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Ordinal.lt_nmul_iff₃ ** a b c d : Ordinal.{u} ⊢ d < a ⨳ b ⨳ c ↔ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** constructor ** case mp a b c d : Ordinal.{u} ⊢ d < a ⨳ b ⨳ c → ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** intro h ** case mp a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** rcases lt_nmul_iff.1 h with ⟨e, he, c', hc, H₁⟩ ** case mp.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** rcases lt_nmul_iff.1 he with ⟨a', ha, b', hb, H₂⟩ ** case mp.intro.intro.intro.intro.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b H₂ : e ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** refine' ⟨a', ha, b', hb, c', hc, _⟩ ** case mp.intro.intro.intro.intro.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b H₂ : e ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' ⊢ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** have := nadd_le_nadd H₁ (nmul_nadd_le H₂ hc.le) ** case mp.intro.intro.intro.intro.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b H₂ : e ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ e ⨳ c' ♯ ((e ♯ a' ⨳ b') ⨳ c ♯ (a' ⨳ b ♯ a ⨳ b') ⨳ c') ≤ e ⨳ c ♯ a ⨳ b ⨳ c' ♯ ((a' ⨳ b ♯ a ⨳ b') ⨳ c ♯ (e ♯ a' ⨳ b') ⨳ c') ⊢ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** simp only [nadd_nmul, nadd_assoc] at this ** case mp.intro.intro.intro.intro.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b H₂ : e ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ (e ⨳ c' ♯ (e ⨳ c ♯ (a' ⨳ b' ⨳ c ♯ (a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c')))) ≤ e ⨳ c ♯ (a ⨳ b ⨳ c' ♯ (a' ⨳ b ⨳ c ♯ (a ⨳ b' ⨳ c ♯ (e ⨳ c' ♯ a' ⨳ b' ⨳ c')))) ⊢ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** rw [nadd_left_comm, nadd_left_comm d, nadd_left_comm, nadd_le_nadd_iff_left,
nadd_left_comm (a ⨳ b' ⨳ c), nadd_left_comm (a' ⨳ b ⨳ c), nadd_left_comm (a ⨳ b ⨳ c'),
nadd_le_nadd_iff_left, nadd_left_comm (a ⨳ b ⨳ c'), nadd_left_comm (a ⨳ b ⨳ c')] at this ** case mp.intro.intro.intro.intro.intro.intro.intro.intro a b c d : Ordinal.{u} h : d < a ⨳ b ⨳ c e : Ordinal.{u} he : e < a ⨳ b c' : Ordinal.{u} hc : c' < c H₁ : d ♯ e ⨳ c' ≤ e ⨳ c ♯ a ⨳ b ⨳ c' a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b H₂ : e ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' this : d ♯ (a' ⨳ b' ⨳ c ♯ (a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c')) ≤ a' ⨳ b ⨳ c ♯ (a ⨳ b' ⨳ c ♯ (a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c')) ⊢ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ** simpa only [nadd_assoc] ** case mpr a b c d : Ordinal.{u} ⊢ (∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c') → d < a ⨳ b ⨳ c ** rintro ⟨a', ha, b', hb, c', hc, h⟩ ** case mpr.intro.intro.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c h : d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ⊢ d < a ⨳ b ⨳ c ** have := h.trans_lt (nmul_nadd_lt₃ ha hb hc) ** case mpr.intro.intro.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c h : d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' this : d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ⊢ d < a ⨳ b ⨳ c ** repeat' rw [nadd_lt_nadd_iff_right] at this ** case mpr.intro.intro.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c h : d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' this : d < a ⨳ b ⨳ c ⊢ d < a ⨳ b ⨳ c ** assumption ** case mpr.intro.intro.intro.intro.intro.intro a b c d a' : Ordinal.{u} ha : a' < a b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c h : d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' this : d ♯ a' ⨳ b' ⨳ c < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ⊢ d < a ⨳ b ⨳ c ** rw [nadd_lt_nadd_iff_right] at this ** Qed | |
