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Cardinal.mk_list_le_max ** α : Type u ⊢ #(List α) ≤ max ℵ₀ #α ** cases finite_or_infinite α ** case inl α : Type u h✝ : Finite α ⊢ #(List α) ≤ max ℵ₀ #α ** exact mk_le_aleph0.trans (le_max_left _ _) ** case inr α : Type u h✝ : Infinite α ⊢ #(List α) ≤ max ℵ₀ #α ** rw [mk_list_eq_mk] ** case inr α : Type u h✝ : Infinite α ⊢ #α ≤ max ℵ₀ #α ** apply le_max_right ** Qed | |
Cardinal.mk_finsupp_lift_of_infinite ** α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ #(α →₀ β) = max (lift.{v, u} #α) (lift.{u, v} #β) ** apply le_antisymm ** case a α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ #(α →₀ β) ≤ max (lift.{v, u} #α) (lift.{u, v} #β) ** calc
#(α →₀ β) ≤ #(Finset (α × β)) := mk_le_of_injective (Finsupp.graph_injective α β)
_ = #(α × β) := mk_finset_of_infinite _
_ = max (lift.{v} #α) (lift.{u} #β) :=
by rw [mk_prod, mul_eq_max_of_aleph0_le_left] <;> simp ** α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ #(α × β) = max (lift.{v, u} #α) (lift.{u, v} #β) ** rw [mk_prod, mul_eq_max_of_aleph0_le_left] <;> simp ** case a α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ max (lift.{v, u} #α) (lift.{u, v} #β) ≤ #(α →₀ β) ** apply max_le <;> rw [← lift_id #(α →₀ β), ← lift_umax] ** case a.h₁ α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ lift.{max u v, u} #α ≤ lift.{max u v, max u v} #(α →₀ β) ** cases' exists_ne (0 : β) with b hb ** case a.h₁.intro α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β b : β hb : b ≠ 0 ⊢ lift.{max u v, u} #α ≤ lift.{max u v, max u v} #(α →₀ β) ** exact lift_mk_le.{v}.2 ⟨⟨_, Finsupp.single_left_injective hb⟩⟩ ** case a.h₂ α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ lift.{max v u, v} #β ≤ lift.{max u v, max u v} #(α →₀ β) ** inhabit α ** case a.h₂ α : Type u β : Type v inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β inhabited_h : Inhabited α ⊢ lift.{max v u, v} #β ≤ lift.{max u v, max u v} #(α →₀ β) ** exact lift_mk_le.{u}.2 ⟨⟨_, Finsupp.single_injective default⟩⟩ ** Qed | |
Cardinal.mk_finsupp_of_infinite ** α β : Type u inst✝² : Infinite α inst✝¹ : Zero β inst✝ : Nontrivial β ⊢ #(α →₀ β) = max #α #β ** simp ** Qed | |
Cardinal.mk_finsupp_lift_of_infinite' ** α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β ⊢ #(α →₀ β) = max (lift.{v, u} #α) (lift.{u, v} #β) ** cases fintypeOrInfinite α ** case inl α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Fintype α ⊢ #(α →₀ β) = max (lift.{v, u} #α) (lift.{u, v} #β) ** rw [mk_finsupp_lift_of_fintype] ** case inl α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Fintype α ⊢ lift.{u, v} #β ^ Fintype.card α = max (lift.{v, u} #α) (lift.{u, v} #β) ** have : ℵ₀ ≤ (#β).lift := aleph0_le_lift.2 (aleph0_le_mk β) ** case inl α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Fintype α this : ℵ₀ ≤ lift.{?u.190424, v} #β ⊢ lift.{u, v} #β ^ Fintype.card α = max (lift.{v, u} #α) (lift.{u, v} #β) ** rw [max_eq_right (le_trans _ this), power_nat_eq this] ** case inl α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Fintype α this : ℵ₀ ≤ lift.{u, v} #β ⊢ 1 ≤ Fintype.card α α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Fintype α this : ℵ₀ ≤ lift.{u, v} #β ⊢ lift.{v, u} #α ≤ ℵ₀ ** exacts [Fintype.card_pos, lift_le_aleph0.2 (lt_aleph0_of_finite _).le] ** case inr α : Type u β : Type v inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β val✝ : Infinite α ⊢ #(α →₀ β) = max (lift.{v, u} #α) (lift.{u, v} #β) ** apply mk_finsupp_lift_of_infinite ** Qed | |
Cardinal.mk_finsupp_of_infinite' ** α β : Type u inst✝² : Nonempty α inst✝¹ : Zero β inst✝ : Infinite β ⊢ #(α →₀ β) = max #α #β ** simp ** Qed | |
Cardinal.mk_finsupp_nat ** α : Type u inst✝ : Nonempty α ⊢ #(α →₀ ℕ) = max #α ℵ₀ ** simp ** Qed | |
Cardinal.mk_multiset_of_infinite ** α : Type u inst✝ : Infinite α ⊢ #(Multiset α) = #α ** simp ** Qed | |
Cardinal.mk_multiset_of_isEmpty ** α : Type u inst✝ : IsEmpty α ⊢ #(α →₀ ℕ) = 1 ** simp ** Qed | |
Cardinal.mk_multiset_of_countable ** α : Type u inst✝¹ : Countable α inst✝ : Nonempty α ⊢ #(α →₀ ℕ) = ℵ₀ ** simp ** Qed | |
Cardinal.mk_bounded_set_le_of_infinite ** α : Type u inst✝ : Infinite α c : Cardinal.{u} ⊢ #{ t // #↑t ≤ c } ≤ #α ^ c ** refine' le_trans _ (by rw [← add_one_eq (aleph0_le_mk α)]) ** α : Type u inst✝ : Infinite α c : Cardinal.{u} ⊢ #{ t // #↑t ≤ c } ≤ (#α + 1) ^ c ** induction' c using Cardinal.inductionOn with β ** case h α : Type u inst✝ : Infinite α β : Type u ⊢ #{ t // #↑t ≤ #β } ≤ (#α + 1) ^ #β ** fapply mk_le_of_surjective ** case h.hf α : Type u inst✝ : Infinite α β : Type u ⊢ Surjective fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) } ** rintro ⟨s, ⟨g⟩⟩ ** case h.hf.mk.intro α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β ⊢ ∃ a, (fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) a = { val := s, property := (_ : Nonempty (↑s ↪ β)) } ** use fun y => if h : ∃ x : s, g x = y then Sum.inl (Classical.choose h).val
else Sum.inr (ULift.up 0) ** case h α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β ⊢ ((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) = { val := s, property := (_ : Nonempty (↑s ↪ β)) } ** apply Subtype.eq ** case h.a α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β ⊢ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) = ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** ext x ** case h.a.h α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α ⊢ x ∈ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) ↔ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** constructor ** α : Type u inst✝ : Infinite α c : Cardinal.{u} ⊢ ?m.199552 ≤ #α ^ c ** rw [← add_one_eq (aleph0_le_mk α)] ** case h.f α : Type u inst✝ : Infinite α β : Type u ⊢ (fun α β => β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β → { t // #↑t ≤ #β } ** intro f ** case h.f α : Type u inst✝ : Infinite α β : Type u f : (fun α β => β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β ⊢ { t // #↑t ≤ #β } ** use Sum.inl ⁻¹' range f ** case property α : Type u inst✝ : Infinite α β : Type u f : (fun α β => β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β ⊢ #↑(Sum.inl ⁻¹' range f) ≤ #β ** refine' le_trans (mk_preimage_of_injective _ _ fun x y => Sum.inl.inj) _ ** case property α : Type u inst✝ : Infinite α β : Type u f : (fun α β => β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β ⊢ #↑(range f) ≤ #β ** apply mk_range_le ** case h.a.h.mp α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α ⊢ x ∈ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) → x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** rintro ⟨y, h⟩ ** case h.a.h.mp.intro α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h : (fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) y = Sum.inl x ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** dsimp only at h ** case h.a.h.mp.intro α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h : (if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) = Sum.inl x ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** by_cases h' : ∃ z : s, g z = y ** case pos α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h : (if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) = Sum.inl x h' : ∃ z, ↑g z = y ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** rw [dif_pos h'] at h ** case pos α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h' : ∃ z, ↑g z = y h : Sum.inl ↑(choose h') = Sum.inl x ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** cases Sum.inl.inj h ** case pos.refl α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β y : β h' : ∃ z, ↑g z = y h : Sum.inl ↑(choose h') = Sum.inl ↑(choose h') ⊢ ↑(choose h') ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** exact (Classical.choose h').2 ** case neg α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h : (if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) = Sum.inl x h' : ¬∃ z, ↑g z = y ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** rw [dif_neg h'] at h ** case neg α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α y : β h : Sum.inr { down := 0 } = Sum.inl x h' : ¬∃ z, ↑g z = y ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ** cases h ** case h.a.h.mpr α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α ⊢ x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } → x ∈ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) ** intro h ** case h.a.h.mpr α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } ⊢ x ∈ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) ** have : ∃ z : s, g z = g ⟨x, h⟩ := ⟨⟨x, h⟩, rfl⟩ ** case h.a.h.mpr α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ x ∈ ↑((fun f => { val := Sum.inl ⁻¹' range f, property := (_ : #↑(Sum.inl ⁻¹' range f) ≤ #β) }) fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) ** use g ⟨x, h⟩ ** case h α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ (fun y => if h : ∃ x, ↑g x = y then Sum.inl ↑(choose h) else Sum.inr { down := 0 }) (↑g { val := x, property := h }) = Sum.inl x ** dsimp only ** case h α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ (if h_1 : ∃ x_1, ↑g x_1 = ↑g { val := x, property := h } then Sum.inl ↑(choose h_1) else Sum.inr { down := 0 }) = Sum.inl x ** rw [dif_pos this] ** case h α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ Sum.inl ↑(choose this) = Sum.inl x ** congr ** case h.e_val α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ ↑(choose this) = x ** suffices : Classical.choose this = ⟨x, h⟩ ** case h.e_val α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this✝ : ∃ z, ↑g z = ↑g { val := x, property := h } this : choose this✝ = { val := x, property := h } ⊢ ↑(choose this✝) = x case this α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ choose this = { val := x, property := h } ** exact congr_arg Subtype.val this ** case this α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ choose this = { val := x, property := h } ** apply g.2 ** case this.a α : Type u inst✝ : Infinite α β : Type u s : Set α g : ↑s ↪ β x : α h : x ∈ ↑{ val := s, property := (_ : Nonempty (↑s ↪ β)) } this : ∃ z, ↑g z = ↑g { val := x, property := h } ⊢ Embedding.toFun g (choose this) = Embedding.toFun g { val := x, property := h } ** exact Classical.choose_spec this ** Qed | |
Cardinal.mk_bounded_set_le ** α : Type u c : Cardinal.{u} ⊢ #{ t // #↑t ≤ c } ≤ max #α ℵ₀ ^ c ** trans #{ t : Set (Sum (ULift.{u} ℕ) α) // #t ≤ c } ** α : Type u c : Cardinal.{u} ⊢ #{ t // #↑t ≤ c } ≤ max #α ℵ₀ ^ c ** apply (mk_bounded_set_le_of_infinite (Sum (ULift.{u} ℕ) α) c).trans ** α : Type u c : Cardinal.{u} ⊢ #(ULift.{u, 0} ℕ ⊕ α) ^ c ≤ max #α ℵ₀ ^ c ** rw [max_comm, ← add_eq_max] <;> rfl ** α : Type u c : Cardinal.{u} ⊢ #{ t // #↑t ≤ c } ≤ #{ t // #↑t ≤ c } ** refine' ⟨Embedding.subtypeMap _ _⟩ ** case refine'_1 α : Type u c : Cardinal.{u} ⊢ Set α ↪ Set (ULift.{u, 0} ℕ ⊕ α) case refine'_2 α : Type u c : Cardinal.{u} ⊢ ∀ ⦃x : Set α⦄, #↑x ≤ c → #↑(↑?refine'_1 x) ≤ c ** apply Embedding.image ** case refine'_1.f α : Type u c : Cardinal.{u} ⊢ α ↪ ULift.{u, 0} ℕ ⊕ α case refine'_2 α : Type u c : Cardinal.{u} ⊢ ∀ ⦃x : Set α⦄, #↑x ≤ c → #↑(↑(Embedding.image ?refine'_1.f) x) ≤ c ** use Sum.inr ** case inj' α : Type u c : Cardinal.{u} ⊢ Injective Sum.inr case refine'_2 α : Type u c : Cardinal.{u} ⊢ ∀ ⦃x : Set α⦄, #↑x ≤ c → #↑(↑(Embedding.image { toFun := Sum.inr, inj' := ?inj' }) x) ≤ c ** apply Sum.inr.inj ** case refine'_2 α : Type u c : Cardinal.{u} ⊢ ∀ ⦃x : Set α⦄, #↑x ≤ c → #↑(↑(Embedding.image { toFun := Sum.inr, inj' := (_ : ∀ {val val_1 : α}, Sum.inr val = Sum.inr val_1 → val = val_1) }) x) ≤ c ** intro s hs ** case refine'_2 α : Type u c : Cardinal.{u} s : Set α hs : #↑s ≤ c ⊢ #↑(↑(Embedding.image { toFun := Sum.inr, inj' := (_ : ∀ {val val_1 : α}, Sum.inr val = Sum.inr val_1 → val = val_1) }) s) ≤ c ** exact mk_image_le.trans hs ** Qed | |
Cardinal.mk_bounded_subset_le ** α : Type u s : Set α c : Cardinal.{u} ⊢ #{ t // t ⊆ s ∧ #↑t ≤ c } ≤ max #↑s ℵ₀ ^ c ** refine' le_trans _ (mk_bounded_set_le s c) ** α : Type u s : Set α c : Cardinal.{u} ⊢ #{ t // t ⊆ s ∧ #↑t ≤ c } ≤ #{ t // #↑t ≤ c } ** refine' ⟨Embedding.codRestrict _ _ _⟩ ** case refine'_1 α : Type u s : Set α c : Cardinal.{u} ⊢ { t // t ⊆ s ∧ #↑t ≤ c } ↪ Set ↑s case refine'_2 α : Type u s : Set α c : Cardinal.{u} ⊢ ∀ (a : { t // t ⊆ s ∧ #↑t ≤ c }), ↑?refine'_1 a ∈ fun t => Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b)) c ** use fun t => (↑) ⁻¹' t.1 ** case refine'_2 α : Type u s : Set α c : Cardinal.{u} ⊢ ∀ (a : { t // t ⊆ s ∧ #↑t ≤ c }), ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t, inj' := (_ : ∀ ⦃a₁ a₂ : { t // t ⊆ s ∧ #↑t ≤ c }⦄, (fun t => Subtype.val ⁻¹' ↑t) a₁ = (fun t => Subtype.val ⁻¹' ↑t) a₂ → a₁ = a₂) } a ∈ fun t => Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b)) c ** rintro ⟨t, _, h2t⟩ ** case refine'_2.mk.intro α : Type u s : Set α c : Cardinal.{u} t : Set α left✝ : t ⊆ s h2t : #↑t ≤ c ⊢ ↑{ toFun := fun t => Subtype.val ⁻¹' ↑t, inj' := (_ : ∀ ⦃a₁ a₂ : { t // t ⊆ s ∧ #↑t ≤ c }⦄, (fun t => Subtype.val ⁻¹' ↑t) a₁ = (fun t => Subtype.val ⁻¹' ↑t) a₂ → a₁ = a₂) } { val := t, property := (_ : t ⊆ s ∧ #↑t ≤ c) } ∈ fun t => Quot.lift ((fun α β => Nonempty (α ↪ β)) ↑t) (_ : ∀ (a b : Type u), a ≈ b → Nonempty (↑t ↪ a) = Nonempty (↑t ↪ b)) c ** exact (mk_preimage_of_injective _ _ Subtype.val_injective).trans h2t ** case inj' α : Type u s : Set α c : Cardinal.{u} ⊢ Injective fun t => Subtype.val ⁻¹' ↑t ** rintro ⟨t, ht1, ht2⟩ ⟨t', h1t', h2t'⟩ h ** case inj'.mk.intro.mk.intro α : Type u s : Set α c : Cardinal.{u} t : Set α ht1 : t ⊆ s ht2 : #↑t ≤ c t' : Set α h1t' : t' ⊆ s h2t' : #↑t' ≤ c h : (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ #↑t ≤ c) } = (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ #↑t' ≤ c) } ⊢ { val := t, property := (_ : t ⊆ s ∧ #↑t ≤ c) } = { val := t', property := (_ : t' ⊆ s ∧ #↑t' ≤ c) } ** apply Subtype.eq ** case inj'.mk.intro.mk.intro.a α : Type u s : Set α c : Cardinal.{u} t : Set α ht1 : t ⊆ s ht2 : #↑t ≤ c t' : Set α h1t' : t' ⊆ s h2t' : #↑t' ≤ c h : (fun t => Subtype.val ⁻¹' ↑t) { val := t, property := (_ : t ⊆ s ∧ #↑t ≤ c) } = (fun t => Subtype.val ⁻¹' ↑t) { val := t', property := (_ : t' ⊆ s ∧ #↑t' ≤ c) } ⊢ ↑{ val := t, property := (_ : t ⊆ s ∧ #↑t ≤ c) } = ↑{ val := t', property := (_ : t' ⊆ s ∧ #↑t' ≤ c) } ** dsimp only at h ⊢ ** case inj'.mk.intro.mk.intro.a α : Type u s : Set α c : Cardinal.{u} t : Set α ht1 : t ⊆ s ht2 : #↑t ≤ c t' : Set α h1t' : t' ⊆ s h2t' : #↑t' ≤ c h : Subtype.val ⁻¹' t = Subtype.val ⁻¹' t' ⊢ t = t' ** refine' (preimage_eq_preimage' _ _).1 h <;> rw [Subtype.range_coe] <;> assumption ** Qed | |
Cardinal.mk_compl_of_infinite ** α : Type u_1 inst✝ : Infinite α s : Set α h2 : #↑s < #α ⊢ #↑sᶜ = #α ** refine' eq_of_add_eq_of_aleph0_le _ h2 (aleph0_le_mk α) ** α : Type u_1 inst✝ : Infinite α s : Set α h2 : #↑s < #α ⊢ #↑s + #↑sᶜ = #α ** exact mk_sum_compl s ** Qed | |
