.continuous_im", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 22]}, {"full_name": "continuous_zero", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [34, 3], "def_end_pos": [34, 14]}]], "state_before": "a x z : \u2102\nhx_re : x.re < 0\nhx_im : x.im < 0\n\u22a2 IsOpen fun y => y.re < 0 \u2227 y.im < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean", "full_name": "CategoryTheory.InjectiveResolution.homotopyEquiv_hom_\u03b9", "start": [206, 1], "end": [207, 67], "traced_tactics": [{"tactic": "simp [homotopyEquiv]", "annotated_tactic": ["simp [<a>homotopyEquiv</a>]", [{"full_name": "CategoryTheory.InjectiveResolution.homotopyEquiv", "def_path": "Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean", "def_pos": [195, 5], "def_end_pos": [195, 18]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX : C\nI J : InjectiveResolution X\n\u22a2 I.\u03b9 \u226b (I.homotopyEquiv J).hom = J.\u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectSum/Ring.lean", "full_name": "DirectSum.of_zero_one", "start": [411, 1], "end": [412, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "full_name": "Polynomial.mul_coeff_zero", "start": [135, 1], "end": [135, 101], "traced_tactics": [{"tactic": "simp [coeff_mul]", "annotated_tactic": ["simp [<a>coeff_mul</a>]", [{"full_name": "Polynomial.coeff_mul", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\n\u22a2 (p * q).coeff 0 = p.coeff 0 * q.coeff 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "Monotone.cauchySeq_alternating_series_of_tendsto_zero", "start": [711, 1], "end": [714, 88], "traced_tactics": [{"tactic": "simp_rw [mul_comm]", "annotated_tactic": ["simp_rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq fun n => \u2211 i \u2208 Finset.range n, (-1) ^ i * f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq fun n => \u2211 x \u2208 Finset.range n, f x * (-1) ^ x"}, {"tactic": "exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le", "annotated_tactic": ["exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 <a>norm_sum_neg_one_pow_le</a>", [{"full_name": "norm_sum_neg_one_pow_le", "def_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "def_pos": [704, 9], "def_end_pos": [704, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq fun n => \u2211 x \u2208 Finset.range n, f x * (-1) ^ x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "full_name": "WeierstrassCurve.Jacobian.smul_equiv", "start": [155, 1], "end": [156, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "full_name": "MeasureTheory.restrict_trim", "start": [96, 1], "end": [101, 67], "traced_tactics": [{"tactic": "refine @Measure.ext _ m _ _ (fun t ht => ?_)", "annotated_tactic": ["refine @<a>Measure.ext</a> _ m _ _ (fun t ht => ?_)", [{"full_name": "MeasureTheory.Measure.ext", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [136, 9], "def_end_pos": [136, 12]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\n\u22a2 (\u03bc.trim hm).restrict s = (\u03bc.restrict s).trim hm", "state_after": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 ((\u03bc.trim hm).restrict s) t = ((\u03bc.restrict s).trim hm) t"}, {"tactic": "rw [@Measure.restrict_apply \u03b1 m _ _ _ ht, trim_measurableSet_eq hm ht,\n  Measure.restrict_apply (hm t ht),\n  trim_measurableSet_eq hm (@MeasurableSet.inter \u03b1 m t s ht hs)]", "annotated_tactic": ["rw [@<a>Measure.restrict_apply</a> \u03b1 m _ _ _ ht, <a>trim_measurableSet_eq</a> hm ht,\n    <a>Measure.restrict_apply</a> (hm t ht),\n    <a>trim_measurableSet_eq</a> hm (@<a>MeasurableSet.inter</a> \u03b1 m t s ht hs)]", [{"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [196, 19], "def_end_pos": [196, 38]}]], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 ((\u03bc.trim hm).restrict s) t = ((\u03bc.restrict s).trim hm) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "SecondCountableTopology.of_separableSpace_orderTopology", "start": [545, 1], "end": [549, 40], "traced_tactics": [{"tactic": "rcases exists_countable_dense \u03b1 with \u27e8s, hc, hd\u27e9", "annotated_tactic": ["rcases <a>exists_countable_dense</a> \u03b1 with \u27e8s, hc, hd\u27e9", [{"full_name": "TopologicalSpace.exists_countable_dense", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [328, 9], "def_end_pos": [328, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u22a2 SecondCountableTopology \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhc : s.Countable\nhd : Dense s\n\u22a2 SecondCountableTopology \u03b1"}, {"tactic": "refine \u27e8\u27e8_, ?