Ordinal.nmul_le_iff₃ ** a b c d : Ordinal.{u} ⊢ a ⨳ b ⨳ c ≤ d ↔ ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → ∀ (c' : Ordinal.{u}), c' < c → a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ** rw [← not_iff_not] ** a b c d : Ordinal.{u} ⊢ ¬a ⨳ b ⨳ c ≤ d ↔ ¬∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → ∀ (c' : Ordinal.{u}), c' < c → a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ** simp [lt_nmul_iff₃] ** Qed | |
Ordinal.lt_nmul_iff₃' ** a b c d : Ordinal.{u} ⊢ d < a ⨳ (b ⨳ c) ↔ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ≤ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') ** simp only [nmul_comm _ (_ ⨳ _), lt_nmul_iff₃, nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal] ** a b c d : Ordinal.{u} ⊢ (∃ a', a' < b ∧ ∃ b', b' < c ∧ ∃ c', c' < a ∧ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (a' ⨳ b' ⨳ a) + ↑toNatOrdinal (a' ⨳ c ⨳ c') + ↑toNatOrdinal (b ⨳ b' ⨳ c')) ≤ ↑toOrdinal (↑toNatOrdinal (a' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ b' ⨳ a) + ↑toNatOrdinal (b ⨳ c ⨳ c') + ↑toNatOrdinal (a' ⨳ b' ⨳ c'))) ↔ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) ≤ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a') + ↑toNatOrdinal (b' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a')) ** constructor <;> rintro ⟨b', hb, c', hc, a', ha, h⟩ ** case mp.intro.intro.intro.intro.intro.intro a b c d b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c a' : Ordinal.{u} ha : a' < a h : ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) ≤ ↑toOrdinal (↑toNatOrdinal (b' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ c' ⨳ a) + ↑toNatOrdinal (b ⨳ c ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a')) ⊢ ∃ a', a' < a ∧ ∃ b', b' < b ∧ ∃ c', c' < c ∧ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) ≤ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a') + ↑toNatOrdinal (b' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a')) ** use a', ha, b', hb, c', hc ** case right a b c d b' : Ordinal.{u} hb : b' < b c' : Ordinal.{u} hc : c' < c a' : Ordinal.{u} ha : a' < a h : ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (b' ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a')) ≤ ↑toOrdinal (↑toNatOrdinal (b' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ c' ⨳ a) + ↑toNatOrdinal (b ⨳ c ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a')) ⊢ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (b' ⨳ c ⨳ a') + ↑toNatOrdinal (b ⨳ c' ⨳ a') + ↑toNatOrdinal (b' ⨳ c' ⨳ a)) ≤ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ a') + ↑toNatOrdinal (b' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ c' ⨳ a) + ↑toNatOrdinal (b' ⨳ c' ⨳ a')) ** convert h using 1 <;> abel_nf ** case mpr.intro.intro.intro.intro.intro.intro a b c d b' : Ordinal.{u} hb : b' < a c' : Ordinal.{u} hc : c' < b a' : Ordinal.{u} ha : a' < c h : ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (c' ⨳ c ⨳ b') + ↑toNatOrdinal (b ⨳ a' ⨳ b') + ↑toNatOrdinal (c' ⨳ a' ⨳ a)) ≤ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ b') + ↑toNatOrdinal (c' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ a' ⨳ a) + ↑toNatOrdinal (c' ⨳ a' ⨳ b')) ⊢ ∃ a', a' < b ∧ ∃ b', b' < c ∧ ∃ c', c' < a ∧ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (a' ⨳ b' ⨳ a) + ↑toNatOrdinal (a' ⨳ c ⨳ c') + ↑toNatOrdinal (b ⨳ b' ⨳ c')) ≤ ↑toOrdinal (↑toNatOrdinal (a' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ b' ⨳ a) + ↑toNatOrdinal (b ⨳ c ⨳ c') + ↑toNatOrdinal (a' ⨳ b' ⨳ c')) ** use c', hc, a', ha, b', hb ** case right a b c d b' : Ordinal.{u} hb : b' < a c' : Ordinal.{u} hc : c' < b a' : Ordinal.{u} ha : a' < c h : ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (c' ⨳ c ⨳ b') + ↑toNatOrdinal (b ⨳ a' ⨳ b') + ↑toNatOrdinal (c' ⨳ a' ⨳ a)) ≤ ↑toOrdinal (↑toNatOrdinal (b ⨳ c ⨳ b') + ↑toNatOrdinal (c' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ a' ⨳ a) + ↑toNatOrdinal (c' ⨳ a' ⨳ b')) ⊢ ↑toOrdinal (↑toNatOrdinal d + ↑toNatOrdinal (c' ⨳ a' ⨳ a) + ↑toNatOrdinal (c' ⨳ c ⨳ b') + ↑toNatOrdinal (b ⨳ a' ⨳ b')) ≤ ↑toOrdinal (↑toNatOrdinal (c' ⨳ c ⨳ a) + ↑toNatOrdinal (b ⨳ a' ⨳ a) + ↑toNatOrdinal (b ⨳ c ⨳ b') + ↑toNatOrdinal (c' ⨳ a' ⨳ b')) ** convert h using 1 <;> abel_nf ** Qed | |