Cardinal.mk_compl_finset_of_infinite ** α : Type u_1 inst✝ : Infinite α s : Finset α ⊢ #↑(↑s)ᶜ = #α ** apply mk_compl_of_infinite ** case h2 α : Type u_1 inst✝ : Infinite α s : Finset α ⊢ #↑↑s < #α ** exact (finset_card_lt_aleph0 s).trans_le (aleph0_le_mk α) ** Qed | |
Cardinal.mk_compl_eq_mk_compl_infinite ** α : Type u_1 inst✝ : Infinite α s t : Set α hs : #↑s < #α ht : #↑t < #α ⊢ #↑sᶜ = #↑tᶜ ** rw [mk_compl_of_infinite s hs, mk_compl_of_infinite t ht] ** Qed | |
Cardinal.mk_compl_eq_mk_compl_finite_lift ** α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h1 : lift.{max v w, u} #α = lift.{max u w, v} #β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** cases nonempty_fintype α ** case intro α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h1 : lift.{max v w, u} #α = lift.{max u w, v} #β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t val✝ : Fintype α ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** rcases lift_mk_eq.{u, v, w}.1 h1 with ⟨e⟩ ** case intro.intro α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h1 : lift.{max v w, u} #α = lift.{max u w, v} #β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t val✝ : Fintype α e : α ≃ β ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** letI : Fintype β := Fintype.ofEquiv α e ** case intro.intro α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h1 : lift.{max v w, u} #α = lift.{max u w, v} #β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** replace h1 : Fintype.card α = Fintype.card β := (Fintype.ofEquiv_card _).symm ** case intro.intro α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e h1 : Fintype.card α = Fintype.card β ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** classical
lift s to Finset α using s.toFinite
lift t to Finset β using t.toFinite
simp only [Finset.coe_sort_coe, mk_fintype, Fintype.card_coe, lift_natCast, Nat.cast_inj] at h2
simp only [← Finset.coe_compl, Finset.coe_sort_coe, mk_coe_finset, Finset.card_compl,
lift_natCast, Nat.cast_inj, h1, h2] ** case intro.intro α : Type u β : Type v inst✝ : Finite α s : Set α t : Set β h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e h1 : Fintype.card α = Fintype.card β ⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ ** lift s to Finset α using s.toFinite ** case intro.intro.intro α : Type u β : Type v inst✝ : Finite α t : Set β val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e h1 : Fintype.card α = Fintype.card β s : Finset α h2 : lift.{max v w, u} #↑↑s = lift.{max u w, v} #↑t ⊢ lift.{max v w, u} #↑(↑s)ᶜ = lift.{max u w, v} #↑tᶜ ** lift t to Finset β using t.toFinite ** case intro.intro.intro.intro α : Type u β : Type v inst✝ : Finite α val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e h1 : Fintype.card α = Fintype.card β s : Finset α t : Finset β h2 : lift.{max v w, u} #↑↑s = lift.{max u w, v} #↑↑t ⊢ lift.{max v w, u} #↑(↑s)ᶜ = lift.{max u w, v} #↑(↑t)ᶜ ** simp only [Finset.coe_sort_coe, mk_fintype, Fintype.card_coe, lift_natCast, Nat.cast_inj] at h2 ** case intro.intro.intro.intro α : Type u β : Type v inst✝ : Finite α val✝ : Fintype α e : α ≃ β this : Fintype β := Fintype.ofEquiv α e h1 : Fintype.card α = Fintype.card β s : Finset α t : Finset β h2 : Finset.card s = Finset.card t ⊢ lift.{max v w, u} #↑(↑s)ᶜ = lift.{max u w, v} #↑(↑t)ᶜ ** simp only [← Finset.coe_compl, Finset.coe_sort_coe, mk_coe_finset, Finset.card_compl,
lift_natCast, Nat.cast_inj, h1, h2] ** Qed | |
Cardinal.mk_compl_eq_mk_compl_finite ** α β : Type u inst✝ : Finite α s : Set α t : Set β h1 : #α = #β h : #↑s = #↑t ⊢ #↑sᶜ = #↑tᶜ ** rw [← lift_inj.{u, max u v}] ** α β : Type u inst✝ : Finite α s : Set α t : Set β h1 : #α = #β h : #↑s = #↑t ⊢ lift.{max u v, u} #↑sᶜ = lift.{max u v, u} #↑tᶜ ** apply mk_compl_eq_mk_compl_finite_lift.{u, u, max u v}
<;> rwa [lift_inj] ** Qed | |
Cardinal.extend_function ** α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** have := h ** α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h this : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** cases' this with g ** case intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) g : ↑sᶜ ≃ ↑(range ↑f)ᶜ ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** let h : α ≃ β :=
(Set.sumCompl (s : Set α)).symm.trans
((sumCongr (Equiv.ofInjective f f.2) g).trans (Set.sumCompl (range f))) ** case intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h✝ : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) g : ↑sᶜ ≃ ↑(range ↑f)ᶜ h : α ≃ β := (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f))) ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** refine' ⟨h, _⟩ ** case intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h✝ : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) g : ↑sᶜ ≃ ↑(range ↑f)ᶜ h : α ≃ β := (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f))) ⊢ ∀ (x : ↑s), ↑h ↑x = ↑f x ** rintro ⟨x, hx⟩ ** case intro.mk α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β h✝ : Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) g : ↑sᶜ ≃ ↑(range ↑f)ᶜ h : α ≃ β := (Set.sumCompl s).symm.trans ((sumCongr (ofInjective ↑f (_ : Injective f.toFun)) g).trans (Set.sumCompl (range ↑f))) x : α hx : x ∈ s ⊢ ↑h ↑{ val := x, property := hx } = ↑f { val := x, property := hx } ** simp [Set.sumCompl_symm_apply_of_mem, hx] ** Qed | |
Cardinal.extend_function_finite ** α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : Nonempty (α ≃ β) ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** apply extend_function.{v, u} f ** α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : Nonempty (α ≃ β) ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** cases' id h with g ** case intro α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : Nonempty (α ≃ β) g : α ≃ β ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** rw [← lift_mk_eq.{u, v, max u v}] at h ** case intro α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : lift.{max u v, u} #α = lift.{max u v, v} #β g : α ≃ β ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** rw [← lift_mk_eq.{u, v, max u v}, mk_compl_eq_mk_compl_finite_lift.{u, v, max u v} h] ** case intro α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : lift.{max u v, u} #α = lift.{max u v, v} #β g : α ≃ β ⊢ lift.{max u v, u} #↑s = lift.{max u v, v} #↑(range ↑f) ** rw [mk_range_eq_lift.{u, v, max u v}] ** case intro α : Type u β : Type v inst✝ : Finite α s : Set α f : ↑s ↪ β h : lift.{max u v, u} #α = lift.{max u v, v} #β g : α ≃ β ⊢ Injective ↑f ** exact f.2 ** Qed | |
Cardinal.extend_function_of_lt ** α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** cases fintypeOrInfinite α ** case inl α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) val✝ : Fintype α ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** exact extend_function_finite f h ** case inr α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) val✝ : Infinite α ⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x ** apply extend_function f ** case inr α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) val✝ : Infinite α ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** cases' id h with g ** case inr.intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) val✝ : Infinite α g : α ≃ β ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** haveI := Infinite.of_injective _ g.injective ** case inr.intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : Nonempty (α ≃ β) val✝ : Infinite α g : α ≃ β this : Infinite β ⊢ Nonempty (↑sᶜ ≃ ↑(range ↑f)ᶜ) ** rw [← lift_mk_eq'] at h ⊢ ** case inr.intro α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : lift.{u_2, u_1} #α = lift.{u_1, u_2} #β val✝ : Infinite α g : α ≃ β this : Infinite β ⊢ lift.{u_2, u_1} #↑sᶜ = lift.{u_1, u_2} #↑(range ↑f)ᶜ ** rwa [mk_compl_of_infinite s hs, mk_compl_of_infinite] ** case inr.intro.h2 α : Type u_1 β : Type u_2 s : Set α f : ↑s ↪ β hs : #↑s < #α h : lift.{u_2, u_1} #α = lift.{u_1, u_2} #β val✝ : Infinite α g : α ≃ β this : Infinite β ⊢ #↑(range ↑f) < #β ** rwa [← lift_lt, mk_range_eq_of_injective f.injective, ← h, lift_lt] ** Qed | |
Ordinal.isOpen_singleton_iff ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ IsOpen {a} ↔ ¬IsLimit a ** refine' ⟨fun h ⟨h₀, hsucc⟩ => _, fun ha => _⟩ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} h : IsOpen {a} x✝ : IsLimit a h₀ : a ≠ 0 hsucc : ∀ (a_1 : Ordinal.{u}), a_1 < a → succ a_1 < a ⊢ False ** obtain ⟨b, c, hbc, hbc'⟩ :=
(mem_nhds_iff_exists_Ioo_subset' ⟨0, Ordinal.pos_iff_ne_zero.2 h₀⟩ ⟨_, lt_succ a⟩).1
(h.mem_nhds rfl) ** case refine'_1.intro.intro.intro s : Set Ordinal.{u} a : Ordinal.{u} h : IsOpen {a} x✝ : IsLimit a h₀ : a ≠ 0 hsucc : ∀ (a_1 : Ordinal.{u}), a_1 < a → succ a_1 < a b c : Ordinal.{u} hbc : a ∈ Set.Ioo b c hbc' : Set.Ioo b c ⊆ {a} ⊢ False ** have hba := hsucc b hbc.1 ** case refine'_1.intro.intro.intro s : Set Ordinal.{u} a : Ordinal.{u} h : IsOpen {a} x✝ : IsLimit a h₀ : a ≠ 0 hsucc : ∀ (a_1 : Ordinal.{u}), a_1 < a → succ a_1 < a b c : Ordinal.{u} hbc : a ∈ Set.Ioo b c hbc' : Set.Ioo b c ⊆ {a} hba : succ b < a ⊢ False ** exact hba.ne (hbc' ⟨lt_succ b, hba.trans hbc.2⟩) ** case refine'_2 s : Set Ordinal.{u} a : Ordinal.{u} ha : ¬IsLimit a ⊢ IsOpen {a} ** rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha') ** case refine'_2.inl s : Set Ordinal.{u} ha : ¬IsLimit 0 ⊢ IsOpen {0} ** rw [← bot_eq_zero, ← Set.Iic_bot, ← Iio_succ] ** case refine'_2.inl s : Set Ordinal.{u} ha : ¬IsLimit 0 ⊢ IsOpen (Set.Iio (succ ⊥)) ** exact isOpen_Iio ** case refine'_2.inr.inl.intro s : Set Ordinal.{u} b : Ordinal.{u} ha : ¬IsLimit (succ b) ⊢ IsOpen {succ b} ** rw [← Set.Icc_self, Icc_succ_left, ← Ioo_succ_right] ** case refine'_2.inr.inl.intro s : Set Ordinal.{u} b : Ordinal.{u} ha : ¬IsLimit (succ b) ⊢ IsOpen (Set.Ioo b (succ (succ b))) ** exact isOpen_Ioo ** case refine'_2.inr.inr s : Set Ordinal.{u} a : Ordinal.{u} ha : ¬IsLimit a ha' : IsLimit a ⊢ IsOpen {a} ** exact (ha ha').elim ** Qed | |
Ordinal.nhds_left'_eq_nhds_ne ** s : Set Ordinal.{u} a✝ : Ordinal.{u} a : Ordinal.{u_1} ⊢ 𝓝[Set.Iio a] a = 𝓝[{a}ᶜ] a ** rw [← nhds_left'_sup_nhds_right', nhds_right', sup_bot_eq] ** Qed | |
Ordinal.nhds_left_eq_nhds ** s : Set Ordinal.{u} a✝ : Ordinal.{u} a : Ordinal.{u_1} ⊢ 𝓝[Set.Iic a] a = 𝓝 a ** rw [← nhds_left_sup_nhds_right', nhds_right', sup_bot_eq] ** Qed | |
Ordinal.isOpen_iff ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ IsOpen s ↔ ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s ** refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho => ?_ ** s : Set Ordinal.{u} a o : Ordinal.{u} ho : o ∈ s ⊢ s ∈ 𝓝 o ↔ IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s ** by_cases ho' : IsLimit o ** case pos s : Set Ordinal.{u} a o : Ordinal.{u} ho : o ∈ s ho' : IsLimit o ⊢ s ∈ 𝓝 o ↔ IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s ** simp only [(nhdsBasis_Ioc ho'.1).mem_iff, ho', true_implies] ** case pos s : Set Ordinal.{u} a o : Ordinal.{u} ho : o ∈ s ho' : IsLimit o ⊢ (∃ i, i < o ∧ Set.Ioc i o ⊆ s) ↔ ∃ a, a < o ∧ Set.Ioo a o ⊆ s ** refine exists_congr fun a => and_congr_right fun ha => ?_ ** case pos s : Set Ordinal.{u} a✝ o : Ordinal.{u} ho : o ∈ s ho' : IsLimit o a : Ordinal.{u} ha : a < o ⊢ Set.Ioc a o ⊆ s ↔ Set.Ioo a o ⊆ s ** simp only [← Set.Ioo_insert_right ha, Set.insert_subset_iff, ho, true_and] ** case neg s : Set Ordinal.{u} a o : Ordinal.{u} ho : o ∈ s ho' : ¬IsLimit o ⊢ s ∈ 𝓝 o ↔ IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s ** simp [nhds_eq_pure.2 ho', ho, ho'] ** Qed | |
Ordinal.mem_closure_tfae ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 1 → 2 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 2 → 3 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 3 → 4 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 4 → 5 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 5 → 6 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a tfae_5_to_6 : (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) → ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_have 6 → 1 ** s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a tfae_5_to_6 : (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) → ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a tfae_6_to_1 : (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) → a ∈ closure s ⊢ TFAE [a ∈ closure s, a ∈ closure (s ∩ Iic a), Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a, ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a, ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a, ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a] ** tfae_finish ** case tfae_1_to_2 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} ⊢ a ∈ closure s → a ∈ closure (s ∩ Iic a) ** simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds] ** case tfae_1_to_2 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} ⊢ Filter.NeBot (𝓝[s] a) → Filter.NeBot (𝓝 a ⊓ Filter.principal s) ** exact id ** case tfae_2_to_3 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) ⊢ a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ** intro h ** case tfae_2_to_3 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) h : a ∈ closure (s ∩ Iic a) ⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ** cases' (s ∩ Iic a).eq_empty_or_nonempty with he hne ** case tfae_2_to_3.inl s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) h : a ∈ closure (s ∩ Iic a) he : s ∩ Iic a = ∅ ⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ** simp [he] at h ** case tfae_2_to_3.inr s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) h : a ∈ closure (s ∩ Iic a) hne : Set.Nonempty (s ∩ Iic a) ⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ** refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩ ** case tfae_2_to_3.inr s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) h : a ∈ closure (s ∩ Iic a) hne : Set.Nonempty (s ∩ Iic a) ⊢ a ∈ upperBounds (s ∩ Iic a) ** exact fun x hx => hx.2 ** case tfae_3_to_4 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a ⊢ Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a ** exact fun h => ⟨_, inter_subset_left _ _, h.1, bddAbove_Iic.mono (inter_subset_right _ _), h.2⟩ ** case tfae_4_to_5 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a ⊢ (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a ** rintro ⟨t, hts, hne, hbdd, rfl⟩ ** case tfae_4_to_5.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t ⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t ** have hlub : IsLUB t (sSup t) := isLUB_csSup hne hbdd ** case tfae_4_to_5.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) ⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t ** let ⟨y, hyt⟩ := hne ** case tfae_4_to_5.