_, hd.topology_eq_generateFrom\u27e9\u27e9", "annotated_tactic": ["refine \u27e8\u27e8_, ?_, hd.topology_eq_generateFrom\u27e9\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhc : s.Countable\nhd : Dense s\n\u22a2 SecondCountableTopology \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhc : s.Countable\nhd : Dense s\n\u22a2 (Ioi '' s \u222a Iio '' s).Countable"}, {"tactic": "exact (hc.image _).union (hc.image _)", "annotated_tactic": ["exact (hc.image _).<a>union</a> (hc.image _)", [{"full_name": "Set.Countable.union", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [256, 9], "def_end_pos": [256, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhc : s.Countable\nhd : Dense s\n\u22a2 (Ioi '' s \u222a Iio '' s).Countable", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "IsLeftRegular.isSMulRegular", "start": [39, 1], "end": [40, 4], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.add_def", "start": [188, 1], "end": [201, 72], "traced_tactics": [{"tactic": "show Rat.add .. = _", "annotated_tactic": ["show <a>Rat.add</a> .. = _", [{"full_name": "Rat.add", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [216, 30], "def_end_pos": [216, 33]}]], "state_before": "a b : Rat\n\u22a2 a + b = normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "a b : Rat\n\u22a2 a.add b = normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "delta Rat.add", "annotated_tactic": ["delta <a>Rat.add</a>", [{"full_name": "Rat.add", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [216, 30], "def_end_pos": [216, 33]}]], "state_before": "a b : Rat\n\u22a2 a.add b = normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "a b : Rat\n\u22a2 (let g := a.den.gcd b.den;\n    if hg : g = 1 then\n      let_fun den_nz := \u22ef;\n      let_fun reduced := \u22ef;\n      { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := den_nz, reduced := reduced }\n    else\n      let den := a.den / g * b.den;\n      let num := a.num * \u2191(b.den / g) + b.num * \u2191(a.den / g);\n      let g1 := num.natAbs.gcd g;\n      let_fun den_nz := \u22ef;\n      let_fun e := \u22ef;\n      maybeNormalize num den g1 \u22ef \u22ef) =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "a b : Rat\n\u22a2 (let g := a.den.gcd b.den;\n    if hg : g = 1 then\n      let_fun den_nz := \u22ef;\n      let_fun reduced := \u22ef;\n      { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := den_nz, reduced := reduced }\n    else\n      let den := a.den / g * b.den;\n      let num := a.num * \u2191(b.den / g) + b.num * \u2191(a.den / g);\n      let g1 := num.natAbs.gcd g;\n      let_fun den_nz := \u22ef;\n      let_fun e := \u22ef;\n      maybeNormalize num den g1 \u22ef \u22ef) =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "a b : Rat\n\u22a2 (if hg : a.den.gcd b.den = 1 then\n      { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := \u22ef, reduced := \u22ef }\n    else\n      maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n        (a.den / a.den.gcd b.den * b.den)\n        ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef) =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "a b : Rat\n\u22a2 (if hg : a.den.gcd b.den = 1 then\n      { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := \u22ef, reduced := \u22ef }\n    else\n      maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n        (a.den / a.den.gcd b.den * b.den)\n        ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef) =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "case isTrue\na b : Rat\nh\u271d : a.den.gcd b.den = 1\n\u22a2 { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := \u22ef, reduced := \u22ef } =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef\n\ncase isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\n\u22a2 maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den)\n      ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "exact (normalize_self _).symm", "annotated_tactic": ["exact (<a>normalize_self</a> _).