Ordinal.nmul_le_iff₃' ** a b c d : Ordinal.{u} ⊢ a ⨳ (b ⨳ c) ≤ d ↔ ∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → ∀ (c' : Ordinal.{u}), c' < c → a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ** rw [← not_iff_not] ** a b c d : Ordinal.{u} ⊢ ¬a ⨳ (b ⨳ c) ≤ d ↔ ¬∀ (a' : Ordinal.{u}), a' < a → ∀ (b' : Ordinal.{u}), b' < b → ∀ (c' : Ordinal.{u}), c' < c → a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ** simp [lt_nmul_iff₃'] ** Qed | |
Ordinal.nmul_assoc ** a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ⨳ b ⨳ c = a ⨳ (b ⨳ c) ** apply le_antisymm ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ⨳ b ⨳ c ≤ a ⨳ (b ⨳ c) ** rw [nmul_le_iff₃] ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → ∀ (c' : Ordinal.{u_1}), c' < c → a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ (b ⨳ c) ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ** intro a' ha b' hb c' hc ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b c' : Ordinal.{u_1} hc : c' < c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ (b ⨳ c) ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ** rw [nmul_assoc a' b c, nmul_assoc a b' c, nmul_assoc a b c', nmul_assoc a' b' c',
nmul_assoc a' b' c, nmul_assoc a' b c', nmul_assoc a b' c'] ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b c' : Ordinal.{u_1} hc : c' < c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ** exact nmul_nadd_lt₃' ha hb hc ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ a ⨳ (b ⨳ c) ≤ a ⨳ b ⨳ c ** rw [nmul_le_iff₃'] ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c : Ordinal.{u_1} ⊢ ∀ (a' : Ordinal.{u_1}), a' < a → ∀ (b' : Ordinal.{u_1}), b' < b → ∀ (c' : Ordinal.{u_1}), c' < c → a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < a ⨳ b ⨳ c ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ** intro a' ha b' hb c' hc ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b c' : Ordinal.{u_1} hc : c' < c ⊢ a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') < a ⨳ b ⨳ c ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ** rw [← nmul_assoc a' b c, ← nmul_assoc a b' c, ← nmul_assoc a b c', ← nmul_assoc a' b' c',
← nmul_assoc a' b' c, ← nmul_assoc a' b c', ← nmul_assoc a b' c'] ** case a a✝ b✝ c✝ d : Ordinal.{u} a b c a' : Ordinal.{u_1} ha : a' < a b' : Ordinal.{u_1} hb : b' < b c' : Ordinal.{u_1} hc : c' < c ⊢ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' < a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ** exact nmul_nadd_lt₃ ha hb hc ** Qed | |
Ordinal.nmul_succ ** a b : Ordinal.{u_1} ⊢ a ⨳ succ b = a ⨳ b ♯ a ** rw [← nadd_one, nmul_nadd_one] ** Qed | |
Ordinal.succ_nmul ** a b : Ordinal.{u_1} ⊢ succ a ⨳ b = a ⨳ b ♯ b ** rw [← nadd_one, nadd_one_nmul] ** Qed | |
NatOrdinal.mul_le_nmul ** a b : Ordinal.{u} ⊢ a * b ≤ a ⨳ b ** refine b.limitRecOn ?_ ?_ ?_ ** case refine_1 a b : Ordinal.{u} ⊢ a * 0 ≤ a ⨳ 0 ** simp ** case refine_2 a b : Ordinal.{u} ⊢ ∀ (o : Ordinal.{u}), a * o ≤ a ⨳ o → a * succ o ≤ a ⨳ succ o ** intro c h ** case refine_2 a b c : Ordinal.{u} h : a * c ≤ a ⨳ c ⊢ a * succ c ≤ a ⨳ succ c ** rw [mul_succ, nmul_succ] ** case refine_2 a b c : Ordinal.{u} h : a * c ≤ a ⨳ c ⊢ a * c + a ≤ a ⨳ c ♯ a ** exact (add_le_nadd _ a).trans (nadd_le_nadd_right h a) ** case refine_3 a b : Ordinal.{u} ⊢ ∀ (o : Ordinal.{u}), IsLimit o → (∀ (o' : Ordinal.