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t ⊢ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = sSup t ** refine ⟨succ (sSup t), succ_ne_zero _, fun x _ => if x ∈ t then x else y, fun x _ => ?_, ?_⟩ ** case tfae_4_to_5.intro.intro.intro.intro.refine_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t x : Ordinal.{u} x✝ : x < succ (sSup t) ⊢ (fun x x_1 => if x ∈ t then x else y) x x✝ ∈ s ** simp only ** case tfae_4_to_5.intro.intro.intro.intro.refine_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t x : Ordinal.{u} x✝ : x < succ (sSup t) ⊢ (if x ∈ t then x else y) ∈ s ** split_ifs with h <;> exact hts ‹_› ** case tfae_4_to_5.intro.intro.intro.intro.refine_2 s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t ⊢ (bsup (succ (sSup t)) fun x x_1 => if x ∈ t then x else y) = sSup t ** refine le_antisymm (bsup_le fun x _ => ?_) (csSup_le hne fun x hx => ?_) ** case tfae_4_to_5.intro.intro.intro.intro.refine_2.refine_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t x : Ordinal.{u} x✝ : x < succ (sSup t) ⊢ (if x ∈ t then x else y) ≤ sSup t ** split_ifs <;> exact hlub.1 ‹_› ** case tfae_4_to_5.intro.intro.intro.intro.refine_2.refine_2 s✝ : Set Ordinal.{u} a : Ordinal.{u} s t : Set Ordinal.{u} hts : t ⊆ s hne : Set.Nonempty t hbdd : BddAbove t tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t)) tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sSup t)) ∧ sSup (s ∩ Iic (sSup t)) = sSup t → ∃ t_1, t_1 ⊆ s ∧ Set.Nonempty t_1 ∧ BddAbove t_1 ∧ sSup t_1 = sSup t hlub : IsLUB t (sSup t) y : Ordinal.{u} hyt : y ∈ t x : Ordinal.{u} hx : x ∈ t ⊢ x ≤ bsup (succ (sSup t)) fun x x_1 => if x ∈ t then x else y ** refine (if_pos hx).symm.trans_le (le_bsup _ _ <| (hlub.1 hx).trans_lt (lt_succ _)) ** case tfae_5_to_6 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a ⊢ (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) → ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a ** rintro ⟨o, h₀, f, hfs, rfl⟩ ** case tfae_5_to_6.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s : Set Ordinal.{u} o : Ordinal.{u} h₀ : o ≠ 0 f : (x : Ordinal.{u}) → x < o → Ordinal.{u} hfs : ∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s tfae_1_to_2 : bsup o f ∈ closure s → bsup o f ∈ closure (s ∩ Iic (bsup o f)) tfae_2_to_3 : bsup o f ∈ closure (s ∩ Iic (bsup o f)) → Set.Nonempty (s ∩ Iic (bsup o f)) ∧ sSup (s ∩ Iic (bsup o f)) = bsup o f tfae_3_to_4 : Set.Nonempty (s ∩ Iic (bsup o f)) ∧ sSup (s ∩ Iic (bsup o f)) = bsup o f → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = bsup o f tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = bsup o f) → ∃ o_1, o_1 ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal.{u}) (hx : x < o_1), f_1 x hx ∈ s) ∧ bsup o_1 f_1 = bsup o f ⊢ ∃ ι, Nonempty ι ∧ ∃ f_1, (∀ (i : ι), f_1 i ∈ s) ∧ sup f_1 = bsup o f ** exact ⟨_, out_nonempty_iff_ne_zero.2 h₀, familyOfBFamily o f, fun _ => hfs _ _, rfl⟩ ** case tfae_6_to_1 s✝ : Set Ordinal.{u} a✝ a : Ordinal.{u} s : Set Ordinal.{u} tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a) tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a tfae_3_to_4 : Set.Nonempty (s ∩ Iic a) ∧ sSup (s ∩ Iic a) = a → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = a) → ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a tfae_5_to_6 : (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) → ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a ⊢ (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) → a ∈ closure s ** rintro ⟨ι, hne, f, hfs, rfl⟩ ** case tfae_6_to_1.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s : Set Ordinal.{u} ι : Type u hne : Nonempty ι f : ι → Ordinal.{u} hfs : ∀ (i : ι), f i ∈ s tfae_1_to_2 : sup f ∈ closure s → sup f ∈ closure (s ∩ Iic (sup f)) tfae_2_to_3 : sup f ∈ closure (s ∩ Iic (sup f)) → Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f) → ∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal.{u}) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f tfae_5_to_6 : (∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal.{u}) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f) → ∃ ι_1, Nonempty ι_1 ∧ ∃ f_1, (∀ (i : ι_1), f_1 i ∈ s) ∧ sup f_1 = sup f ⊢ sup f ∈ closure s ** rw [sup, iSup] ** case tfae_6_to_1.intro.intro.intro.intro s✝ : Set Ordinal.{u} a : Ordinal.{u} s : Set Ordinal.{u} ι : Type u hne : Nonempty ι f : ι → Ordinal.{u} hfs : ∀ (i : ι), f i ∈ s tfae_1_to_2 : sup f ∈ closure s → sup f ∈ closure (s ∩ Iic (sup f)) tfae_2_to_3 : sup f ∈ closure (s ∩ Iic (sup f)) → Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f tfae_3_to_4 : Set.Nonempty (s ∩ Iic (sup f)) ∧ sSup (s ∩ Iic (sup f)) = sup f → ∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f tfae_4_to_5 : (∃ t, t ⊆ s ∧ Set.Nonempty t ∧ BddAbove t ∧ sSup t = sup f) → ∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal.{u}) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f tfae_5_to_6 : (∃ o, o ≠ 0 ∧ ∃ f_1, (∀ (x : Ordinal.{u}) (hx : x < o), f_1 x hx ∈ s) ∧ bsup o f_1 = sup f) → ∃ ι_1, Nonempty ι_1 ∧ ∃ f_1, (∀ (i : ι_1), f_1 i ∈ s) ∧ sup f_1 = sup f ⊢ sSup (Set.range f) ∈ closure s ** exact closure_mono (range_subset_iff.2 hfs) <| csSup_mem_closure (range_nonempty f)
(bddAbove_range.{u, u} f) ** Qed | |
Ordinal.mem_closure_iff_sup ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ (∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) ↔ ∃ ι x f, (∀ (i : ι), f i ∈ s) ∧ sup f = a ** simp only [exists_prop] ** Qed | |
Ordinal.mem_closed_iff_sup ** s : Set Ordinal.{u} a : Ordinal.{u} hs : IsClosed s ⊢ a ∈ s ↔ ∃ ι _hι f, (∀ (i : ι), f i ∈ s) ∧ sup f = a ** rw [← mem_closure_iff_sup, hs.closure_eq] ** Qed | |
Ordinal.mem_closure_iff_bsup ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ (∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ bsup o f = a) ↔ ∃ o _ho f, (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) ∧ bsup o f = a ** simp only [exists_prop] ** Qed | |
Ordinal.mem_closed_iff_bsup ** s : Set Ordinal.{u} a : Ordinal.{u} hs : IsClosed s ⊢ a ∈ s ↔ ∃ o _ho f, (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) ∧ bsup o f = a ** rw [← mem_closure_iff_bsup, hs.closure_eq] ** Qed | |
Ordinal.isClosed_iff_sup ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ IsClosed s ↔ ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s ** use fun hs ι hι f hf => (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩ ** case mpr s : Set Ordinal.{u} a : Ordinal.{u} ⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → IsClosed s ** rw [← closure_subset_iff_isClosed] ** case mpr s : Set Ordinal.{u} a : Ordinal.{u} ⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) → closure s ⊆ s ** intro h x hx ** case mpr s : Set Ordinal.{u} a : Ordinal.{u} h : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s x : Ordinal.{u} hx : x ∈ closure s ⊢ x ∈ s ** rcases mem_closure_iff_sup.1 hx with ⟨ι, hι, f, hf, rfl⟩ ** case mpr.intro.intro.intro.intro s : Set Ordinal.{u} a : Ordinal.{u} h : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s hx : sup f ∈ closure s ⊢ sup f ∈ s ** exact h hι f hf ** Qed | |
Ordinal.isClosed_iff_bsup ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ IsClosed s ↔ ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s ** rw [isClosed_iff_sup] ** s : Set Ordinal.{u} a : Ordinal.{u} ⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s) ↔ ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s ** refine' ⟨fun H o ho f hf => H (out_nonempty_iff_ne_zero.2 ho) _ _, fun H ι hι f hf => _⟩ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} H : ∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → sup f ∈ s o : Ordinal.{u} ho : o ≠ 0 f : (a : Ordinal.{u}) → a < o → Ordinal.{u} hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s ⊢ ∀ (i : (Quotient.out o).α), familyOfBFamily o f i ∈ s ** exact fun i => hf _ _ ** case refine'_2 s : Set Ordinal.{u} a : Ordinal.{u} H : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s ⊢ sup f ∈ s ** rw [← bsup_eq_sup] ** case refine'_2 s : Set Ordinal.{u} a : Ordinal.{u} H : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s ⊢ bsup (type WellOrderingRel) (bfamilyOfFamily f) ∈ s ** apply H (type_ne_zero_iff_nonempty.2 hι) ** case refine'_2.a s : Set Ordinal.{u} a : Ordinal.{u} H : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s ⊢ ∀ (i : Ordinal.{u}) (hi : i < type WellOrderingRel), bfamilyOfFamily f i hi ∈ s ** exact fun i hi => hf _ ** Qed | |
Ordinal.isLimit_of_mem_frontier ** s : Set Ordinal.{u} a : Ordinal.{u} ha : a ∈ frontier s ⊢ IsLimit a ** simp only [frontier_eq_closure_inter_closure, Set.mem_inter_iff, mem_closure_iff] at ha ** s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) ⊢ IsLimit a ** by_contra h ** s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) h : ¬IsLimit a ⊢ False ** rw [← isOpen_singleton_iff] at h ** s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {a} ⊢ False ** rcases ha.1 _ h rfl with ⟨b, hb, hb'⟩ ** case intro.intro s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {a} b : Ordinal.{u} hb : b ∈ {a} hb' : b ∈ s ⊢ False ** rcases ha.2 _ h rfl with ⟨c, hc, hc'⟩ ** case intro.intro.intro.intro s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {a} b : Ordinal.{u} hb : b ∈ {a} hb' : b ∈ s c : Ordinal.{u} hc : c ∈ {a} hc' : c ∈ sᶜ ⊢ False ** rw [Set.mem_singleton_iff] at * ** case intro.intro.intro.intro s : Set Ordinal.{u} a : Ordinal.{u} ha : (∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → a ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {a} b : Ordinal.{u} hb : b = a hb' : b ∈ s c : Ordinal.{u} hc : c = a hc' : c ∈ sᶜ ⊢ False ** subst hb ** case intro.intro.intro.intro s : Set Ordinal.{u} b : Ordinal.{u} hb' : b ∈ s c : Ordinal.{u} hc' : c ∈ sᶜ ha : (∀ (o : Set Ordinal.{u}), IsOpen o → b ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → b ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {b} hc : c = b ⊢ False ** subst hc ** case intro.intro.intro.intro s : Set Ordinal.{u} c : Ordinal.{u} hc' : c ∈ sᶜ hb' : c ∈ s ha : (∀ (o : Set Ordinal.{u}), IsOpen o → c ∈ o → Set.Nonempty (o ∩ s)) ∧ ∀ (o : Set Ordinal.{u}), IsOpen o → c ∈ o → Set.Nonempty (o ∩ sᶜ) h : IsOpen {c} ⊢ False ** exact hc' hb' ** Qed | |
Ordinal.isNormal_iff_strictMono_and_continuous ** s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} ⊢ IsNormal f ↔ StrictMono f ∧ Continuous f ** refine' ⟨fun h => ⟨h.strictMono, _⟩, _⟩ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f ⊢ Continuous f ** rw [continuous_def] ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f ⊢ ∀ (s : Set Ordinal.{u}), IsOpen s → IsOpen (f ⁻¹' s) ** intro s hs ** case refine'_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : IsOpen s ⊢ IsOpen (f ⁻¹' s) ** rw [isOpen_iff] at * ** case refine'_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s ⊢ ∀ (o : Ordinal.{u}), o ∈ f ⁻¹' s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ f ⁻¹' s ** intro o ho ho' ** case refine'_1 s✝ : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s o : Ordinal.{u} ho : o ∈ f ⁻¹' s ho' : IsLimit o ⊢ ∃ a, a < o ∧ Set.Ioo a o ⊆ f ⁻¹' s ** rcases hs _ ho (h.isLimit ho') with ⟨a, ha, has⟩ ** case refine'_1.intro.intro s✝ : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s o : Ordinal.{u} ho : o ∈ f ⁻¹' s ho' : IsLimit o a : Ordinal.{u} ha : a < f o has : Set.Ioo a (f o) ⊆ s ⊢ ∃ a, a < o ∧ Set.Ioo a o ⊆ f ⁻¹' s ** rw [← IsNormal.bsup_eq.{u, u} h ho', lt_bsup] at ha ** case refine'_1.intro.intro s✝ : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s o : Ordinal.{u} ho : o ∈ f ⁻¹' s ho' : IsLimit o a : Ordinal.{u} ha : ∃ i hi, a < f i has : Set.Ioo a (f o) ⊆ s ⊢ ∃ a, a < o ∧ Set.Ioo a o ⊆ f ⁻¹' s ** rcases ha with ⟨b, hb, hab⟩ ** case refine'_1.intro.intro.intro.intro s✝ : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : IsNormal f s : Set Ordinal.{u} hs : ∀ (o : Ordinal.{u}), o ∈ s → IsLimit o → ∃ a, a < o ∧ Set.Ioo a o ⊆ s o : Ordinal.{u} ho : o ∈ f ⁻¹' s ho' : IsLimit o a : Ordinal.{u} has : Set.Ioo a (f o) ⊆ s b : Ordinal.{u} hb : b < o hab : a < f b ⊢ ∃ a, a < o ∧ Set.Ioo a o ⊆ f ⁻¹' s ** exact
⟨b, hb, fun c hc =>
Set.mem_preimage.2 (has ⟨hab.trans (h.strictMono hc.1), h.strictMono hc.2⟩)⟩ ** case refine'_2 s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} ⊢ StrictMono f ∧ Continuous f → IsNormal f ** rw [isNormal_iff_strictMono_limit] ** case refine'_2 s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} ⊢ StrictMono f ∧ Continuous f → StrictMono f ∧ ∀ (o : Ordinal.{u}), IsLimit o → ∀ (a : Ordinal.{u}), (∀ (b : Ordinal.{u}), b < o → f b ≤ a) → f o ≤ a ** rintro ⟨h, h'⟩ ** case refine'_2.intro s : Set Ordinal.{u} a : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h : StrictMono f h' : Continuous f ⊢ StrictMono f ∧ ∀ (o : Ordinal.{u}), IsLimit o → ∀ (a : Ordinal.{u}), (∀ (b : Ordinal.{u}), b < o → f b ≤ a) → f o ≤ a ** refine' ⟨h, fun o ho a h => _⟩ ** case refine'_2.intro s : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h✝ : StrictMono f h' : Continuous f o : Ordinal.{u} ho : IsLimit o a : Ordinal.{u} h : ∀ (b : Ordinal.{u}), b < o → f b ≤ a ⊢ f o ≤ a ** suffices : o ∈ f ⁻¹' Set.Iic a ** case refine'_2.intro s : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h✝ : StrictMono f h' : Continuous f o : Ordinal.{u} ho : IsLimit o a : Ordinal.{u} h : ∀ (b : Ordinal.{u}), b < o → f b ≤ a this : o ∈ f ⁻¹' Set.Iic a ⊢ f o ≤ a case this s : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h✝ : StrictMono f h' : Continuous f o : Ordinal.{u} ho : IsLimit o a : Ordinal.{u} h : ∀ (b : Ordinal.{u}), b < o → f b ≤ a ⊢ o ∈ f ⁻¹' Set.Iic a ** exact Set.mem_preimage.1 this ** case this s : Set Ordinal.{u} a✝ : Ordinal.{u} f : Ordinal.{u} → Ordinal.{u} h✝ : StrictMono f h' : Continuous f o : Ordinal.{u} ho : IsLimit o a : Ordinal.{u} h : ∀ (b : Ordinal.{u}), b < o → f b ≤ a ⊢ o ∈ f ⁻¹' Set.Iic a ** rw [mem_closed_iff_sup (IsClosed.preimage h' (@isClosed_Iic _ _ _ _ a))] ** Qed | |
Ordinal.enumOrd_isNormal_iff_isClosed ** s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s ⊢ IsNormal (enumOrd s) ↔ IsClosed s ** have Hs := enumOrd_strictMono hs ** s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) ⊢ IsNormal (enumOrd s) ↔ IsClosed s ** refine'
⟨fun h => isClosed_iff_sup.2 fun {ι} hι f hf => _, fun h =>
(isNormal_iff_strictMono_limit _).2 ⟨Hs, fun a ha o H => _⟩⟩ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s ⊢ sup f ∈ s ** let g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso hs).symm ⟨_, hf i⟩ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } ⊢ sup f ∈ s ** suffices enumOrd s (sup.{u, u} g) = sup.{u, u} f by
rw [← this]