<a>symm</a>", [{"full_name": "Rat.normalize_self", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 23]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case isTrue\na b : Rat\nh\u271d : a.den.gcd b.den = 1\n\u22a2 { num := a.num * \u2191b.den + b.num * \u2191a.den, den := a.den * b.den, den_nz := \u22ef, reduced := \u22ef } =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "no goals"}, {"tactic": "have : a.den.gcd b.den \u2260 0 := Nat.gcd_ne_zero_left a.den_nz", "annotated_tactic": ["have : a.den.gcd b.den \u2260 0 := <a>Nat.gcd_ne_zero_left</a> a.den_nz", [{"full_name": "Nat.gcd_ne_zero_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [158, 9], "def_end_pos": [158, 25]}]], "state_before": "case isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\n\u22a2 maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den)\n      ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "case isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den)\n      ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "rw [maybeNormalize_eq_normalize _ _\n    (Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left ..)\n    (Nat.dvd_trans (Nat.gcd_dvd_right ..) <|\n     Nat.dvd_trans (Nat.gcd_dvd_right ..) (Nat.dvd_mul_left ..)),\n  \u2190 normalize_mul_right _ this]", "annotated_tactic": ["rw [<a>maybeNormalize_eq_normalize</a> _ _\n        (<a>Int.ofNat_dvd_left</a>.2 <| <a>Nat.gcd_dvd_left</a> ..)\n        (<a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_right</a> ..) <|\n         <a>Nat.dvd_trans</a> (<a>Nat.gcd_dvd_right</a> ..) (<a>Nat.dvd_mul_left</a> ..)),\n      \u2190 <a>normalize_mul_right</a> _ this]", [{"full_name": "Rat.maybeNormalize_eq_normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [76, 9], "def_end_pos": [76, 36]}, {"full_name": "Int.ofNat_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [86, 9], "def_end_pos": [86, 21]}, {"full_name": "Nat.dvd_trans", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [19, 19], "def_end_pos": [19, 28]}, {"full_name": "Nat.gcd_dvd_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}, {"full_name": "Nat.dvd_trans", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [19, 19], "def_end_pos": [19, 28]}, {"full_name": "Nat.gcd_dvd_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}, {"full_name": "Nat.dvd_mul_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [16, 19], "def_end_pos": [16, 31]}, {"full_name": "Rat.normalize_mul_right", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [58, 9], "def_end_pos": [58, 28]}]], "state_before": "case isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 maybeNormalize (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den)\n      ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)).natAbs.gcd (a.den.gcd b.den)) \u22ef \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "case isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 normalize ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)) * \u2191(a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den * a.den.gcd b.den) \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case isFalse\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 normalize ((a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)) * \u2191(a.den.gcd b.den))\n      (a.den / a.den.gcd b.den * b.den * a.den.gcd b.den) \u22ef =\n    normalize (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den) \u22ef", "state_after": "case isFalse.e_num\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)) * \u2191(a.den.gcd b.den) =\n    a.num * \u2191b.den + b.num * \u2191a.den\n\ncase isFalse.e_den\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 a.den / a.den.gcd b.den * b.den * a.den.gcd b.den = a.den * b.den"}, {"tactic": "simp only [Int.add_mul, Int.mul_assoc, Int.ofNat_mul_ofNat,\n  Nat.div_mul_cancel (Nat.gcd_dvd_left ..), Nat.div_mul_cancel (Nat.gcd_dvd_right ..)]", "annotated_tactic": ["simp only [<a>Int.add_mul</a>, <a>Int.mul_assoc</a>, <a>Int.ofNat_mul_ofNat</a>,\n        <a>Nat.div_mul_cancel</a> (<a>Nat.gcd_dvd_left</a> ..), <a>Nat.div_mul_cancel</a> (<a>Nat.gcd_dvd_right</a> ..)]", [{"full_name": "Int.add_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [438, 