{u}), o' < o → a * o' ≤ a ⨳ o') → a * o ≤ a ⨳ o ** intro c hc H ** case refine_3 a b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a * o' ≤ a ⨳ o' ⊢ a * c ≤ a ⨳ c ** rcases eq_zero_or_pos a with (rfl | ha) ** case refine_3.inl b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → 0 * o' ≤ 0 ⨳ o' ⊢ 0 * c ≤ 0 ⨳ c ** simp ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a * o' ≤ a ⨳ o' ha : 0 < a ⊢ a * c ≤ a ⨳ c ** have := IsNormal.blsub_eq.{u, u} (mul_isNormal ha) hc ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a * o' ≤ a ⨳ o' ha : 0 < a this : (blsub c fun x x_1 => (fun x x_2 => x * x_2) a x) = (fun x x_1 => x * x_1) a c ⊢ a * c ≤ a ⨳ c ** dsimp at this ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a * o' ≤ a ⨳ o' ha : 0 < a this : (blsub c fun x x_1 => a * x) = a * c ⊢ a * c ≤ a ⨳ c ** rw [← this, blsub_le_iff] ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : ∀ (o' : Ordinal.{u}), o' < c → a * o' ≤ a ⨳ o' ha : 0 < a this : (blsub c fun x x_1 => a * x) = a * c ⊢ ∀ (i : Ordinal.{u}), i < c → a * i < a ⨳ c ** exact fun i hi => (H i hi).trans_lt (nmul_lt_nmul_of_pos_left hi ha) ** Qed | |
SetTheory.PGame.birthday_def ** x : PGame ⊢ birthday x = max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) ** cases x ** case mk α✝ β✝ : Type u a✝¹ : α✝ → PGame a✝ : β✝ → PGame ⊢ birthday (mk α✝ β✝ a✝¹ a✝) = max (lsub fun i => birthday (moveLeft (mk α✝ β✝ a✝¹ a✝) i)) (lsub fun i => birthday (moveRight (mk α✝ β✝ a✝¹ a✝) i)) ** rw [birthday] ** case mk α✝ β✝ : Type u a✝¹ : α✝ → PGame a✝ : β✝ → PGame ⊢ max (lsub fun i => birthday (a✝¹ i)) (lsub fun i => birthday (a✝ i)) = max (lsub fun i => birthday (moveLeft (mk α✝ β✝ a✝¹ a✝) i)) (lsub fun i => birthday (moveRight (mk α✝ β✝ a✝¹ a✝) i)) ** rfl ** Qed | |
SetTheory.PGame.birthday_moveLeft_lt ** x : PGame i : LeftMoves x ⊢ birthday (moveLeft x i) < birthday x ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame i : LeftMoves (mk α✝ β✝ a✝¹ a✝) ⊢ birthday (moveLeft (mk α✝ β✝ a✝¹ a✝) i) < birthday (mk α✝ β✝ a✝¹ a✝) ** rw [birthday] ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame i : LeftMoves (mk α✝ β✝ a✝¹ a✝) ⊢ birthday (moveLeft (mk α✝ β✝ a✝¹ a✝) i) < max (lsub fun i => birthday (a✝¹ i)) (lsub fun i => birthday (a✝ i)) ** exact lt_max_of_lt_left (lt_lsub _ i) ** Qed | |
SetTheory.PGame.birthday_moveRight_lt ** x : PGame i : RightMoves x ⊢ birthday (moveRight x i) < birthday x ** cases x ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame i : RightMoves (mk α✝ β✝ a✝¹ a✝) ⊢ birthday (moveRight (mk α✝ β✝ a✝¹ a✝) i) < birthday (mk α✝ β✝ a✝¹ a✝) ** rw [birthday] ** case mk α✝ β✝ : Type u_1 a✝¹ : α✝ → PGame a✝ : β✝ → PGame i : RightMoves (mk α✝ β✝ a✝¹ a✝) ⊢ birthday (moveRight (mk α✝ β✝ a✝¹ a✝) i) < max (lsub fun i => birthday (a✝¹ i)) (lsub fun i => birthday (a✝ i)) ** exact lt_max_of_lt_right (lt_lsub _ i) ** Qed | |
SetTheory.PGame.lt_birthday_iff ** x : PGame o : Ordinal.{u_1} ⊢ o < birthday x ↔ (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** constructor ** case mp x : PGame o : Ordinal.{u_1} ⊢ o < birthday x → (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** rw [birthday_def] ** case mp x : PGame o : Ordinal.{u_1} ⊢ o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) → (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** intro h ** case mp x : PGame o : Ordinal.{u_1} h : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) ⊢ (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** cases' lt_max_iff.1 h with h' h' ** case mp.inl x : PGame o : Ordinal.{u_1} h : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) h' : o < lsub fun i => birthday (moveLeft x i) ⊢ (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** left ** case mp.inl.h x : PGame o : Ordinal.