exact enumOrd_mem hs _ ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } ⊢ enumOrd s (sup g) = sup f ** rw [@IsNormal.sup.{u, u, u} _ h ι g hι] ** case refine'_1 s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } ⊢ sup (enumOrd s ∘ g) = sup f ** congr ** case refine'_1.e_f s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } ⊢ enumOrd s ∘ g = f ** ext x ** case refine'_1.e_f.h s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } x : ι ⊢ (enumOrd s ∘ g) x = f x ** change ((enumOrdOrderIso hs) _).val = f x ** case refine'_1.e_f.h s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } x : ι ⊢ ↑(↑(enumOrdOrderIso hs) (g x)) = f x ** rw [OrderIso.apply_symm_apply] ** s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } this : enumOrd s (sup g) = sup f ⊢ sup f ∈ s ** rw [← this] ** s : Set Ordinal.{u} a : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsNormal (enumOrd s) ι : Type u hι : Nonempty ι f : ι → Ordinal.{u} hf : ∀ (i : ι), f i ∈ s g : ι → Ordinal.{u} := fun i => ↑(OrderIso.symm (enumOrdOrderIso hs)) { val := f i, property := (_ : f i ∈ s) } this : enumOrd s (sup g) = sup f ⊢ enumOrd s (sup g) ∈ s ** exact enumOrd_mem hs _ ** case refine'_2 s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : IsClosed s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o ⊢ enumOrd s a ≤ o ** rw [isClosed_iff_bsup] at h ** case refine'_2 s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o ⊢ enumOrd s a ≤ o ** suffices : enumOrd s a ≤ bsup.{u, u} a fun b (_ : b < a) => enumOrd s b ** case refine'_2 s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o this : enumOrd s a ≤ bsup a fun b x => enumOrd s b ⊢ enumOrd s a ≤ o case this s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o ⊢ enumOrd s a ≤ bsup a fun b x => enumOrd s b ** exact this.trans (bsup_le H) ** case this s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o ⊢ enumOrd s a ≤ bsup a fun b x => enumOrd s b ** cases' enumOrd_surjective hs _
(h ha.1 (fun b _ => enumOrd s b) fun b _ => enumOrd_mem hs b) with
b hb ** case this.intro s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b ⊢ enumOrd s a ≤ bsup a fun b x => enumOrd s b ** rw [← hb] ** case this.intro s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b ⊢ enumOrd s a ≤ enumOrd s b ** apply Hs.monotone ** case this.intro.a s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b ⊢ a ≤ b ** by_contra' hba ** case this.intro.a s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b hba : b < a ⊢ False ** apply (Hs (lt_succ b)).not_le ** case this.intro.a s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b hba : b < a ⊢ enumOrd s (succ b) ≤ enumOrd s b ** rw [hb] ** case this.intro.a s : Set Ordinal.{u} a✝ : Ordinal.{u} hs : Set.Unbounded (fun x x_1 => x < x_1) s Hs : StrictMono (enumOrd s) h : ∀ {o : Ordinal.{u}}, o ≠ 0 → ∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → bsup o f ∈ s a : Ordinal.{u} ha : IsLimit a o : Ordinal.{u} H : ∀ (b : Ordinal.{u}), b < a → enumOrd s b ≤ o b : Ordinal.{u} hb : enumOrd s b = bsup a fun b x => enumOrd s b hba : b < a ⊢ enumOrd s (succ b) ≤ bsup a fun b x => enumOrd s b ** exact le_bsup.{u, u} _ _ (ha.2 _ hba) ** Qed | |
Pretrivialization.coe_linearMapAt ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B ⊢ ↑(Pretrivialization.linearMapAt R e b) = fun y => if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0 ** rw [Pretrivialization.linearMapAt] ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B ⊢ ↑(if hb : b ∈ e.baseSet then ↑(linearEquivAt R e b hb) else 0) = fun y => if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0 ** split_ifs <;> rfl ** Qed | |
Pretrivialization.coe_linearMapAt_of_mem ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B hb : b ∈ e.baseSet ⊢ ↑(Pretrivialization.linearMapAt R e b) = fun y => (↑e { proj := b, snd := y }).2 ** simp_rw [coe_linearMapAt, if_pos hb] ** Qed | |
Pretrivialization.linearMapAt_apply ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B y : E b ⊢ ↑(Pretrivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0 ** rw [coe_linearMapAt] ** Qed | |
Pretrivialization.symmₗ_linearMapAt ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B hb : b ∈ e.baseSet y : E b ⊢ ↑(Pretrivialization.symmₗ R e b) (↑(Pretrivialization.linearMapAt R e b) y) = y ** rw [e.linearMapAt_def_of_mem hb] ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B hb : b ∈ e.baseSet y : E b ⊢ ↑(Pretrivialization.symmₗ R e b) (↑↑(linearEquivAt R e b hb) y) = y ** exact (e.linearEquivAt R b hb).left_inv y ** Qed | |
Pretrivialization.linearMapAt_symmₗ ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B hb : b ∈ e.baseSet y : F ⊢ ↑(Pretrivialization.linearMapAt R e b) (↑(Pretrivialization.symmₗ R e b) y) = y ** rw [e.linearMapAt_def_of_mem hb] ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁷ : Semiring R inst✝⁶ : TopologicalSpace F inst✝⁵ : TopologicalSpace B e✝ : Pretrivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Pretrivialization F TotalSpace.proj inst✝ : Pretrivialization.IsLinear R e b : B hb : b ∈ e.baseSet y : F ⊢ ↑↑(linearEquivAt R e b hb) (↑(Pretrivialization.symmₗ R e b) y) = y ** exact (e.linearEquivAt R b hb).right_inv y ** Qed | |
Trivialization.coe_linearMapAt_of_mem ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : Semiring R inst✝⁷ : TopologicalSpace F inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet ⊢ ↑(Trivialization.linearMapAt R e b) = fun y => (↑e { proj := b, snd := y }).2 ** simp_rw [coe_linearMapAt, if_pos hb] ** Qed | |
Trivialization.linearMapAt_apply ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : Semiring R inst✝⁷ : TopologicalSpace F inst✝⁶ : TopologicalSpace B inst✝⁵ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁴ : AddCommMonoid F inst✝³ : Module R F inst✝² : (x : B) → AddCommMonoid (E x) inst✝¹ : (x : B) → Module R (E x) e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B y : E b ⊢ ↑(Trivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0 ** rw [coe_linearMapAt] ** Qed | |
Trivialization.symm_coordChangeL ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e'.baseSet ∩ e.baseSet ⊢ ContinuousLinearEquiv.symm (coordChangeL R e e' b) = coordChangeL R e' e b ** apply ContinuousLinearEquiv.toLinearEquiv_injective ** case a R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e'.baseSet ∩ e.baseSet ⊢ (ContinuousLinearEquiv.symm (coordChangeL R e e' b)).toLinearEquiv = (coordChangeL R e' e b).toLinearEquiv ** rw [coe_coordChangeL' e' e hb, (coordChangeL R e e' b).symm_toLinearEquiv,
coe_coordChangeL' e e' hb.symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm] ** Qed | |
Trivialization.mk_coordChangeL ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ (b, ↑(coordChangeL R e e' b) y) = ↑e' { proj := b, snd := Trivialization.symm e b y } ** ext ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ (b, ↑(coordChangeL R e e' b) y).1 = (↑e' { proj := b, snd := Trivialization.symm e b y }).1 ** rw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1] ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y)).proj ∈ e'.baseSet ** rw [e.proj_symm_apply' hb.1] ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ b ∈ e'.baseSet ** exact hb.2 ** case h₂ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ (b, ↑(coordChangeL R e e' b) y).2 = (↑e' { proj := b, snd := Trivialization.symm e b y }).2 ** exact e.coordChangeL_apply e' hb y ** Qed | |
Trivialization.apply_symm_apply_eq_coordChangeL ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ ↑e' (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, v)) = (b, ↑(coordChangeL R e e' b) v) ** rw [e.mk_coordChangeL e' hb, e.mk_symm hb.1] ** Qed | |
Trivialization.coordChangeL_apply' ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : Semiring R inst✝⁸ : TopologicalSpace F inst✝⁷ : TopologicalSpace B inst✝⁶ : TopologicalSpace (TotalSpace F E) e✝ : Trivialization F TotalSpace.proj x : TotalSpace F E b✝ : B y✝ : E b✝ inst✝⁵ : AddCommMonoid F inst✝⁴ : Module R F inst✝³ : (x : B) → AddCommMonoid (E x) inst✝² : (x : B) → Module R (E x) e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet y : F ⊢ ↑(coordChangeL R e e' b) y = (↑e' (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y))).2 ** rw [e.coordChangeL_apply e' hb, e.mk_symm hb.1] ** Qed | |
Trivialization.apply_eq_prod_continuousLinearEquivAt ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : E b ⊢ ↑e { proj := b, snd := z } = (b, ↑(continuousLinearEquivAt R e b hb) z) ** ext ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : E b ⊢ (↑e { proj := b, snd := z }).1 = (b, ↑(continuousLinearEquivAt R e b hb) z).1 ** refine' e.coe_fst _ ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : E b ⊢ { proj := b, snd := z } ∈ e.source ** rw [e.source_eq] ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : E b ⊢ { proj := b, snd := z } ∈ TotalSpace.proj ⁻¹' e.baseSet ** exact hb ** case h₂ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : E b ⊢ (↑e { proj := b, snd := z }).2 = (b, ↑(continuousLinearEquivAt R e b hb) z).2 ** simp only [coe_coe, continuousLinearEquivAt_apply] ** Qed | |
Trivialization.zeroSection ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e x : B hx : x ∈ e.baseSet ⊢ ↑e (zeroSection F E x) = (x, 0) ** simp_rw [zeroSection, e.apply_eq_prod_continuousLinearEquivAt R x hx 0, map_zero] ** Qed | |
Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F ⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) = { proj := b, snd := ↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z } ** have h : (b, z) ∈ e.target ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F h : (b, z) ∈ e.target ⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) = { proj := b, snd := ↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z } ** apply e.injOn (e.map_target h) ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F ⊢ (b, z) ∈ e.target ** rw [e.target_eq] ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F ⊢ (b, z) ∈ e.baseSet ×ˢ univ ** exact ⟨hb, mem_univ _⟩ ** case a R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F h : (b, z) ∈ e.target ⊢ { proj := b, snd := ↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z } ∈ e.source ** simpa only [e.source_eq, mem_preimage] ** case a R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁹ : NontriviallyNormedField R inst✝⁸ : (x : B) → AddCommMonoid (E x) inst✝⁷ : (x : B) → Module R (E x) inst✝⁶ : NormedAddCommGroup F inst✝⁵ : NormedSpace R F inst✝⁴ : TopologicalSpace B inst✝³ : TopologicalSpace (TotalSpace F E) inst✝² : (x : B) → TopologicalSpace (E x) inst✝¹ : FiberBundle F E e : Trivialization F TotalSpace.proj inst✝ : Trivialization.IsLinear R e b : B hb : b ∈ e.baseSet z : F h : (b, z) ∈ e.target ⊢ ↑e.toLocalHomeomorph (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z)) = ↑e.toLocalHomeomorph { proj := b, snd := ↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z } ** simp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuousLinearEquivAt R b hb,
ContinuousLinearEquiv.apply_symm_apply] ** Qed | |
Trivialization.comp_continuousLinearEquivAt_eq_coord_change ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝¹⁰ : NontriviallyNormedField R inst✝⁹ : (x : B) → AddCommMonoid (E x) inst✝⁸ : (x : B) → Module R (E x) inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace R F inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (x : B) → TopologicalSpace (E x) inst✝² : FiberBundle F E e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet ⊢ ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet))) (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)) = coordChangeL R e e' b ** ext v ** case h.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝¹⁰ : NontriviallyNormedField R inst✝⁹ : (x : B) → AddCommMonoid (E x) inst✝⁸ : (x : B) → Module R (E x) inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace R F inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (x : B) → TopologicalSpace (E x) inst✝² : FiberBundle F E e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet))) (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet))) v = ↑(coordChangeL R e e' b) v ** rw [coordChangeL_apply e e' hb] ** case h.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝¹⁰ : NontriviallyNormedField R inst✝⁹ : (x : B) → AddCommMonoid (E x) inst✝⁸ : (x : B) → Module R (E x) inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace R F inst✝⁵ : TopologicalSpace B inst✝⁴ : TopologicalSpace (TotalSpace F E) inst✝³ : (x : B) → TopologicalSpace (E x) inst✝² : FiberBundle F E e e' : Trivialization F TotalSpace.proj inst✝¹ : Trivialization.IsLinear R e inst✝ : Trivialization.IsLinear R e' b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet))) (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet))) v = (↑e' { proj := b, snd := Trivialization.symm e b v }).2 ** rfl ** Qed | |
VectorBundleCore.coordChange_linear_comp ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι i j k : ι x : B hx : x ∈ baseSet Z i ∩ baseSet Z j ∩ baseSet Z k ⊢ ContinuousLinearMap.comp (coordChange Z j k x) (coordChange Z i j x) = coordChange Z i k x ** ext v ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι i j k : ι x : B hx : x ∈ baseSet Z i ∩ baseSet Z j ∩ baseSet Z k v : F ⊢ ↑(ContinuousLinearMap.comp (coordChange Z j k x) (coordChange Z i j x)) v = ↑(coordChange Z i k x) v ** exact Z.coordChange_comp i j k x hx v ** Qed | |
VectorBundleCore.localTriv_symm_apply ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : F ⊢ Trivialization.symm (localTriv Z i) b v = ↑(coordChange Z i (indexAt Z b) b) v ** apply (Z.localTriv i).symm_apply hb v ** Qed | |
VectorBundleCore.localTriv_coordChange_eq ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i ∩ baseSet Z j v : F ⊢ ↑(Trivialization.coordChangeL R (localTriv Z i) (localTriv Z j) b) v = ↑(coordChange Z i j b) v ** rw [Trivialization.coordChangeL_apply', localTriv_symm_fst, localTriv_apply, coordChange_comp] ** case a R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i ∩ baseSet Z j v : F ⊢ { proj := (b, v).1, snd := ↑(coordChange Z i (indexAt Z (b, v).1) (b, v).1) (b, v).2 }.proj ∈ baseSet Z i ∩ baseSet Z (indexAt Z { proj := (b, v).1, snd := ↑(coordChange Z i (indexAt Z (b, v).1) (b, v).1) (b, v).2 }.proj) ∩ baseSet Z j case hb R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i ∩ baseSet Z j v : F ⊢ b ∈ (localTriv Z i).baseSet ∩ (localTriv Z j).baseSet ** exacts [⟨⟨hb.1, Z.mem_baseSet_at b⟩, hb.2⟩, hb] ** Qed | |
VectorBundleCore.mem_source_at ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b : B a : F i j : ι ⊢ { proj := b, snd := a } ∈ (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source ** rw [localTrivAt, mem_localTriv_source] ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b : B a : F i j : ι ⊢ { proj := b, snd := a }.proj ∈ baseSet Z (indexAt Z b) ** exact Z.mem_baseSet_at b ** Qed | |