19], "def_end_pos": [438, 26]}, {"full_name": "Int.mul_assoc", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [376, 19], "def_end_pos": [376, 28]}, {"full_name": "Int.ofNat_mul_ofNat", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [48, 23], "def_end_pos": [48, 38]}, {"full_name": "Nat.div_mul_cancel", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [91, 19], "def_end_pos": [91, 33]}, {"full_name": "Nat.gcd_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [86, 9], "def_end_pos": [86, 21]}, {"full_name": "Nat.div_mul_cancel", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [91, 19], "def_end_pos": [91, 33]}, {"full_name": "Nat.gcd_dvd_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}]], "state_before": "case isFalse.e_num\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 (a.num * \u2191(b.den / a.den.gcd b.den) + b.num * \u2191(a.den / a.den.gcd b.den)) * \u2191(a.den.gcd b.den) =\n    a.num * \u2191b.den + b.num * \u2191a.den", "state_after": "no goals"}, {"tactic": "rw [Nat.mul_right_comm, Nat.div_mul_cancel (Nat.gcd_dvd_left ..)]", "annotated_tactic": ["rw [<a>Nat.mul_right_comm</a>, <a>Nat.div_mul_cancel</a> (<a>Nat.gcd_dvd_left</a> ..)]", [{"full_name": "Nat.mul_right_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [424, 19], "def_end_pos": [424, 33]}, {"full_name": "Nat.div_mul_cancel", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [91, 19], "def_end_pos": [91, 33]}, {"full_name": "Nat.gcd_dvd_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [86, 9], "def_end_pos": [86, 21]}]], "state_before": "case isFalse.e_den\na b : Rat\nh\u271d : \u00aca.den.gcd b.den = 1\nthis : a.den.gcd b.den \u2260 0\n\u22a2 a.den / a.den.gcd b.den * b.den * a.den.gcd b.den = a.den * b.den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "of_quotient_center_nilpotent", "start": [649, 1], "end": [652, 56], "traced_tactics": [{"tactic": "obtain \u27e8n, hn\u27e9 := h.nilpotent", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := h.nilpotent", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nh : Group.IsNilpotent (G \u29f8 center G)\n\u22a2 Group.IsNilpotent G", "state_after": "case intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nh : Group.IsNilpotent (G \u29f8 center G)\nn : \u2115\nhn : upperCentralSeries (G \u29f8 center G) n = \u22a4\n\u22a2 Group.IsNilpotent G"}, {"tactic": "use n.succ", "annotated_tactic": ["use n.succ", []], "state_before": "case intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nh : Group.IsNilpotent (G \u29f8 center G)\nn : \u2115\nhn : upperCentralSeries (G \u29f8 center G) n = \u22a4\n\u22a2 Group.IsNilpotent G", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nh : Group.IsNilpotent (G \u29f8 center G)\nn : \u2115\nhn : upperCentralSeries (G \u29f8 center G) n = \u22a4\n\u22a2 upperCentralSeries G n.succ = \u22a4"}, {"tactic": "simp [\u2190 comap_upperCentralSeries_quotient_center, hn]", "annotated_tactic": ["simp [\u2190 <a>comap_upperCentralSeries_quotient_center</a>, hn]", [{"full_name": "comap_upperCentralSeries_quotient_center", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [590, 9], "def_end_pos": [590, 49]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nh : Group.IsNilpotent (G \u29f8 center G)\nn : \u2115\nhn : upperCentralSeries (G \u29f8 center G) n = \u22a4\n\u22a2 upperCentralSeries G n.succ = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Thickening.lean", "full_name": "Metric.cthickening_closure", "start": [429, 1], "end": [430, 42], "traced_tactics": [{"tactic": "simp_rw [cthickening, infEdist_closure]", "annotated_tactic": ["simp_rw [<a>cthickening</a>, <a>infEdist_closure</a>]", [{"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/Thickening.lean", "def_pos": [195, 5], "def_end_pos": [195, 16]}, {"full_name": "EMetric.infEdist_closure", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [154, 9], "def_end_pos": [154, 25]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u22a2 cthickening \u03b4 (closure s) = cthickening \u03b4 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "isOpenMap_smul", "start": [257, 1], "end": [258, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/Ring/Basic.lean", "full_name": "SemiRingCat.ofHom_apply", "start": [160, 