{u_1} h : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) h' : o < lsub fun i => birthday (moveLeft x i) ⊢ ∃ i, o ≤ birthday (moveLeft x i) ** rwa [lt_lsub_iff] at h' ** case mp.inr x : PGame o : Ordinal.{u_1} h : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) h' : o < lsub fun i => birthday (moveRight x i) ⊢ (∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i) ** right ** case mp.inr.h x : PGame o : Ordinal.{u_1} h : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) h' : o < lsub fun i => birthday (moveRight x i) ⊢ ∃ i, o ≤ birthday (moveRight x i) ** rwa [lt_lsub_iff] at h' ** case mpr x : PGame o : Ordinal.{u_1} ⊢ ((∃ i, o ≤ birthday (moveLeft x i)) ∨ ∃ i, o ≤ birthday (moveRight x i)) → o < birthday x ** rintro (⟨i, hi⟩ | ⟨i, hi⟩) ** case mpr.inl.intro x : PGame o : Ordinal.{u_1} i : LeftMoves x hi : o ≤ birthday (moveLeft x i) ⊢ o < birthday x ** exact hi.trans_lt (birthday_moveLeft_lt i) ** case mpr.inr.intro x : PGame o : Ordinal.{u_1} i : RightMoves x hi : o ≤ birthday (moveRight x i) ⊢ o < birthday x ** exact hi.trans_lt (birthday_moveRight_lt i) ** Qed | |
SetTheory.PGame.Relabelling.birthday_congr ** xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ birthday (PGame.mk xl xr xL xR) = birthday (PGame.mk yl yr yL yR) ** unfold birthday ** xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ max (lsub fun i => birthday (xL i)) (lsub fun i => birthday (xR i)) = max (lsub fun i => birthday (yL i)) (lsub fun i => birthday (yR i)) ** congr 1 ** case e_a xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ (lsub fun i => birthday (xL i)) = lsub fun i => birthday (yL i) case e_a xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ (lsub fun i => birthday (xR i)) = lsub fun i => birthday (yR i) ** all_goals
apply lsub_eq_of_range_eq.{u, u, u}
ext i; constructor ** case e_a.h.mp xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (xL i)) → i ∈ Set.range fun i => birthday (yL i) case e_a.h.mpr xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (yL i)) → i ∈ Set.range fun i => birthday (xL i) case e_a.h.mp xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (xR i)) → i ∈ Set.range fun i => birthday (yR i) case e_a.h.mpr xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (yR i)) → i ∈ Set.range fun i => birthday (xR i) ** all_goals rintro ⟨j, rfl⟩ ** case e_a xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ (lsub fun i => birthday (xR i)) = lsub fun i => birthday (yR i) ** apply lsub_eq_of_range_eq.{u, u, u} ** case e_a xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR ⊢ (Set.range fun i => birthday (xR i)) = Set.range fun i => birthday (yR i) ** ext i ** case e_a.h xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (xR i)) ↔ i ∈ Set.range fun i => birthday (yR i) ** constructor ** case e_a.h.mpr xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR i : Ordinal.{u} ⊢ (i ∈ Set.range fun i => birthday (yR i)) → i ∈ Set.range fun i => birthday (xR i) ** rintro ⟨j, rfl⟩ ** case e_a.h.mp.intro xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR j : xl ⊢ (fun i => birthday (xL i)) j ∈ Set.range fun i => birthday (yL i) ** exact ⟨_, (r.moveLeft j).birthday_congr.symm⟩ ** case e_a.h.mpr.intro xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR j : yl ⊢ (fun i => birthday (yL i)) j ∈ Set.range fun i => birthday (xL i) ** exact ⟨_, (r.moveLeftSymm j).birthday_congr⟩ ** case e_a.h.mp.intro xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR j : xr ⊢ (fun i => birthday (xR i)) j ∈ Set.range fun i => birthday (yR i) ** exact ⟨_, (r.moveRight j).birthday_congr.symm⟩ ** case e_a.h.mpr.intro xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame r : PGame.mk xl xr xL xR ≡r PGame.mk yl yr yL yR j : yr ⊢ (fun i => birthday (yR i)) j ∈ Set.range fun i => birthday (xR i) ** exact ⟨_, (r.moveRightSymm j).birthday_congr⟩ ** Qed | |