VectorBundleCore.localTriv_continuousLinearMapAt ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i ⊢ Trivialization.continuousLinearMapAt R (localTriv Z i) b = coordChange Z (indexAt Z b) i b ** ext1 v ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : Fiber Z b ⊢ ↑(Trivialization.continuousLinearMapAt R (localTriv Z i) b) v = ↑(coordChange Z (indexAt Z b) i b) v ** rw [(Z.localTriv i).continuousLinearMapAt_apply R, (Z.localTriv i).coe_linearMapAt_of_mem] ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : Fiber Z b ⊢ (fun y => (↑(localTriv Z i) { proj := b, snd := y }).2) v = ↑(coordChange Z (indexAt Z b) i b) v case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : Fiber Z b ⊢ b ∈ (localTriv Z i).baseSet ** exacts [rfl, hb] ** Qed | |
VectorBundleCore.localTriv_symmL ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i ⊢ Trivialization.symmL R (localTriv Z i) b = coordChange Z i (indexAt Z b) b ** ext1 v ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : F ⊢ ↑(Trivialization.symmL R (localTriv Z i) b) v = ↑(coordChange Z i (indexAt Z b) b) v ** rw [(Z.localTriv i).symmL_apply R, (Z.localTriv i).symm_apply] ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : F ⊢ cast (_ : Fiber Z (↑(LocalHomeomorph.symm (localTriv Z i).toLocalHomeomorph) (b, v)).proj = Fiber Z b) (↑(LocalHomeomorph.symm (localTriv Z i).toLocalHomeomorph) (b, v)).snd = ↑(coordChange Z i (indexAt Z b) b) v case h.hb R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁸ : NontriviallyNormedField R inst✝⁷ : (x : B) → AddCommMonoid (E x) inst✝⁶ : (x : B) → Module R (E x) inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace R F inst✝³ : TopologicalSpace B inst✝² : TopologicalSpace (TotalSpace F E) inst✝¹ : (x : B) → TopologicalSpace (E x) inst✝ : FiberBundle F E ι : Type u_5 Z : VectorBundleCore R B F ι b✝ : B a : F i j : ι b : B hb : b ∈ baseSet Z i v : F ⊢ b ∈ (localTriv Z i).baseSet ** exacts [rfl, hb] ** Qed | |
VectorPrebundle.mk_coordChange ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E e e' : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas he' : e' ∈ a.pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ (b, ↑(coordChange a he he' b) v) = ↑e' { proj := b, snd := Pretrivialization.symm e b v } ** ext ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E e e' : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas he' : e' ∈ a.pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ (b, ↑(coordChange a he he' b) v).1 = (↑e' { proj := b, snd := Pretrivialization.symm e b v }).1 ** rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1] ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E e e' : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas he' : e' ∈ a.pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ (↑(LocalEquiv.symm e.toLocalEquiv) (b, v)).proj ∈ e'.baseSet ** rw [e.proj_symm_apply' hb.1] ** case h₁ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E e e' : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas he' : e' ∈ a.pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ b ∈ e'.baseSet ** exact hb.2 ** case h₂ R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E e e' : Pretrivialization F TotalSpace.proj he : e ∈ a.pretrivializationAtlas he' : e' ∈ a.pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ (b, ↑(coordChange a he he' b) v).2 = (↑e' { proj := b, snd := Pretrivialization.symm e b v }).2 ** exact a.coordChange_apply he he' hb v ** Qed | |
VectorPrebundle.toVectorBundle ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a ⊢ ∀ (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear R e ** rintro _ ⟨e, he, rfl⟩ ** case mk.intro.intro R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas ⊢ Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) ** apply linear_trivializationOfMemPretrivializationAtlas ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a ⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'], ContinuousOn (fun b => ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet) ** rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩ ** case mk.intro.intro.mk.intro.intro R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas ⊢ ContinuousOn (fun b => ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)) ((FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩ (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet) ** refine (a.continuousOn_coordChange he he').congr fun b hb ↦ ?_ ** case mk.intro.intro.mk.intro.intro R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet ⊢ ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) = coordChange a he he' b ** ext v ** case mk.intro.intro.mk.intro.intro.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F ⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) v = ↑(coordChange a he he' b) v ** haveI h₁ := a.linear_trivializationOfMemPretrivializationAtlas he ** case mk.intro.intro.mk.intro.intro.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F h₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he) ⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) v = ↑(coordChange a he he' b) v ** haveI h₂ := a.linear_trivializationOfMemPretrivializationAtlas he' ** case mk.intro.intro.mk.intro.intro.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F h₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he) h₂ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he') ⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) v = ↑(coordChange a he he' b) v ** rw [trivializationOfMemPretrivializationAtlas] at h₁ h₂ ** case mk.intro.intro.mk.intro.intro.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F h₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) h₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') ⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) v = ↑(coordChange a he he' b) v ** rw [a.coordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
Trivialization.coordChangeL_apply] ** case mk.intro.intro.mk.intro.intro.h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F h₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) h₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') ⊢ (↑(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') { proj := b, snd := Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b v }).2 = (↑e' { proj := b, snd := Pretrivialization.symm e b v }).2 case mk.intro.intro.mk.intro.intro.h.hb R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝⁶ : NontriviallyNormedField R inst✝⁵ : (x : B) → AddCommMonoid (E x) inst✝⁴ : (x : B) → Module R (E x) inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace R F inst✝¹ : TopologicalSpace B inst✝ : (x : B) → TopologicalSpace (E x) a : VectorPrebundle R F E this✝ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a this : FiberBundle F E := toFiberBundle a e : Pretrivialization F TotalSpace.proj he : e ∈ (toFiberPrebundle a).pretrivializationAtlas e' : Pretrivialization F TotalSpace.proj he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas b : B hb : b ∈ e.baseSet ∩ e'.baseSet v : F h₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) h₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') ⊢ b ∈ (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩ (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet ** exacts [rfl, hb] ** Qed | |
ContinuousLinearMap.inCoordinates_eq ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝²² : NontriviallyNormedField R inst✝²¹ : (x : B) → AddCommMonoid (E x) inst✝²⁰ : (x : B) → Module R (E x) inst✝¹⁹ : NormedAddCommGroup F inst✝¹⁸ : NormedSpace R F inst✝¹⁷ : TopologicalSpace B inst✝¹⁶ : (x : B) → TopologicalSpace (E x) 𝕜₁ : Type u_5 𝕜₂ : Type u_6 inst✝¹⁵ : NontriviallyNormedField 𝕜₁ inst✝¹⁴ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ B' : Type u_7 inst✝¹³ : TopologicalSpace B' inst✝¹² : NormedSpace 𝕜₁ F inst✝¹¹ : (x : B) → Module 𝕜₁ (E x) inst✝¹⁰ : TopologicalSpace (TotalSpace F E) F' : Type u_8 inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜₂ F' E' : B' → Type u_9 inst✝⁷ : (x : B') → AddCommMonoid (E' x) inst✝⁶ : (x : B') → Module 𝕜₂ (E' x) inst✝⁵ : TopologicalSpace (TotalSpace F' E') inst✝⁴ : FiberBundle F E inst✝³ : VectorBundle 𝕜₁ F E inst✝² : (x : B') → TopologicalSpace (E' x) inst✝¹ : FiberBundle F' E' inst✝ : VectorBundle 𝕜₂ F' E' x₀ x : B y₀ y : B' ϕ : E x →SL[σ] E' y hx : x ∈ (trivializationAt F E x₀).baseSet hy : y ∈ (trivializationAt F' E' y₀).baseSet ⊢ inCoordinates F E F' E' x₀ x y₀ y ϕ = comp (↑(Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F' E' y₀) y hy)) (comp ϕ ↑(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F E x₀) x hx))) ** ext ** case h R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝²² : NontriviallyNormedField R inst✝²¹ : (x : B) → AddCommMonoid (E x) inst✝²⁰ : (x : B) → Module R (E x) inst✝¹⁹ : NormedAddCommGroup F inst✝¹⁸ : NormedSpace R F inst✝¹⁷ : TopologicalSpace B inst✝¹⁶ : (x : B) → TopologicalSpace (E x) 𝕜₁ : Type u_5 𝕜₂ : Type u_6 inst✝¹⁵ : NontriviallyNormedField 𝕜₁ inst✝¹⁴ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ B' : Type u_7 inst✝¹³ : TopologicalSpace B' inst✝¹² : NormedSpace 𝕜₁ F inst✝¹¹ : (x : B) → Module 𝕜₁ (E x) inst✝¹⁰ : TopologicalSpace (TotalSpace F E) F' : Type u_8 inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜₂ F' E' : B' → Type u_9 inst✝⁷ : (x : B') → AddCommMonoid (E' x) inst✝⁶ : (x : B') → Module 𝕜₂ (E' x) inst✝⁵ : TopologicalSpace (TotalSpace F' E') inst✝⁴ : FiberBundle F E inst✝³ : VectorBundle 𝕜₁ F E inst✝² : (x : B') → TopologicalSpace (E' x) inst✝¹ : FiberBundle F' E' inst✝ : VectorBundle 𝕜₂ F' E' x₀ x : B y₀ y : B' ϕ : E x →SL[σ] E' y hx : x ∈ (trivializationAt F E x₀).baseSet hy : y ∈ (trivializationAt F' E' y₀).baseSet x✝ : F ⊢ ↑(inCoordinates F E F' E' x₀ x y₀ y ϕ) x✝ = ↑(comp (↑(Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F' E' y₀) y hy)) (comp ϕ ↑(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F E x₀) x hx)))) x✝ ** simp_rw [inCoordinates, ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe,
Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq] ** Qed | |
VectorBundleCore.inCoordinates_eq ** R : Type u_1 B : Type u_2 F : Type u_3 E : B → Type u_4 inst✝²² : NontriviallyNormedField R inst✝²¹ : (x : B) → AddCommMonoid (E x) inst✝²⁰ : (x : B) → Module R (E x) inst✝¹⁹ : NormedAddCommGroup F inst✝¹⁸ : NormedSpace R F inst✝¹⁷ : TopologicalSpace B inst✝¹⁶ : (x : B) → TopologicalSpace (E x) 𝕜₁ : Type u_5 𝕜₂ : Type u_6 inst✝¹⁵ : NontriviallyNormedField 𝕜₁ inst✝¹⁴ : NontriviallyNormedField 𝕜₂ σ : 𝕜₁ →+* 𝕜₂ B' : Type u_7 inst✝¹³ : TopologicalSpace B' inst✝¹² : NormedSpace 𝕜₁ F inst✝¹¹ : (x : B) → Module 𝕜₁ (E x) inst✝¹⁰ : TopologicalSpace (TotalSpace F E) F' : Type u_8 inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜₂ F' E' : B' → Type u_9 inst✝⁷ : (x : B') → AddCommMonoid (E' x) inst✝⁶ : (x : B') → Module 𝕜₂ (E' x) inst✝⁵ : TopologicalSpace (TotalSpace F' E') inst✝⁴ : FiberBundle F E inst✝³ : VectorBundle 𝕜₁ F E inst✝² : (x : B') → TopologicalSpace (E' x) inst✝¹ : FiberBundle F' E' inst✝ : VectorBundle 𝕜₂ F' E' ι : Type u_10 ι' : Type u_11 Z : VectorBundleCore 𝕜₁ B F ι Z' : VectorBundleCore 𝕜₂ B' F' ι' x₀ x : B y₀ y : B' ϕ : F →SL[σ] F' hx : x ∈ VectorBundleCore.baseSet Z (VectorBundleCore.indexAt Z x₀) hy : y ∈ VectorBundleCore.baseSet Z' (VectorBundleCore.indexAt Z' y₀) ⊢ inCoordinates F (VectorBundleCore.Fiber Z) F' (VectorBundleCore.Fiber Z') x₀ x y₀ y ϕ = comp (VectorBundleCore.coordChange Z' (VectorBundleCore.indexAt Z' y) (VectorBundleCore.indexAt Z' y₀) y) (comp ϕ (VectorBundleCore.coordChange Z (VectorBundleCore.indexAt Z x₀) (VectorBundleCore.indexAt Z x) x)) ** simp_rw [inCoordinates, Z'.trivializationAt_continuousLinearMapAt hy,
Z.trivializationAt_symmL hx] ** Qed | |
SetTheory.PGame.numeric_def ** x : PGame ⊢ Numeric x ↔ (∀ (i : LeftMoves x) (j : RightMoves x), moveLeft x i < moveRight x j) ∧ (∀ (i : LeftMoves x), Numeric (moveLeft x i)) ∧ ∀ (j : RightMoves x), Numeric (moveRight x j) ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame ⊢ Numeric (mk α✝ β✝ a✝¹ a✝) ↔ (∀ (i : LeftMoves (mk α✝ β✝ a✝¹ a✝)) (j : RightMoves (mk α✝ β✝ a✝¹ a✝)), moveLeft (mk α✝ β✝ a✝¹ a✝) i < moveRight (mk α✝ β✝ a✝¹ a✝) j) ∧ (∀ (i : LeftMoves (mk α✝ β✝ a✝¹ a✝)), Numeric (moveLeft (mk α✝ β✝ a✝¹ a✝) i)) ∧ ∀ (j : RightMoves (mk α✝ β✝ a✝¹ a✝)), Numeric (moveRight (mk α✝ β✝ a✝¹ a✝) j) ** rfl ** Qed | |
SetTheory.PGame.Numeric.left_lt_right ** x : PGame o : Numeric x i : LeftMoves x j : RightMoves x ⊢ moveLeft x i < moveRight x j ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame o : Numeric (PGame.mk α✝ β✝ a✝¹ a✝) i : LeftMoves (PGame.mk α✝ β✝ a✝¹ a✝) j : RightMoves (PGame.mk α✝ β✝ a✝¹ a✝) ⊢ moveLeft (PGame.mk α✝ β✝ a✝¹ a✝) i < moveRight (PGame.mk α✝ β✝ a✝¹ a✝) j ** exact o.1 i j ** Qed | |
SetTheory.PGame.Numeric.moveLeft ** x : PGame o : Numeric x i : LeftMoves x ⊢ Numeric (PGame.moveLeft x i) ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame o : Numeric (PGame.mk α✝ β✝ a✝¹ a✝) i : LeftMoves (PGame.mk α✝ β✝ a✝¹ a✝) ⊢ Numeric (PGame.moveLeft (PGame.mk α✝ β✝ a✝¹ a✝) i) ** exact o.2.1 i ** Qed | |
SetTheory.PGame.Numeric.moveRight ** x : PGame o : Numeric x j : RightMoves x ⊢ Numeric (PGame.moveRight x j) ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame o : Numeric (PGame.mk α✝ β✝ a✝¹ a✝) j : RightMoves (PGame.mk α✝ β✝ a✝¹ a✝) ⊢ Numeric (PGame.moveRight (PGame.mk α✝ β✝ a✝¹ a✝) j) ** exact o.2.2 j ** Qed | |
SetTheory.PGame.Relabelling.numeric_imp ** x y : PGame r : x ≡r y ox : Numeric x ⊢ Numeric y ** induction' x using PGame.moveRecOn with x IHl IHr generalizing y ** case IH x✝ y✝ : PGame r✝ : x✝ ≡r y✝ ox✝ : Numeric x✝ x : PGame IHl : ∀ (i : LeftMoves x) {y : PGame}, PGame.moveLeft x i ≡r y → Numeric (PGame.moveLeft x i) → Numeric y IHr : ∀ (j : RightMoves x) {y : PGame}, PGame.moveRight x j ≡r y → Numeric (PGame.moveRight x j) → Numeric y y : PGame r : x ≡r y ox : Numeric x ⊢ Numeric y ** apply Numeric.mk (fun i j => ?_) (fun i => ?_) fun j => ?_ ** x✝ y✝ : PGame r✝ : x✝ ≡r y✝ ox✝ : Numeric x✝ x : PGame IHl : ∀ (i : LeftMoves x) {y : PGame}, PGame.moveLeft x i ≡r y → Numeric (PGame.moveLeft x i) → Numeric y IHr : ∀ (j : RightMoves x) {y : PGame}, PGame.moveRight x j ≡r y → Numeric (PGame.moveRight x j) → Numeric y y : PGame r : x ≡r y ox : Numeric x i : LeftMoves y j : RightMoves y ⊢ PGame.moveLeft y i < PGame.moveRight y j ** rw [← lt_congr (r.moveLeftSymm i).equiv (r.moveRightSymm j).equiv] ** x✝ y✝ : PGame r✝ : x✝ ≡r y✝ ox✝ : Numeric x✝ x : PGame IHl : ∀ (i : LeftMoves x) {y : PGame}, PGame.moveLeft x i ≡r y → Numeric (PGame.moveLeft x i) → Numeric y IHr : ∀ (j : RightMoves x) {y : PGame}, PGame.moveRight x j ≡r y → Numeric (PGame.moveRight x j) → Numeric y y : PGame r : x ≡r y ox : Numeric x i : LeftMoves y j : RightMoves y ⊢ PGame.moveLeft x (↑(leftMovesEquiv r).symm i) < PGame.moveRight x (↑(rightMovesEquiv r).symm j) ** apply ox.left_lt_right ** x✝ y✝ : PGame r✝ : x✝ ≡r y✝ ox✝ : Numeric x✝ x : PGame IHl : ∀ (i : LeftMoves x) {y : PGame}, PGame.moveLeft x i ≡r y → Numeric (PGame.moveLeft x i) → Numeric y IHr : ∀ (j : RightMoves x) {y : PGame}, PGame.moveRight x j ≡r y → Numeric (PGame.moveRight x j) → Numeric y y : PGame r : x ≡r y ox : Numeric x i : LeftMoves y ⊢ Numeric (PGame.moveLeft y i) ** exact IHl _ (r.moveLeftSymm i) (ox.moveLeft _) ** x✝ y✝ : PGame r✝ : x✝ ≡r y✝ ox✝ : Numeric x✝ x : PGame IHl : ∀ (i : LeftMoves x) {y : PGame}, PGame.moveLeft x i ≡r y → Numeric (PGame.moveLeft x i) → Numeric y IHr : ∀ (j : RightMoves x) {y : PGame}, PGame.moveRight x j ≡r y → Numeric (PGame.moveRight x j) → Numeric y y : PGame r : x ≡r y ox : Numeric x j : RightMoves y ⊢ Numeric (PGame.moveRight y j) ** exact IHr _ (r.moveRightSymm j) (ox.moveRight _) ** Qed | |