1], "end": [162, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.lift_add", "start": [74, 1], "end": [78, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.cos_pi_div_two_sub", "start": [474, 1], "end": [476, 34], "traced_tactics": [{"tactic": "induction \u03b8 using Real.Angle.induction_on", "annotated_tactic": ["induction \u03b8 using <a>Real.Angle.induction_on</a>", [{"full_name": "Real.Angle.induction_on", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [74, 19], "def_end_pos": [74, 31]}]], "state_before": "\u03b8 : Angle\n\u22a2 (\u2191(\u03c0 / 2) - \u03b8).cos = \u03b8.sin", "state_after": "case h\nx\u271d : \u211d\n\u22a2 (\u2191(\u03c0 / 2) - \u2191x\u271d).cos = (\u2191x\u271d).sin"}, {"tactic": "exact Real.cos_pi_div_two_sub _", "annotated_tactic": ["exact <a>Real.cos_pi_div_two_sub</a> _", [{"full_name": "Real.cos_pi_div_two_sub", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [517, 9], "def_end_pos": [517, 27]}]], "state_before": "case h\nx\u271d : \u211d\n\u22a2 (\u2191(\u03c0 / 2) - \u2191x\u271d).cos = (\u2191x\u271d).sin", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "full_name": "SMulMemClass.coeSubtype", "start": [53, 11], "end": [54, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.isMaximal_iff", "start": [308, 1], "end": [318, 50], "traced_tactics": [{"tactic": "rw [lt_iff_le_not_le]", "annotated_tactic": ["rw [<a>lt_iff_le_not_le</a>]", [{"full_name": "lt_iff_le_not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [60, 9], "def_end_pos": [60, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI J : Ideal \u03b1\n\u22a2 I < J \u2192 J = \u22a4 \u2194 \u2200 (x : \u03b1), I \u2264 J \u2192 x \u2209 I \u2192 x \u2208 J \u2192 1 \u2208 J", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI J : Ideal \u03b1\n\u22a2 I \u2264 J \u2227 \u00acJ \u2264 I \u2192 J = \u22a4 \u2194 \u2200 (x : \u03b1), I \u2264 J \u2192 x \u2209 I \u2192 x \u2208 J \u2192 1 \u2208 J"}, {"tactic": "exact\n  \u27e8fun H x h hx\u2081 hx\u2082 => J.eq_top_iff_one.1 <| H \u27e8h, not_subset.2 \u27e8_, hx\u2082, hx\u2081\u27e9\u27e9,\n    fun H \u27e8h\u2081, h\u2082\u27e9 =>\n    let \u27e8x, xJ, xI\u27e9 := not_subset.1 h\u2082\n    J.eq_top_iff_one.2 <| H x h\u2081 xI xJ\u27e9", "annotated_tactic": ["exact\n            \u27e8fun H x h hx\u2081 hx\u2082 => J.eq_top_iff_one.1 <| H \u27e8h, <a>not_subset</a>.2 \u27e8_, hx\u2082, hx\u2081\u27e9\u27e9,\n              fun H \u27e8h\u2081, h\u2082\u27e9 =>\n              let \u27e8x, xJ, xI\u27e9 := <a>not_subset</a>.1 h\u2082\n              J.eq_top_iff_one.2 <| H x h\u2081 xI xJ\u27e9", [{"full_name": "Set.not_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}, {"full_name": "Set.not_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI J : Ideal \u03b1\n\u22a2 I \u2264 J \u2227 \u00acJ \u2264 I \u2192 J = \u22a4 \u2194 \u2200 (x : \u03b1), I \u2264 J \u2192 x \u2209 I \u2192 x \u2208 J \u2192 1 \u2208 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelIso/Set.lean", "full_name": "RelIso.preimage_eq_image_symm", "start": [111, 1], "end": [112, 42], "traced_tactics": [{"tactic": "rw [e.symm.image_eq_preimage_symm]", "annotated_tactic": ["rw [e.symm.image_eq_preimage_symm]", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\nu : \u03b4 \u2192 \u03b4 \u2192 Prop\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : r \u2243r s\nt : Set \u03b2\n\u22a2 \u21d1e \u207b\u00b9' t = \u21d1e.symm '' t", "state_after": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\nu : \u03b4 \u2192 \u03b4 \u2192 Prop\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : r \u2243r s\nt : Set \u03b2\n\u22a2 \u21d1e \u207b\u00b9' t = \u21d1e.symm.symm \u207b\u00b9' t"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\nu : \u03b4 \u2192 \u03b4 \u2192 Prop\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : r \u2243r s\nt : Set \u03b2\n\u22a2 \u21d1e \u207b\u00b9' t = \u21d1e.symm.symm \u207b\u00b9' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.singularPart_of_not_haveLebesgueDecomposition", "start": [97, 1], "end": [99, 31], "traced_tactics": [{"tactic": "rw [singularPart, dif_neg h]", "annotated_tactic": ["rw [<a>singularPart</a>, <a>dif_neg</a> h]", [{"full_name": "MeasureTheory.Measure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [72, 31], "def_end_pos": [72, 43]}, {"full_name": "dif_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [954, 9], "def_end_pos": [954, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nh : \u00ac\u03bc.HaveLebesgueDecomposition \u03bd\n\u22a2 \u03bc.singularPart \u03bd = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arcsin_eq_neg_pi_div_two", "start": [259, 1], "end": [260, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Range.lean", "full_name": "Multiset.range_subset", "start": [43, 1], "end": [44, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_univ", "start": [134, 1], "end": [135, 51], "traced_tactics": [{"tactic": "simp [cond, measure_univ, Measure.restrict_univ]", "annotated_tactic": ["simp [<a>cond</a>, <a>measure_univ</a>, <a>Measure.restrict_univ</a>]", [{"full_name": "ProbabilityTheory.cond", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [73, 5], "def_end_pos": [73, 9]}, {"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [234, 3], "def_end_pos": [234, 15]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [239, 9], "def_end_pos": [239, 22]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 \u03bc[|univ] = \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "full_name": "AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_fst_snd", "start": [302, 1], "end": [304, 27], "traced_tactics": [{"tactic": "delta gluedCoverT'", "annotated_tactic": ["delta <a>gluedCoverT'</a>", [{"full_name": "AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [282, 5], "def_end_pos": [282, 17]}]], "state_before": "X : Scheme\n\ud835\udcb0 : X.OpenCover\nx y z : \ud835\udcb0.J\n\u22a2 \ud835\udcb0.gluedCoverT' x y z \u226b pullback.fst \u226b pullback.snd = pullback.snd \u226b pullback.snd", "state_after": "X : Scheme\n\ud835\udcb0 : X.OpenCover\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (\ud835\udcb0.map x) (\ud835\udcb0.map z) pullback.fst).hom \u226b\n        (pullback.map (pullback.fst \u226b \ud835\udcb0.map x) (\ud835\udcb0.map z) (pullback.fst \u226b \ud835\udcb0.map y) (\ud835\udcb0.map z)\n              (pullbackSymmetry (\ud835\udcb0.map x) (\ud835\udcb0.map y)).hom (\ud835\udfd9 (\ud835\udcb0.obj z)) (\ud835\udfd9 X) \u22ef \u22ef \u226b\n            (pullbackRightPullbackFstIso (\ud835\udcb0.map y) (\ud835\udcb0.map z) pullback.fst).inv) \u226b\n          (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n      pullback.fst \u226b pullback.snd =\n    pullback.snd \u226b pullback.snd"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "X : Scheme\n\ud835\udcb0 : X.OpenCover\nx y z : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (\ud835\udcb0.map x) (\ud835\udcb0.map z) pullback.fst).hom \u226b\n        (pullback.map (pullback.fst \u226b \ud835\udcb0.map x) (\ud835\udcb0.map z) (pullback.fst \u226b \ud835\udcb0.map y) (\ud835\udcb0.map z)\n              (pullbackSymmetry (\ud835\udcb0.map x) (\ud835\udcb0.map y)).hom (\ud835\udfd9 (\ud835\udcb0.obj z)) (\ud835\udfd9 X) \u22ef \u22ef \u226b\n            (pullbackRightPullbackFstIso (\ud835\udcb0.map y) (\ud835\udcb0.map z) pullback.fst).inv) \u226b\n          (pullbackSymmetry pullback.fst pullback.fst).hom) \u226b\n      pullback.fst \u226b pullback.snd =\n    pullback.snd \u226b pullback.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSet.cond", "start": [235, 11], "end": [238, 18], "traced_tactics": [{"tactic": "cases i", "annotated_tactic": ["cases i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\ni : Bool\n\u22a2 MeasurableSet (bif i then s\u2081 else s\u2082)", "state_after": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif false then s\u2081 else s\u2082)\n\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif true then s\u2081 else s\u2082)"}, {"tactic": "exacts [h\u2082, h\u2081]", "annotated_tactic": ["exacts [h\u2082, h\u2081]", []], "state_before": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif false then s\u2081 else s\u2082)\n\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif true then s\u2081 else s\u2082)", "state_after": "no goals"}]}]
\ No newline at end of file