SetTheory.PGame.birthday_eq_zero ** x : PGame ⊢ birthday x = 0 ↔ IsEmpty (LeftMoves x) ∧ IsEmpty (RightMoves x) ** rw [birthday_def, max_eq_zero, lsub_eq_zero_iff, lsub_eq_zero_iff] ** Qed | |
SetTheory.PGame.birthday_zero ** ⊢ birthday 0 = 0 ** simp [inferInstanceAs (IsEmpty PEmpty)] ** Qed | |
SetTheory.PGame.birthday_one ** ⊢ birthday 1 = 1 ** rw [birthday_def] ** ⊢ max (lsub fun i => birthday (moveLeft 1 i)) (lsub fun i => birthday (moveRight 1 i)) = 1 ** simp ** Qed | |
SetTheory.PGame.birthday_star ** ⊢ birthday star = 1 ** rw [birthday_def] ** ⊢ max (lsub fun i => birthday (moveLeft star i)) (lsub fun i => birthday (moveRight star i)) = 1 ** simp ** Qed | |
SetTheory.PGame.neg_birthday ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame ⊢ birthday (-mk xl xr xL xR) = birthday (mk xl xr xL xR) ** rw [birthday_def, birthday_def, max_comm] ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame ⊢ max (lsub fun i => birthday (moveRight (-mk xl xr xL xR) i)) (lsub fun i => birthday (moveLeft (-mk xl xr xL xR) i)) = max (lsub fun i => birthday (moveLeft (mk xl xr xL xR) i)) (lsub fun i => birthday (moveRight (mk xl xr xL xR) i)) ** congr <;> funext <;> apply neg_birthday ** Qed | |
SetTheory.PGame.toPGame_birthday ** o : Ordinal.{u_1} ⊢ birthday (toPGame o) = o ** induction' o using Ordinal.induction with o IH ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (toPGame k) = k ⊢ birthday (toPGame o) = o ** rw [toPGame_def, PGame.birthday] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (toPGame k) = k ⊢ max (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) (lsub fun i => birthday (PEmpty.elim i)) = o ** simp only [lsub_empty, max_zero_right] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (toPGame k) = k ⊢ (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) = o ** conv_rhs => rw [← lsub_typein o] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (toPGame k) = k ⊢ (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) = lsub (typein fun x x_1 => x < x_1) ** congr with x ** case h.e_f.h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (toPGame k) = k x : (Quotient.out o).α ⊢ birthday (toPGame (typein (fun x x_1 => x < x_1) x)) = typein (fun x x_1 => x < x_1) x ** exact IH _ (typein_lt_self x) ** Qed | |
SetTheory.PGame.le_birthday ** xl β✝ : Type u_1 xL : xl → PGame a✝ : β✝ → PGame i : LeftMoves (mk xl β✝ xL a✝) ⊢ moveLeft (mk xl β✝ xL a✝) i ≤ moveLeft (toPGame (birthday (mk xl β✝ xL a✝))) (↑toLeftMovesToPGame { val := birthday (moveLeft (mk xl β✝ xL a✝) i), property := (_ : birthday (moveLeft (mk xl β✝ xL a✝) i) < birthday (mk xl β✝ xL a✝)) }) ** simp [le_birthday (xL i)] ** Qed | |
SetTheory.PGame.neg_birthday_le ** a b x : PGame ⊢ -toPGame (birthday x) ≤ x ** simpa only [neg_birthday, ← neg_le_iff] using le_birthday (-x) ** Qed | |
SetTheory.PGame.birthday_add ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ birthday (mk xl xr xL xR + mk yl yr yL yR) = birthday (mk xl xr xL xR) ♯ birthday (mk yl yr yL yR) ** rw [birthday_def, nadd_def] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (lsub fun i => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) i)) (lsub fun i => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) i)) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** erw [lsub_sum, lsub_sum] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) (Sum.inl a))) (lsub fun b => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) (Sum.inr b)))) (max (lsub fun a => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) (Sum.inl a))) (lsub fun b => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) (Sum.inr b)))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** simp only [lsub_sum, mk_add_moveLeft_inl, moveLeft_mk, mk_add_moveLeft_inr,