SetTheory.PGame.lf_asymm ** x y : PGame ox : Numeric x oy : Numeric y ⊢ x ⧏ y → ¬y ⧏ x ** refine' numeric_rec (C := fun x => ∀ z (_oz : Numeric z), x ⧏ z → ¬z ⧏ x)
(fun xl xr xL xR hx _oxl _oxr IHxl IHxr => _) x ox y oy ** x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) ⊢ (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (mk xl xr xL xR) ** refine' numeric_rec fun yl yr yL yR hy oyl oyr _IHyl _IHyr => _ ** x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR ⊢ mk xl xr xL xR ⧏ mk yl yr yL yR → ¬mk yl yr yL yR ⧏ mk xl xr xL xR ** rw [mk_lf_mk, mk_lf_mk] ** x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR ⊢ ((∃ i, mk xl xr xL xR ≤ yL i) ∨ ∃ j, xR j ≤ mk yl yr yL yR) → ¬((∃ i, mk yl yr yL yR ≤ xL i) ∨ ∃ j, yR j ≤ mk xl xr xL xR) ** rintro (⟨i, h₁⟩ | ⟨j, h₁⟩) (⟨i, h₂⟩ | ⟨j, h₂⟩) ** case inl.intro.inl.intro x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR i✝ : yl h₁ : mk xl xr xL xR ≤ yL i✝ i : xl h₂ : mk yl yr yL yR ≤ xL i ⊢ False ** exact IHxl _ _ (oyl _) (h₁.moveLeft_lf _) (h₂.moveLeft_lf _) ** case inl.intro.inr.intro x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR i : yl h₁ : mk xl xr xL xR ≤ yL i j : yr h₂ : yR j ≤ mk xl xr xL xR ⊢ False ** exact (le_trans h₂ h₁).not_gf (lf_of_lt (hy _ _)) ** case inr.intro.inl.intro x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR j : xr h₁ : xR j ≤ mk yl yr yL yR i : xl h₂ : mk yl yr yL yR ≤ xL i ⊢ False ** exact (le_trans h₁ h₂).not_gf (lf_of_lt (hx _ _)) ** case inr.intro.inr.intro x y : PGame ox : Numeric x oy : Numeric y xl xr : Type u_1 xL : xl → PGame xR : xr → PGame hx : ∀ (i : xl) (j : xr), xL i < xR j _oxl : ∀ (i : xl), Numeric (xL i) _oxr : ∀ (i : xr), Numeric (xR i) IHxl : ∀ (i : xl), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xL i) IHxr : ∀ (i : xr), (fun x => ∀ (z : PGame), Numeric z → x ⧏ z → ¬z ⧏ x) (xR i) yl yr : Type u_1 yL : yl → PGame yR : yr → PGame hy : ∀ (i : yl) (j : yr), yL i < yR j oyl : ∀ (i : yl), Numeric (yL i) oyr : ∀ (i : yr), Numeric (yR i) _IHyl : ∀ (i : yl), mk xl xr xL xR ⧏ yL i → ¬yL i ⧏ mk xl xr xL xR _IHyr : ∀ (i : yr), mk xl xr xL xR ⧏ yR i → ¬yR i ⧏ mk xl xr xL xR j✝ : xr h₁ : xR j✝ ≤ mk yl yr yL yR j : yr h₂ : yR j ≤ mk xl xr xL xR ⊢ False ** exact IHxr _ _ (oyr _) (h₁.lf_moveRight _) (h₂.lf_moveRight _) ** Qed | |
SetTheory.PGame.le_iff_forall_lt ** x y : PGame ox : Numeric x oy : Numeric y ⊢ x ≤ y ↔ (∀ (i : LeftMoves x), moveLeft x i < y) ∧ ∀ (j : RightMoves y), x < moveRight y j ** refine' le_iff_forall_lf.trans (and_congr _ _) <;>
refine' forall_congr' fun i => lf_iff_lt _ _ <;>
apply_rules [Numeric.moveLeft, Numeric.moveRight] ** Qed | |
SetTheory.PGame.lt_iff_exists_le ** x y : PGame ox : Numeric x oy : Numeric y ⊢ x < y ↔ (∃ i, x ≤ moveLeft y i) ∨ ∃ j, moveRight x j ≤ y ** rw [← lf_iff_lt ox oy, lf_iff_exists_le] ** Qed | |
SetTheory.PGame.lt_def ** x y : PGame ox : Numeric x oy : Numeric y ⊢ x < y ↔ (∃ i, (∀ (i' : LeftMoves x), moveLeft x i' < moveLeft y i) ∧ ∀ (j : RightMoves (moveLeft y i)), x < moveRight (moveLeft y i) j) ∨ ∃ j, (∀ (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i < y) ∧ ∀ (j' : RightMoves y), moveRight x j < moveRight y j' ** rw [← lf_iff_lt ox oy, lf_def] ** x y : PGame ox : Numeric x oy : Numeric y ⊢ ((∃ i, (∀ (i' : LeftMoves x), moveLeft x i' ⧏ moveLeft y i) ∧ ∀ (j : RightMoves (moveLeft y i)), x ⧏ moveRight (moveLeft y i) j) ∨ ∃ j, (∀ (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i ⧏ y) ∧ ∀ (j' : RightMoves y), moveRight x j ⧏ moveRight y j') ↔ (∃ i, (∀ (i' : LeftMoves x), moveLeft x i' < moveLeft y i) ∧ ∀ (j : RightMoves (moveLeft y i)), x < moveRight (moveLeft y i) j) ∨ ∃ j, (∀ (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i < y) ∧ ∀ (j' : RightMoves y), moveRight x j < moveRight y j' ** refine' or_congr _ _ <;> refine' exists_congr fun x_1 => _ <;> refine' and_congr _ _ <;>
refine' forall_congr' fun i => lf_iff_lt _ _ <;>
apply_rules [Numeric.moveLeft, Numeric.moveRight] ** Qed | |
SetTheory.PGame.Numeric.add ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ⊢ ∀ (i : xl ⊕ yl) (j : xr ⊕ yr), (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) i < (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) j ** rintro (ix | iy) (jx | jy) ** case inl.inl xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ix : xl jx : xr ⊢ (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inl ix) < (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inl jx) ** exact add_lt_add_right (ox.1 ix jx) _ ** case inl.inr xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ix : xl jy : yr ⊢ (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inl ix) < (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inr jy) ** exact (add_lf_add_of_lf_of_le (lf_mk _ _ ix) (oy.le_moveRight jy)).lt
((ox.moveLeft ix).add oy) (ox.add (oy.moveRight jy)) ** case inr.inl xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) iy : yl jx : xr ⊢ (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inr iy) < (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inl jx) ** exact (add_lf_add_of_lf_of_le (mk_lf _ _ jx) (oy.moveLeft_le iy)).lt
(ox.add (oy.moveLeft iy)) ((ox.moveRight jx).add oy) ** case inr.inr xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) iy : yl jy : yr ⊢ (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inr iy) < (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inr jy) ** exact add_lt_add_left (oy.1 iy jy) ⟨xl, xr, xL, xR⟩ ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ⊢ (∀ (i : xl ⊕ yl), Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) i)) ∧ ∀ (j : xr ⊕ yr), Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) j) ** constructor ** case left xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ⊢ ∀ (i : xl ⊕ yl), Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) i) ** rintro (ix | iy) ** case left.inl xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ix : xl ⊢ Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inl ix)) ** exact (ox.moveLeft ix).add oy ** case left.inr xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) iy : yl ⊢ Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yL a)) t) (Sum.inr iy)) ** exact ox.add (oy.moveLeft iy) ** case right xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) ⊢ ∀ (j : xr ⊕ yr), Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) j) ** rintro (jx | jy) ** case right.inl xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) jx : xr ⊢ Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inl jx)) ** apply (ox.moveRight jx).add oy ** case right.inr xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) jy : yr ⊢ Numeric ((fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i (PGame.mk yl yr yL yR)) (fun a => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xL a)) i y) IHyl t) fun t => Sum.rec (fun i => (fun a => rec (motive := fun x => PGame → PGame) (fun xl xr a a IHxl IHxr y => rec (fun yl yr yL yR IHyl IHyr => let_fun y := PGame.mk yl yr yL yR; PGame.mk (xl ⊕ yl) (xr ⊕ yr) (fun t => Sum.rec (fun i => IHxl i y) IHyl t) fun t => Sum.rec (fun i => IHxr i y) IHyr t) y) (xR a)) i y) IHyr t) (yR a)) t) (Sum.inr jy)) ** apply ox.add (oy.moveRight jy) ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame yl yr : Type u_1 yL : yl → PGame yR : yr → PGame ox : Numeric (PGame.mk xl xr xL xR) oy : Numeric (PGame.mk yl yr yL yR) x✝ : ∀ (y : (x : PGame) ×' (x_1 : PGame) ×' (_ : Numeric x) ×' Numeric x_1), (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => PSigma.casesOn snd fun x_1 snd => (x, y)) Prod.instWellFoundedRelationProd).1 y { fst := PGame.mk xl xr xL xR, snd := { fst := PGame.mk yl yr yL yR, snd := { fst := ox, snd := oy } } } → Numeric (y.1 + y.2.1) jy : yr ⊢ (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => PSigma.casesOn snd fun x_1 snd => (x, y)) Prod.instWellFoundedRelationProd).1 { fst := PGame.mk xl xr xL xR, snd := { fst := PGame.moveRight (PGame.mk yl yr yL yR) jy, snd := { fst := ox, snd := (_ : Numeric (PGame.moveRight (PGame.mk yl yr yL yR) jy)) } } } { fst := PGame.mk xl xr xL xR, snd := { fst := PGame.mk yl yr yL yR, snd := { fst := ox, snd := oy } } } ** pgame_wf_tac ** Qed | |
SetTheory.PGame.numeric_toPGame ** o : Ordinal.{u_1} ⊢ Numeric (Ordinal.toPGame o) ** induction' o using Ordinal.induction with o IH ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → Numeric (Ordinal.toPGame k) ⊢ Numeric (Ordinal.toPGame o) ** apply numeric_of_isEmpty_rightMoves ** case h.H o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → Numeric (Ordinal.toPGame k) ⊢ ∀ (i : LeftMoves (Ordinal.toPGame o)), Numeric (moveLeft (Ordinal.toPGame o) i) ** simpa using fun i => IH _ (Ordinal.toLeftMovesToPGame_symm_lt i) ** Qed | |
Ordinal.principal_iff_principal_swap ** op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} o : Ordinal.{u_1} ⊢ Principal op o ↔ Principal (Function.swap op) o ** constructor <;> exact fun h a b ha hb => h hb ha ** Qed | |
Ordinal.principal_one_iff ** op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} ⊢ Principal op 1 ↔ op 0 0 = 0 ** refine' ⟨fun h => _, fun h a b ha hb => _⟩ ** case refine'_1 op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} h : Principal op 1 ⊢ op 0 0 = 0 ** rw [← lt_one_iff_zero] ** case refine'_1 op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} h : Principal op 1 ⊢ op 0 0 < 1 ** exact h zero_lt_one zero_lt_one ** case refine'_2 op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} h : op 0 0 = 0 a b : Ordinal.{u_1} ha : a < 1 hb : b < 1 ⊢ op a b < 1 ** rwa [lt_one_iff_zero, ha, hb] at * ** Qed | |
Ordinal.Principal.iterate_lt ** op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} a o : Ordinal.{u_1} hao : a < o ho : Principal op o n : ℕ ⊢ (op a)^[n] a < o ** induction' n with n hn ** case zero op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} a o : Ordinal.{u_1} hao : a < o ho : Principal op o ⊢ (op a)^[Nat.zero] a < o ** rwa [Function.iterate_zero] ** case succ op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} a o : Ordinal.{u_1} hao : a < o ho : Principal op o n : ℕ hn : (op a)^[n] a < o ⊢ (op a)^[Nat.succ n] a < o ** rw [Function.iterate_succ'] ** case succ op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1} a o : Ordinal.{u_1} hao : a < o ho : Principal op o n : ℕ hn : (op a)^[n] a < o ⊢ (op a ∘ (op a)^[n]) a < o ** exact ho hao hn ** Qed | |
Ordinal.op_eq_self_of_principal ** op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} a o : Ordinal.{u} hao : a < o H : IsNormal (op a) ho : Principal op o ho' : IsLimit o ⊢ op a o = o ** refine' le_antisymm _ (H.self_le _) ** op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} a o : Ordinal.{u} hao : a < o H : IsNormal (op a) ho : Principal op o ho' : IsLimit o ⊢ op a o ≤ o ** rw [← IsNormal.bsup_eq.{u, u} H ho', bsup_le_iff] ** op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} a o : Ordinal.{u} hao : a < o H : IsNormal (op a) ho : Principal op o ho' : IsLimit o ⊢ ∀ (i : Ordinal.{u}), i < o → op a i ≤ o ** exact fun b hbo => (ho hao hbo).le ** Qed | |
Ordinal.principal_nfp_blsub₂ ** op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} ha : a < nfp (fun o' => blsub₂ o' o' fun a x b x => op a b) o hb : b < nfp (fun o' => blsub₂ o' o' fun a x b x => op a b) o ⊢ op a b < nfp (fun o' => blsub₂ o' o' fun a x b x => op a b) o ** rw [lt_nfp] at * ** op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} ha : ∃ n, a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o hb : ∃ n, b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ⊢ ∃ n, op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ** cases' ha with m hm ** case intro op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} hb : ∃ n, b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ⊢ ∃ n, op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ** cases' hb with n hn ** case intro.intro op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ⊢ ∃ n, op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ** cases' le_total
((fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b))^[m] o)
((fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b))^[n] o) with h h ** case intro.intro.inl op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ⊢ ∃ n, op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ** use n + 1 ** case h op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ⊢ op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n + 1] o ** rw [Function.iterate_succ'] ** case h op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ⊢ op a b < ((fun o' => blsub₂ o' o' fun a x b x => op a b) ∘ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n]) o ** exact lt_blsub₂ (@fun a _ b _ => op a b) (hm.trans_le h) hn ** case intro.intro.inr op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ⊢ ∃ n, op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ** use m + 1 ** case h op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ⊢ op a b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m + 1] o ** rw [Function.iterate_succ'] ** case h op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} o a b : Ordinal.{u} m : ℕ hm : a < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o n : ℕ hn : b < (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o h : (fun o' => blsub₂ o' o' fun a x b x => op a b)^[n] o ≤ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m] o ⊢ op a b < ((fun o' => blsub₂ o' o' fun a x b x => op a b) ∘ (fun o' => blsub₂ o' o' fun a x b x => op a b)^[m]) o ** exact lt_blsub₂ (@fun a _ b _ => op a b) hm (hn.trans_le h) ** Qed | |
Ordinal.principal_add_of_le_one ** o : Ordinal.{u_1} ho : o ≤ 1 ⊢ Principal (fun x x_1 => x + x_1) o ** rcases le_one_iff.1 ho with (rfl | rfl) ** case inl ho : 0 ≤ 1 ⊢ Principal (fun x x_1 => x + x_1) 0 ** exact principal_zero ** case inr ho : 1 ≤ 1 ⊢ Principal (fun x x_1 => x + x_1) 1 ** exact principal_add_one ** Qed | |