mk_add_moveRight_inl, moveRight_mk, mk_add_moveRight_inr] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b))) (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** conv_lhs => left; left; right; intro a; rw [birthday_add (xL a) ⟨yl, yr, yL, yR⟩] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b))) (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** conv_lhs => left; right; right; intro b; rw [birthday_add ⟨xl, xr, xL, xR⟩ (yL b)] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b))) (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** conv_lhs => right; left; right; intro a; rw [birthday_add (xR a) ⟨yl, yr, yL, yR⟩] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b))) (max (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** conv_lhs => right; right; right; intro b; rw [birthday_add ⟨xl, xr, xL, xR⟩ (yR b)] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b))) (max (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** rw [max_max_max_comm] ** a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR))) (max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b))) = max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ** congr <;> apply le_antisymm ** case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) ≤ blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR) case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) ≤ max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ≤ blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b' case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ≤ max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** any_goals
exact
max_le_iff.2
⟨lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),
lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)⟩ ** case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ (blsub (birthday (mk xl xr xL xR)) fun a' x => a' ♯ birthday (mk yl yr yL yR)) ≤ max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ≤ max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** all_goals
refine blsub_le_iff.2 fun i hi => ?_
rcases lt_birthday_iff.1 hi with (⟨j, hj⟩ | ⟨j, hj⟩) ** case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ≤ blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b' ** exact
max_le_iff.2
⟨lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),
lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)⟩ ** case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame ⊢ (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) ♯ b') ≤ max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** refine blsub_le_iff.2 fun i hi => ?_ ** case e_a.a a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame i : Ordinal.{u} hi : i < birthday (mk yl yr yL yR) ⊢ birthday (mk xl xr xL xR) ♯ i < max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** rcases lt_birthday_iff.1 hi with (⟨j, hj⟩ | ⟨j, hj⟩) ** case e_a.a.inl.intro a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame i : Ordinal.{u} hi : i < birthday (mk xl xr xL xR) j : LeftMoves (mk xl xr xL xR) hj : i ≤ birthday (moveLeft (mk xl xr xL xR) j) ⊢ i ♯ birthday (mk yl yr yL yR) < max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) ** exact lt_max_of_lt_left ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _)) ** case e_a.a.inr.intro a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame i : Ordinal.{u} hi : i < birthday (mk xl xr xL xR) j : RightMoves (mk xl xr xL xR) hj : i ≤ birthday (moveRight (mk xl xr xL xR) j) ⊢ i ♯ birthday (mk yl yr yL yR) < max (lsub fun a => birthday (xL a) ♯ birthday (mk yl yr yL yR)) (lsub fun a => birthday (xR a) ♯ birthday (mk yl yr yL yR)) ** exact lt_max_of_lt_right ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _)) ** case e_a.a.inl.intro a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame i : Ordinal.