Ordinal.principal_add_isLimit ** o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o ⊢ IsLimit o ** refine' ⟨fun ho₀ => _, fun a hao => _⟩ ** case refine'_1 o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o ho₀ : o = 0 ⊢ False ** rw [ho₀] at ho₁ ** case refine'_1 o : Ordinal.{u_1} ho₁ : 1 < 0 ho : Principal (fun x x_1 => x + x_1) o ho₀ : o = 0 ⊢ False ** exact not_lt_of_gt zero_lt_one ho₁ ** case refine'_2 o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ⊢ succ a < o ** cases' eq_or_ne a 0 with ha ha ** case refine'_2.inl o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ha : a = 0 ⊢ succ a < o ** rw [ha, succ_zero] ** case refine'_2.inl o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ha : a = 0 ⊢ 1 < o ** exact ho₁ ** case refine'_2.inr o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ha : a ≠ 0 ⊢ succ a < o ** refine' lt_of_le_of_lt _ (ho hao hao) ** case refine'_2.inr o : Ordinal.{u_1} ho₁ : 1 < o ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ha : a ≠ 0 ⊢ succ a ≤ (fun x x_1 => x + x_1) a a ** rwa [← add_one_eq_succ, add_le_add_iff_left, one_le_iff_ne_zero] ** Qed | |
Ordinal.principal_add_iff_add_left_eq_self ** o : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x + x_1) o ↔ ∀ (a : Ordinal.{u_1}), a < o → a + o = o ** refine' ⟨fun ho a hao => _, fun h a b hao hbo => _⟩ ** case refine'_1 o : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ⊢ a + o = o ** cases' lt_or_le 1 o with ho₁ ho₁ ** case refine'_1.inl o : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ho₁ : 1 < o ⊢ a + o = o ** exact op_eq_self_of_principal hao (add_isNormal a) ho (principal_add_isLimit ho₁ ho) ** case refine'_1.inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) o a : Ordinal.{u_1} hao : a < o ho₁ : o ≤ 1 ⊢ a + o = o ** rcases le_one_iff.1 ho₁ with (rfl | rfl) ** case refine'_1.inr.inl a : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) 0 hao : a < 0 ho₁ : 0 ≤ 1 ⊢ a + 0 = 0 ** exact (Ordinal.not_lt_zero a hao).elim ** case refine'_1.inr.inr a : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) 1 hao : a < 1 ho₁ : 1 ≤ 1 ⊢ a + 1 = 1 ** rw [lt_one_iff_zero] at hao ** case refine'_1.inr.inr a : Ordinal.{u_1} ho : Principal (fun x x_1 => x + x_1) 1 hao : a = 0 ho₁ : 1 ≤ 1 ⊢ a + 1 = 1 ** rw [hao, zero_add] ** case refine'_2 o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), a < o → a + o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ (fun x x_1 => x + x_1) a b < o ** rw [← h a hao] ** case refine'_2 o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), a < o → a + o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ (fun x x_1 => x + x_1) a b < a + o ** exact (add_isNormal a).strictMono hbo ** Qed | |
Ordinal.exists_lt_add_of_not_principal_add ** a : Ordinal.{u_1} ha : ¬Principal (fun x x_1 => x + x_1) a ⊢ ∃ b c x x, b + c = a ** unfold Principal at ha ** a : Ordinal.{u_1} ha : ¬∀ ⦃a_1 b : Ordinal.{u_1}⦄, a_1 < a → b < a → (fun x x_1 => x + x_1) a_1 b < a ⊢ ∃ b c x x, b + c = a ** push_neg at ha ** a : Ordinal.{u_1} ha : Exists fun ⦃a_1⦄ => Exists fun ⦃b⦄ => a_1 < a ∧ b < a ∧ a ≤ a_1 + b ⊢ ∃ b c x x, b + c = a ** rcases ha with ⟨b, c, hb, hc, H⟩ ** case intro.intro.intro.intro a b c : Ordinal.{u_1} hb : b < a hc : c < a H : a ≤ b + c ⊢ ∃ b c x x, b + c = a ** refine'
⟨b, _, hb, lt_of_le_of_ne (sub_le_self a b) fun hab => _, Ordinal.add_sub_cancel_of_le hb.le⟩ ** case intro.intro.intro.intro a b c : Ordinal.{u_1} hb : b < a hc : c < a H : a ≤ b + c hab : a - b = a ⊢ False ** rw [← sub_le, hab] at H ** case intro.intro.intro.intro a b c : Ordinal.{u_1} hb : b < a hc : c < a H : a ≤ c hab : a - b = a ⊢ False ** exact H.not_lt hc ** Qed | |
Ordinal.principal_add_iff_add_lt_ne_self ** a : Ordinal.{u_1} H : ∀ ⦃b c : Ordinal.{u_1}⦄, b < a → c < a → b + c ≠ a ⊢ Principal (fun x x_1 => x + x_1) a ** by_contra' ha ** a : Ordinal.{u_1} H : ∀ ⦃b c : Ordinal.{u_1}⦄, b < a → c < a → b + c ≠ a ha : ¬Principal (fun x x_1 => x + x_1) a ⊢ False ** rcases exists_lt_add_of_not_principal_add ha with ⟨b, c, hb, hc, rfl⟩ ** case intro.intro.intro.intro b c : Ordinal.{u_1} H : ∀ ⦃b_1 c_1 : Ordinal.{u_1}⦄, b_1 < b + c → c_1 < b + c → b_1 + c_1 ≠ b + c ha : ¬Principal (fun x x_1 => x + x_1) (b + c) hb : b < b + c hc : c < b + c ⊢ False ** exact (H hb hc).irrefl ** Qed | |
Ordinal.add_omega ** a : Ordinal.{u_1} h : a < ω ⊢ a + ω = ω ** rcases lt_omega.1 h with ⟨n, rfl⟩ ** case intro n : ℕ h : ↑n < ω ⊢ ↑n + ω = ω ** clear h ** case intro n : ℕ ⊢ ↑n + ω = ω ** induction' n with n IH ** case intro.zero ⊢ ↑Nat.zero + ω = ω ** rw [Nat.cast_zero, zero_add] ** case intro.succ n : ℕ IH : ↑n + ω = ω ⊢ ↑(Nat.succ n) + ω = ω ** rwa [Nat.cast_succ, add_assoc, one_add_of_omega_le (le_refl _)] ** Qed | |
Ordinal.add_omega_opow ** a b : Ordinal.{u_1} h : a < ω ^ b ⊢ a + ω ^ b = ω ^ b ** refine' le_antisymm _ (le_add_left _ _) ** a b : Ordinal.{u_1} h : a < ω ^ b ⊢ a + ω ^ b ≤ ω ^ b ** induction' b using limitRecOn with b _ b l IH ** case H₁ a b : Ordinal.{u_1} h✝ : a < ω ^ b h : a < ω ^ 0 ⊢ a + ω ^ 0 ≤ ω ^ 0 ** rw [opow_zero, ← succ_zero, lt_succ_iff, Ordinal.le_zero] at h ** case H₁ a b : Ordinal.{u_1} h✝ : a < ω ^ b h : a = 0 ⊢ a + ω ^ 0 ≤ ω ^ 0 ** rw [h, zero_add] ** case H₂ a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} a✝ : a < ω ^ b → a + ω ^ b ≤ ω ^ b h : a < ω ^ succ b ⊢ a + ω ^ succ b ≤ ω ^ succ b ** rw [opow_succ] at h ** case H₂ a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} a✝ : a < ω ^ b → a + ω ^ b ≤ ω ^ b h : a < ω ^ b * ω ⊢ a + ω ^ succ b ≤ ω ^ succ b ** rcases (lt_mul_of_limit omega_isLimit).1 h with ⟨x, xo, ax⟩ ** case H₂.intro.intro a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} a✝ : a < ω ^ b → a + ω ^ b ≤ ω ^ b h : a < ω ^ b * ω x : Ordinal.{u_1} xo : x < ω ax : a < ω ^ b * x ⊢ a + ω ^ succ b ≤ ω ^ succ b ** refine' le_trans (add_le_add_right (le_of_lt ax) _) _ ** case H₂.intro.intro a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} a✝ : a < ω ^ b → a + ω ^ b ≤ ω ^ b h : a < ω ^ b * ω x : Ordinal.{u_1} xo : x < ω ax : a < ω ^ b * x ⊢ ω ^ b * x + ω ^ succ b ≤ ω ^ succ b ** rw [opow_succ, ← mul_add, add_omega xo] ** case H₃ a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} l : IsLimit b IH : ∀ (o' : Ordinal.{u_1}), o' < b → a < ω ^ o' → a + ω ^ o' ≤ ω ^ o' h : a < ω ^ b ⊢ a + ω ^ b ≤ ω ^ b ** rcases (lt_opow_of_limit omega_ne_zero l).1 h with ⟨x, xb, ax⟩ ** case H₃.intro.intro a b✝ : Ordinal.{u_1} h✝ : a < ω ^ b✝ b : Ordinal.{u_1} l : IsLimit b IH : ∀ (o' : Ordinal.{u_1}), o' < b → a < ω ^ o' → a + ω ^ o' ≤ ω ^ o' h : a < ω ^ b x : Ordinal.{u_1} xb : x < b ax : a < ω ^ x ⊢ a + ω ^ b ≤ ω ^ b ** exact
(((add_isNormal a).trans (opow_isNormal one_lt_omega)).limit_le l).2 fun y yb =>
(add_le_add_left (opow_le_opow_right omega_pos (le_max_right _ _)) _).trans
(le_trans
(IH _ (max_lt xb yb) (ax.trans_le <| opow_le_opow_right omega_pos (le_max_left _ _)))
(opow_le_opow_right omega_pos <| le_of_lt <| max_lt xb yb)) ** Qed | |
Ordinal.principal_add_iff_zero_or_omega_opow ** o : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x + x_1) o ↔ o = 0 ∨ ∃ a, o = ω ^ a ** rcases eq_or_ne o 0 with (rfl | ho) ** case inl ⊢ Principal (fun x x_1 => x + x_1) 0 ↔ 0 = 0 ∨ ∃ a, 0 = ω ^ a ** simp only [principal_zero, Or.inl] ** case inr o : Ordinal.{u_1} ho : o ≠ 0 ⊢ Principal (fun x x_1 => x + x_1) o ↔ o = 0 ∨ ∃ a, o = ω ^ a ** rw [principal_add_iff_add_left_eq_self] ** case inr o : Ordinal.{u_1} ho : o ≠ 0 ⊢ (∀ (a : Ordinal.{u_1}), a < o → a + o = o) ↔ o = 0 ∨ ∃ a, o = ω ^ a ** simp only [ho, false_or_iff] ** case inr o : Ordinal.{u_1} ho : o ≠ 0 ⊢ (∀ (a : Ordinal.{u_1}), a < o → a + o = o) ↔ ∃ a, o = ω ^ a ** refine'
⟨fun H => ⟨_, ((lt_or_eq_of_le (opow_log_le_self _ ho)).resolve_left fun h => _).symm⟩,
fun ⟨b, e⟩ => e.symm ▸ fun a => add_omega_opow⟩ ** case inr o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o ⊢ False ** have := H _ h ** case inr o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o ⊢ False ** have := lt_opow_succ_log_self one_lt_omega o ** case inr o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this✝ : ω ^ log ω o + o = o this : o < ω ^ succ (log ω o) ⊢ False ** rw [opow_succ, lt_mul_of_limit omega_isLimit] at this ** case inr o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this✝ : ω ^ log ω o + o = o this : ∃ c', c' < ω ∧ o < ω ^ log ω o * c' ⊢ False ** rcases this with ⟨a, ao, h'⟩ ** case inr.intro.intro o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o a : Ordinal.{u_1} ao : a < ω h' : o < ω ^ log ω o * a ⊢ False ** rcases lt_omega.1 ao with ⟨n, rfl⟩ ** case inr.intro.intro.intro o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ ao : ↑n < ω h' : o < ω ^ log ω o * ↑n ⊢ False ** clear ao ** case inr.intro.intro.intro o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ h' : o < ω ^ log ω o * ↑n ⊢ False ** revert h' ** case inr.intro.intro.intro o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ ⊢ o < ω ^ log ω o * ↑n → False ** apply not_lt_of_le ** case inr.intro.intro.intro.h o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ ⊢ ω ^ log ω o * ↑n ≤ o ** suffices e : (omega^log omega o) * ↑n + o = o ** case e o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ ⊢ ω ^ log ω o * ↑n + o = o ** induction' n with n IH ** case e.succ o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ IH : ω ^ log ω o * ↑n + o = o ⊢ ω ^ log ω o * ↑(Nat.succ n) + o = o ** simp only [Nat.cast_succ, mul_add_one, add_assoc, this, IH] ** case inr.intro.intro.intro.h o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o n : ℕ e : ω ^ log ω o * ↑n + o = o ⊢ ω ^ log ω o * ↑n ≤ o ** simpa only [e] using le_add_right ((omega^log omega o) * ↑n) o ** case e.zero o : Ordinal.{u_1} ho : o ≠ 0 H : ∀ (a : Ordinal.{u_1}), a < o → a + o = o h : ω ^ log ω o < o this : ω ^ log ω o + o = o ⊢ ω ^ log ω o * ↑Nat.zero + o = o ** simp [Nat.cast_zero, mul_zero, zero_add] ** Qed | |
Ordinal.opow_principal_add_of_principal_add ** a : Ordinal.{u_1} ha : Principal (fun x x_1 => x + x_1) a b : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x + x_1) (a ^ b) ** rcases principal_add_iff_zero_or_omega_opow.1 ha with (rfl | ⟨c, rfl⟩) ** case inl b : Ordinal.{u_1} ha : Principal (fun x x_1 => x + x_1) 0 ⊢ Principal (fun x x_1 => x + x_1) (0 ^ b) ** rcases eq_or_ne b 0 with (rfl | hb) ** case inl.inl ha : Principal (fun x x_1 => x + x_1) 0 ⊢ Principal (fun x x_1 => x + x_1) (0 ^ 0) ** rw [opow_zero] ** case inl.inl ha : Principal (fun x x_1 => x + x_1) 0 ⊢ Principal (fun x x_1 => x + x_1) 1 ** exact principal_add_one ** case inl.inr b : Ordinal.{u_1} ha : Principal (fun x x_1 => x + x_1) 0 hb : b ≠ 0 ⊢ Principal (fun x x_1 => x + x_1) (0 ^ b) ** rwa [zero_opow hb] ** case inr.intro b c : Ordinal.{u_1} ha : Principal (fun x x_1 => x + x_1) (ω ^ c) ⊢ Principal (fun x x_1 => x + x_1) ((ω ^ c) ^ b) ** rw [← opow_mul] ** case inr.intro b c : Ordinal.{u_1} ha : Principal (fun x x_1 => x + x_1) (ω ^ c) ⊢ Principal (fun x x_1 => x + x_1) (ω ^ (c * b)) ** exact principal_add_omega_opow _ ** Qed | |
Ordinal.add_absorp ** a b c : Ordinal.{u_1} h₁ : a < ω ^ b h₂ : ω ^ b ≤ c ⊢ a + c = c ** rw [← Ordinal.add_sub_cancel_of_le h₂, ← add_assoc, add_omega_opow h₁] ** Qed | |
Ordinal.mul_principal_add_is_principal_add ** a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b ⊢ Principal (fun x x_1 => x + x_1) (a * b) ** rcases eq_zero_or_pos a with (rfl | _) ** case inl b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b ⊢ Principal (fun x x_1 => x + x_1) (0 * b) ** rw [zero_mul] ** case inl b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b ⊢ Principal (fun x x_1 => x + x_1) 0 ** exact principal_zero ** case inr a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a ⊢ Principal (fun x x_1 => x + x_1) (a * b) ** rcases eq_zero_or_pos b with (rfl | hb₁') ** case inr.inl a : Ordinal.{u} h✝ : 0 < a hb₁ : 0 ≠ 1 hb : Principal (fun x x_1 => x + x_1) 0 ⊢ Principal (fun x x_1 => x + x_1) (a * 0) ** rw [mul_zero] ** case inr.inl a : Ordinal.{u} h✝ : 0 < a hb₁ : 0 ≠ 1 hb : Principal (fun x x_1 => x + x_1) 0 ⊢ Principal (fun x x_1 => x + x_1) 0 ** exact principal_zero ** case inr.inr a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁' : 0 < b ⊢ Principal (fun x x_1 => x + x_1) (a * b) ** rw [← succ_le_iff, succ_zero] at hb₁' ** case inr.inr a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b ⊢ Principal (fun x x_1 => x + x_1) (a * b) ** intro c d hc hd ** case inr.inr a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d : Ordinal.{u} hc : c < a * b hd : d < a * b ⊢ (fun x x_1 => x + x_1) c d < a * b ** rw [lt_mul_of_limit (principal_add_isLimit (lt_of_le_of_ne hb₁' hb₁.symm) hb)] at * ** case inr.inr a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d : Ordinal.{u} hc : ∃ c', c' < b ∧ c < a * c' hd : ∃ c', c' < b ∧ d < a * c' ⊢ ∃ c', c' < b ∧ (fun x x_1 => x + x_1) c d < a * c' ** rcases hc with ⟨x, hx, hx'⟩ ** case inr.inr.intro.intro a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d : Ordinal.{u} hd : ∃ c', c' < b ∧ d < a * c' x : Ordinal.{u} hx : x < b hx' : c < a * x ⊢ ∃ c', c' < b ∧ (fun x x_1 => x + x_1) c d < a * c' ** rcases hd with ⟨y, hy, hy'⟩ ** case inr.inr.intro.intro.intro.intro a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d x : Ordinal.{u} hx : x < b hx' : c < a * x y : Ordinal.{u} hy : y < b hy' : d < a * y ⊢ ∃ c', c' < b ∧ (fun x x_1 => x + x_1) c d < a * c' ** use x + y, hb hx hy ** case right a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d x : Ordinal.{u} hx : x < b hx' : c < a * x y : Ordinal.{u} hy : y < b hy' : d < a * y ⊢ (fun x x_1 => x + x_1) c d < a * (x + y) ** rw [mul_add] ** case right a b : Ordinal.{u} hb₁ : b ≠ 1 hb : Principal (fun x x_1 => x + x_1) b h✝ : 0 < a hb₁'✝ : 0 < b hb₁' : 1 ≤ b c d x : Ordinal.{u} hx : x < b hx' : c < a * x y : Ordinal.{u} hy : y < b hy' : d < a * y ⊢ (fun x x_1 => x + x_1) c d < a * x + a * y ** exact Left.add_lt_add hx' hy' ** Qed | |
Ordinal.principal_mul_one ** ⊢ Principal (fun x x_1 => x * x_1) 1 ** rw [principal_one_iff] ** ⊢ 0 * 0 = 0 ** exact zero_mul _ ** Qed | |
Ordinal.principal_mul_two ** a b : Ordinal.{u_1} ha : a < 2 hb : b < 2 ⊢ (fun x x_1 => x * x_1) a b < 2 ** have h₂ : succ (1 : Ordinal) = 2 := by simp ** a b : Ordinal.{u_1} ha : a < 2 hb : b < 2 h₂ : succ 1 = 2 ⊢ (fun x x_1 => x * x_1) a b < 2 ** dsimp only ** a b : Ordinal.{u_1} ha : a < 2 hb : b < 2 h₂ : succ 1 = 2 ⊢ a * b < 2 ** rw [← h₂, lt_succ_iff] at ha hb ⊢ ** a b : Ordinal.{u_1} ha : a ≤ 1 hb : b ≤ 1 h₂ : succ 1 = 2 ⊢ a * b ≤ 1 ** convert mul_le_mul' ha hb ** case h.e'_4 a b : Ordinal.{u_1} ha : a ≤ 1 hb : b ≤ 1 h₂ : succ 1 = 2 ⊢ 1 = 1 * 1 ** exact (mul_one 1).symm ** a b : Ordinal.{u_1} ha : a < 2 hb : b < 2 ⊢ succ 1 = 2 ** simp ** Qed | |
Ordinal.principal_mul_of_le_two ** o : Ordinal.{u_1} ho : o ≤ 2 ⊢ Principal (fun x x_1 => x * x_1) o ** rcases lt_or_eq_of_le ho with (ho | rfl) ** case inl o : Ordinal.{u_1} ho✝ : o ≤ 2 ho : o < 2 ⊢ Principal (fun x x_1 => x * x_1) o ** have h₂ : succ (1 : Ordinal) = 2 := by simp ** case inl o : Ordinal.{u_1} ho✝ : o ≤ 2 ho : o < 2 h₂ : succ 1 = 2 ⊢ Principal (fun x x_1 => x * x_1) o ** rw [← h₂, lt_succ_iff] at ho ** case inl o : Ordinal.{u_1} ho✝ : o ≤ 2 ho : o ≤ 1 h₂ : succ 1 = 2 ⊢ Principal (fun x x_1 => x * x_1) o ** rcases lt_or_eq_of_le ho with (ho | rfl) ** o : Ordinal.{u_1} ho✝ : o ≤ 2 ho : o < 2 ⊢ succ 1 = 2 ** simp ** case inl.inl o : Ordinal.{u_1} ho✝¹ : o ≤ 2 ho✝ : o ≤ 1 h₂ : succ 1 = 2 ho : o < 1 ⊢ Principal (fun x x_1 => x * x_1) o ** rw [lt_one_iff_zero.1 ho] ** case inl.inl o : Ordinal.{u_1} ho✝¹ : o ≤ 2 ho✝ : o ≤ 1 h₂ : succ 1 = 2 ho : o < 1 ⊢ Principal (fun x x_1 => x * x_1) 0 ** exact principal_zero ** case inl.inr h₂ : succ 1 = 2 ho✝ : 1 ≤ 2 ho : 1 ≤ 1 ⊢ Principal (fun x x_1 => x * x_1) 1 ** exact principal_mul_one ** case inr ho : 2 ≤ 2 ⊢ Principal (fun x x_1 => x * x_1) 2 ** exact principal_mul_two ** Qed | |
Ordinal.principal_add_of_principal_mul ** o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≠ 2 ⊢ Principal (fun x x_1 => x + x_1) o ** cases' lt_or_gt_of_ne ho₂ with ho₁ ho₂ ** case inl o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≠ 2 ho₁ : o < 2 ⊢ Principal (fun x x_1 => x + x_1) o ** replace ho₁ : o < succ 1 := by simpa using ho₁ ** case inl o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≠ 2 ho₁ : o < succ 1 ⊢ Principal (fun x x_1 => x + x_1) o ** rw [lt_succ_iff] at ho₁ ** case inl o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≠ 2 ho₁ : o ≤ 1 ⊢ Principal (fun x x_1 => x + x_1) o ** exact principal_add_of_le_one ho₁ ** o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≠ 2 ho₁ : o < 2 ⊢ o < succ 1 ** simpa using ho₁ ** case inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂✝ : o ≠ 2 ho₂ : o > 2 ⊢ Principal (fun x x_1 => x + x_1) o ** refine' fun a b hao hbo => lt_of_le_of_lt _ (ho (max_lt hao hbo) ho₂) ** case inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂✝ : o ≠ 2 ho₂ : o > 2 a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ (fun x x_1 => x + x_1) a b ≤ (fun x x_1 => x * x_1) (max a b) 2 ** dsimp only ** case inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂✝ : o ≠ 2 ho₂ : o > 2 a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ a + b ≤ max a b * 2 ** rw [← one_add_one_eq_two, mul_add, mul_one] ** case inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂✝ : o ≠ 2 ho₂ : o > 2 a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ a + b ≤ max a b + max a b ** exact add_le_add (le_max_left a b) (le_max_right a b) ** Qed | |
Ordinal.principal_mul_isLimit ** o : Ordinal.{u} ho₂ : 2 < o ho : Principal (fun x x_1 => x * x_1) o ⊢ succ 1 < o ** simpa using ho₂ ** Qed | |
Ordinal.principal_mul_iff_mul_left_eq ** o : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x * x_1) o ↔ ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o ** refine' ⟨fun h a ha₀ hao => _, fun h a b hao hbo => _⟩ ** case refine'_1 o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ⊢ a * o = o ** cases' le_or_gt o 2 with ho ho ** case refine'_1.inl o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 ⊢ a * o = o ** convert one_mul o ** case h.e'_2.h.e'_5 o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 ⊢ a = 1 ** apply le_antisymm ** case h.e'_2.h.e'_5.a o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 ⊢ a ≤ 1 ** have : a < succ 1 := hao.trans_le (by simpa using ho) ** case h.e'_2.h.e'_5.a o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 this : a < succ 1 ⊢ a ≤ 1 ** rwa [lt_succ_iff] at this ** o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 ⊢ o ≤ succ 1 ** simpa using ho ** case h.e'_2.h.e'_5.a o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o ≤ 2 ⊢ 1 ≤ a ** rwa [← succ_le_iff, succ_zero] at ha₀ ** case refine'_1.inr o : Ordinal.{u_1} h : Principal (fun x x_1 => x * x_1) o a : Ordinal.{u_1} ha₀ : 0 < a hao : a < o ho : o > 2 ⊢ a * o = o ** exact op_eq_self_of_principal hao (mul_isNormal ha₀) h (principal_mul_isLimit ho h) ** case refine'_2 o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ⊢ (fun x x_1 => x * x_1) a b < o ** rcases eq_or_ne a 0 with (rfl | ha) ** case refine'_2.inr o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ha : a ≠ 0 ⊢ (fun x x_1 => x * x_1) a b < o ** rw [← Ordinal.pos_iff_ne_zero] at ha ** case refine'_2.inr o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ha : 0 < a ⊢ (fun x x_1 => x * x_1) a b < o ** rw [← h a ha hao] ** case refine'_2.inr o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o a b : Ordinal.{u_1} hao : a < o hbo : b < o ha : 0 < a ⊢ (fun x x_1 => x * x_1) a b < a * o ** exact (mul_isNormal ha).strictMono hbo ** case refine'_2.inl o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o b : Ordinal.{u_1} hbo : b < o hao : 0 < o ⊢ (fun x x_1 => x * x_1) 0 b < o ** dsimp only ** case refine'_2.inl o : Ordinal.{u_1} h : ∀ (a : Ordinal.{u_1}), 0 < a → a < o → a * o = o b : Ordinal.{u_1} hbo : b < o hao : 0 < o ⊢ 0 * b < o ** rwa [zero_mul] ** Qed | |
Ordinal.principal_mul_omega ** a b : Ordinal.{u_1} m n : ℕ ha : ↑m < ω hb : ↑n < ω ⊢ (fun x x_1 => x * x_1) ↑m ↑n < ω ** dsimp only ** a b : Ordinal.{u_1} m n : ℕ ha : ↑m < ω hb : ↑n < ω ⊢ ↑m * ↑n < ω ** rw [← nat_cast_mul] ** a b : Ordinal.{u_1} m n : ℕ ha : ↑m < ω hb : ↑n < ω ⊢ ↑(m * n) < ω ** apply nat_lt_omega ** Qed | |
Ordinal.mul_lt_omega_opow ** a b c : Ordinal.{u_1} c0 : 0 < c ha : a < ω ^ c hb : b < ω ⊢ a * b < ω ^ c ** rcases zero_or_succ_or_limit c with (rfl | ⟨c, rfl⟩ | l) ** case inl a b : Ordinal.{u_1} hb : b < ω c0 : 0 < 0 ha : a < ω ^ 0 ⊢ a * b < ω ^ 0 ** exact (lt_irrefl _).elim c0 ** case inr.inl.intro a b : Ordinal.{u_1} hb : b < ω c : Ordinal.{u_1} c0 : 0 < succ c ha : a < ω ^ succ c ⊢ a * b < ω ^ succ c ** rw [opow_succ] at ha ** case inr.inl.intro a b : Ordinal.{u_1} hb : b < ω c : Ordinal.{u_1} c0 : 0 < succ c ha : a < ω ^ c * ω ⊢ a * b < ω ^ succ c ** rcases ((mul_isNormal <| opow_pos _ omega_pos).limit_lt omega_isLimit).1 ha with ⟨n, hn, an⟩ ** case inr.inl.intro.intro.intro a b : Ordinal.{u_1} hb : b < ω c : Ordinal.{u_1} c0 : 0 < succ c ha : a < ω ^ c * ω n : Ordinal.{u_1} hn : n < ω an : a < (fun x x_1 => x * x_1) (ω ^ c) n ⊢ a * b < ω ^ succ c ** apply (mul_le_mul_right' (le_of_lt an) _).trans_lt ** case inr.inl.intro.intro.intro a b : Ordinal.{u_1} hb : b < ω c : Ordinal.{u_1} c0 : 0 < succ c ha : a < ω ^ c * ω n : Ordinal.{u_1} hn : n < ω an : a < (fun x x_1 => x * x_1) (ω ^ c) n ⊢ (fun x x_1 => x * x_1) (ω ^ c) n * b < ω ^ succ c ** rw [opow_succ, mul_assoc, mul_lt_mul_iff_left (opow_pos _ omega_pos)] ** case inr.inl.intro.intro.intro a b : Ordinal.{u_1} hb : b < ω c : Ordinal.{u_1} c0 : 0 < succ c ha : a < ω ^ c * ω n : Ordinal.{u_1} hn : n < ω an : a < (fun x x_1 => x * x_1) (ω ^ c) n ⊢ n * b < ω ** exact principal_mul_omega hn hb ** case inr.inr a b c : Ordinal.{u_1} c0 : 0 < c ha : a < ω ^ c hb : b < ω l : IsLimit c ⊢ a * b < ω ^ c ** rcases ((opow_isNormal one_lt_omega).limit_lt l).1 ha with ⟨x, hx, ax⟩ ** case inr.inr.intro.intro a b c : Ordinal.{u_1} c0 : 0 < c ha : a < ω ^ c hb : b < ω l : IsLimit c x : Ordinal.{u_1} hx : x < c ax : a < (fun x x_1 => x ^ x_1) ω x ⊢ a * b < ω ^ c ** refine' (mul_le_mul' (le_of_lt ax) (le_of_lt hb)).trans_lt _ ** case inr.inr.intro.intro a b c : Ordinal.{u_1} c0 : 0 < c ha : a < ω ^ c hb : b < ω l : IsLimit c x : Ordinal.{u_1} hx : x < c ax : a < (fun x x_1 => x ^ x_1) ω x ⊢ (fun x x_1 => x ^ x_1) ω x * ω < ω ^ c ** rw [← opow_succ, opow_lt_opow_iff_right one_lt_omega] ** case inr.inr.intro.intro a b c : Ordinal.{u_1} c0 : 0 < c ha : a < ω ^ c hb : b < ω l : IsLimit c x : Ordinal.{u_1} hx : x < c ax : a < (fun x x_1 => x ^ x_1) ω x ⊢ succ x < c ** exact l.2 _ hx ** Qed | |
Ordinal.mul_omega_opow_opow ** a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b ⊢ a * ω ^ ω ^ b = ω ^ ω ^ b ** by_cases b0 : b = 0 ** case neg a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : ¬b = 0 ⊢ a * ω ^ ω ^ b = ω ^ ω ^ b ** refine'
le_antisymm _
(by simpa only [one_mul] using mul_le_mul_right' (one_le_iff_pos.2 a0) (omega^omega^b)) ** case neg a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : ¬b = 0 ⊢ a * ω ^ ω ^ b ≤ ω ^ ω ^ b ** rcases (lt_opow_of_limit omega_ne_zero (opow_isLimit_left omega_isLimit b0)).1 h with ⟨x, xb, ax⟩ ** case neg.intro.intro a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : ¬b = 0 x : Ordinal.{u_1} xb : x < ω ^ b ax : a < ω ^ x ⊢ a * ω ^ ω ^ b ≤ ω ^ ω ^ b ** apply (mul_le_mul_right' (le_of_lt ax) _).trans ** case neg.intro.intro a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : ¬b = 0 x : Ordinal.{u_1} xb : x < ω ^ b ax : a < ω ^ x ⊢ ω ^ x * ω ^ ω ^ b ≤ ω ^ ω ^ b ** rw [← opow_add, add_omega_opow xb] ** case pos a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : b = 0 ⊢ a * ω ^ ω ^ b = ω ^ ω ^ b ** rw [b0, opow_zero, opow_one] at h ⊢ ** case pos a b : Ordinal.{u_1} a0 : 0 < a h : a < ω b0 : b = 0 ⊢ a * ω = ω ** exact mul_omega a0 h ** a b : Ordinal.{u_1} a0 : 0 < a h : a < ω ^ ω ^ b b0 : ¬b = 0 ⊢ ω ^ ω ^ b ≤ a * ω ^ ω ^ b ** simpa only [one_mul] using mul_le_mul_right' (one_le_iff_pos.2 a0) (omega^omega^b) ** Qed | |
Ordinal.principal_add_of_principal_mul_opow ** o b : Ordinal.{u_1} hb : 1 < b ho : Principal (fun x x_1 => x * x_1) (b ^ o) x y : Ordinal.{u_1} hx : x < o hy : y < o ⊢ (fun x x_1 => x + x_1) x y < o ** have := ho ((opow_lt_opow_iff_right hb).2 hx) ((opow_lt_opow_iff_right hb).2 hy) ** o b : Ordinal.{u_1} hb : 1 < b ho : Principal (fun x x_1 => x * x_1) (b ^ o) x y : Ordinal.{u_1} hx : x < o hy : y < o this : (fun x x_1 => x * x_1) (b ^ x) (b ^ y) < b ^ o ⊢ (fun x x_1 => x + x_1) x y < o ** dsimp only at * ** o b : Ordinal.{u_1} hb : 1 < b ho : Principal (fun x x_1 => x * x_1) (b ^ o) x y : Ordinal.{u_1} hx : x < o hy : y < o this : b ^ x * b ^ y < b ^ o ⊢ x + y < o ** rwa [← opow_add, opow_lt_opow_iff_right hb] at this ** Qed | |
Ordinal.principal_mul_iff_le_two_or_omega_opow_opow ** o : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x * x_1) o ↔ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a ** refine' ⟨fun ho => _, _⟩ ** case refine'_1 o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ⊢ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a ** cases' le_or_lt o 2 with ho₂ ho₂ ** case refine'_1.inr o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : 2 < o ⊢ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a ** rcases principal_add_iff_zero_or_omega_opow.1 (principal_add_of_principal_mul ho ho₂.ne') with
(rfl | ⟨a, rfl⟩) ** case refine'_1.inr.inr.intro a : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) (ω ^ a) ho₂ : 2 < ω ^ a ⊢ ω ^ a ≤ 2 ∨ ∃ a_1, ω ^ a = ω ^ ω ^ a_1 ** rcases principal_add_iff_zero_or_omega_opow.1
(principal_add_of_principal_mul_opow one_lt_omega ho) with
(rfl | ⟨b, rfl⟩) ** case refine'_1.inr.inr.intro.inr.intro b : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) (ω ^ ω ^ b) ho₂ : 2 < ω ^ ω ^ b ⊢ ω ^ ω ^ b ≤ 2 ∨ ∃ a, ω ^ ω ^ b = ω ^ ω ^ a ** exact Or.inr ⟨b, rfl⟩ ** case refine'_1.inl o : Ordinal.{u_1} ho : Principal (fun x x_1 => x * x_1) o ho₂ : o ≤ 2 ⊢ o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a ** exact Or.inl ho₂ ** case refine'_1.inr.inl ho : Principal (fun x x_1 => x * x_1) 0 ho₂ : 2 < 0 ⊢ 0 ≤ 2 ∨ ∃ a, 0 = ω ^ ω ^ a ** exact (Ordinal.not_lt_zero 2 ho₂).elim ** case refine'_1.inr.inr.intro.inl ho : Principal (fun x x_1 => x * x_1) (ω ^ 0) ho₂ : 2 < ω ^ 0 ⊢ ω ^ 0 ≤ 2 ∨ ∃ a, ω ^ 0 = ω ^ ω ^ a ** left ** case refine'_1.inr.inr.intro.inl.h ho : Principal (fun x x_1 => x * x_1) (ω ^ 0) ho₂ : 2 < ω ^ 0 ⊢ ω ^ 0 ≤ 2 ** simpa using one_le_two ** case refine'_2 o : Ordinal.{u_1} ⊢ (o ≤ 2 ∨ ∃ a, o = ω ^ ω ^ a) → Principal (fun x x_1 => x * x_1) o ** rintro (ho₂ | ⟨a, rfl⟩) ** case refine'_2.inl o : Ordinal.{u_1} ho₂ : o ≤ 2 ⊢ Principal (fun x x_1 => x * x_1) o ** exact principal_mul_of_le_two ho₂ ** case refine'_2.inr.intro a : Ordinal.{u_1} ⊢ Principal (fun x x_1 => x * x_1) (ω ^ ω ^ a) ** exact principal_mul_omega_opow_opow a ** Qed | |
Ordinal.mul_omega_dvd ** a : Ordinal.{u_1} a0 : 0 < a ha : a < ω b : Ordinal.{u_1} ⊢ a * (ω * b) = ω * b ** rw [← mul_assoc, mul_omega a0 ha] ** Qed | |
Ordinal.mul_eq_opow_log_succ ** a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b ⊢ a * b = b ^ succ (log b a) ** apply le_antisymm ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b ⊢ a * b ≤ b ^ succ (log b a) ** have hbl := principal_mul_isLimit hb₂ hb ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b ⊢ a * b ≤ b ^ succ (log b a) ** have := IsNormal.bsup_eq.{u, u} (mul_isNormal (Ordinal.pos_iff_ne_zero.2 ha)) hbl ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => (fun x x_2 => x * x_2) a x) = (fun x x_1 => x * x_1) a b ⊢ a * b ≤ b ^ succ (log b a) ** dsimp at this ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b ⊢ a * b ≤ b ^ succ (log b a) ** rw [← this, bsup_le_iff] ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b ⊢ ∀ (i : Ordinal.{u}), i < b → a * i ≤ b ^ succ (log b a) ** intro c hcb ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b ⊢ a * c ≤ b ^ succ (log b a) ** have hb₁ : 1 < b := (lt_succ 1).trans (by simpa using hb₂) ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b ⊢ a * c ≤ b ^ succ (log b a) ** have hbo₀ : (b^b.log a) ≠ 0 := Ordinal.pos_iff_ne_zero.1 (opow_pos _ (zero_lt_one.trans hb₁)) ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b hbo₀ : b ^ log b a ≠ 0 ⊢ a * c ≤ b ^ succ (log b a) ** apply le_trans (mul_le_mul_right' (le_of_lt (lt_mul_succ_div a hbo₀)) c) ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b hbo₀ : b ^ log b a ≠ 0 ⊢ b ^ log b a * succ (a / b ^ log b a) * c ≤ b ^ succ (log b a) ** rw [mul_assoc, opow_succ] ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b hbo₀ : b ^ log b a ≠ 0 ⊢ b ^ log b a * (succ (a / b ^ log b a) * c) ≤ b ^ log b a * b ** refine' mul_le_mul_left' (le_of_lt (hb (hbl.2 _ _) hcb)) _ ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b hbo₀ : b ^ log b a ≠ 0 ⊢ a / b ^ log b a < b ** rw [div_lt hbo₀, ← opow_succ] ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b hb₁ : 1 < b hbo₀ : b ^ log b a ≠ 0 ⊢ a < b ^ succ (log b a) ** exact lt_opow_succ_log_self hb₁ _ ** a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b hbl : IsLimit b this : (bsup b fun x x_1 => a * x) = a * b c : Ordinal.{u} hcb : c < b ⊢ succ 1 < b ** simpa using hb₂ ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b ⊢ b ^ succ (log b a) ≤ a * b ** rw [opow_succ] ** case a a b : Ordinal.{u} ha : a ≠ 0 hb : Principal (fun x x_1 => x * x_1) b hb₂ : 2 < b ⊢ b ^ log b a * b ≤ a * b ** exact mul_le_mul_right' (opow_log_le_self b ha) b ** Qed | |
Ordinal.principal_opow_omega ** a b : Ordinal.{u_1} m n : ℕ ha : ↑m < ω hb : ↑n < ω ⊢ (fun x x_1 => x ^ x_1) ↑m ↑n < ω ** simp_rw [← nat_cast_opow] ** a b : Ordinal.{u_1} m n : ℕ ha : ↑m < ω hb : ↑n < ω ⊢ ↑(m ^ n) < ω ** apply nat_lt_omega ** Qed |
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