{u} hi : i < birthday (mk yl yr yL yR) j : LeftMoves (mk yl yr yL yR) hj : i ≤ birthday (moveLeft (mk yl yr yL yR) j) ⊢ birthday (mk xl xr xL xR) ♯ i < max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** exact lt_max_of_lt_left ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _)) ** case e_a.a.inr.intro a b x : PGame xl xr : Type u xL : xl → PGame xR : xr → PGame yl yr : Type u yL : yl → PGame yR : yr → PGame i : Ordinal.{u} hi : i < birthday (mk yl yr yL yR) j : RightMoves (mk yl yr yL yR) hj : i ≤ birthday (moveRight (mk yl yr yL yR) j) ⊢ birthday (mk xl xr xL xR) ♯ i < max (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yL b)) (lsub fun b => birthday (mk xl xr xL xR) ♯ birthday (yR b)) ** exact lt_max_of_lt_right ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _)) ** Qed | |
SetTheory.PGame.birthday_add_zero ** a b x : PGame ⊢ birthday (a + 0) = birthday a ** simp ** Qed | |
SetTheory.PGame.birthday_zero_add ** a b x : PGame ⊢ birthday (0 + a) = birthday a ** simp ** Qed | |
SetTheory.PGame.birthday_add_one ** a b x : PGame ⊢ birthday (a + 1) = Order.succ (birthday a) ** simp ** Qed | |
SetTheory.PGame.birthday_one_add ** a b x : PGame ⊢ birthday (1 + a) = Order.succ (birthday a) ** simp ** Qed | |
SetTheory.PGame.birthday_nat_cast ** a b x : PGame n : ℕ ⊢ birthday ↑(n + 1) = ↑(n + 1) ** simp [birthday_nat_cast] ** Qed | |
SetTheory.PGame.birthday_add_nat ** a b x : PGame n : ℕ ⊢ birthday (a + ↑n) = birthday a + ↑n ** simp ** Qed | |
SetTheory.PGame.birthday_nat_add ** a b x : PGame n : ℕ ⊢ birthday (↑n + a) = birthday a + ↑n ** simp ** Qed | |
ZFSet.IsTransitive.inter ** x y z✝ : ZFSet hx : IsTransitive x hy : IsTransitive y z : ZFSet hz : z ∈ x ∩ y w : ZFSet hw : w ∈ z ⊢ w ∈ x ∩ y ** rw [mem_inter] at hz ⊢ ** x y z✝ : ZFSet hx : IsTransitive x hy : IsTransitive y z : ZFSet hz : z ∈ x ∧ z ∈ y w : ZFSet hw : w ∈ z ⊢ w ∈ x ∧ w ∈ y ** exact ⟨hx.mem_trans hw hz.1, hy.mem_trans hw hz.2⟩ ** Qed | |
ZFSet.IsTransitive.sUnion ** x y✝ z✝ : ZFSet h : IsTransitive x y : ZFSet hy : y ∈ ⋃₀ x z : ZFSet hz : z ∈ y ⊢ z ∈ ⋃₀ x ** rcases mem_sUnion.1 hy with ⟨w, hw, hw'⟩ ** case intro.intro x y✝ z✝ : ZFSet h : IsTransitive x y : ZFSet hy : y ∈ ⋃₀ x z : ZFSet hz : z ∈ y w : ZFSet hw : w ∈ x hw' : y ∈ w ⊢ z ∈ ⋃₀ x ** exact mem_sUnion_of_mem hz (h.mem_trans hw' hw) ** Qed | |
ZFSet.IsTransitive.sUnion' ** x y✝ z✝ : ZFSet H : ∀ (y : ZFSet), y ∈ x → IsTransitive y y : ZFSet hy : y ∈ ⋃₀ x z : ZFSet hz : z ∈ y ⊢ z ∈ ⋃₀ x ** rcases mem_sUnion.1 hy with ⟨w, hw, hw'⟩ ** case intro.intro x y✝ z✝ : ZFSet H : ∀ (y : ZFSet), y ∈ x → IsTransitive y y : ZFSet hy : y ∈ ⋃₀ x z : ZFSet hz : z ∈ y w : ZFSet hw : w ∈ x hw' : y ∈ w ⊢ z ∈ ⋃₀ x ** exact mem_sUnion_of_mem ((H w hw).mem_trans hz hw') hw ** Qed | |
ZFSet.IsTransitive.union ** x y z : ZFSet hx : IsTransitive x hy : IsTransitive y ⊢ IsTransitive (x ∪ y) ** rw [← sUnion_pair] ** x y z : ZFSet hx : IsTransitive x hy : IsTransitive y ⊢ IsTransitive (⋃₀ {x, y}) ** apply IsTransitive.sUnion' fun z => _ ** x y z : ZFSet hx : IsTransitive x hy : IsTransitive y ⊢ ∀ (z : ZFSet), z ∈ {x, y} → IsTransitive z ** intro ** x y z : ZFSet hx : IsTransitive x hy : IsTransitive y z✝ : ZFSet ⊢ z✝ ∈ {x, y} → IsTransitive z✝ ** rw [mem_pair] ** x y z : ZFSet hx : IsTransitive x hy : IsTransitive y z✝ : ZFSet ⊢ z✝ = x ∨ z✝ = y → IsTransitive z✝ ** rintro (rfl | rfl) ** case inl y z : ZFSet hy : IsTransitive y z✝ : ZFSet hx : IsTransitive z✝ ⊢ IsTransitive z✝ case inr x z : ZFSet hx : IsTransitive x z✝ : ZFSet hy : IsTransitive z✝ ⊢ IsTransitive z✝ ** assumption' ** Qed | |
ZFSet.IsTransitive.powerset ** x y✝ z✝ : ZFSet h : IsTransitive x y : ZFSet hy : y ∈ powerset x z : ZFSet hz : z ∈ y ⊢ z ∈ powerset x ** rw [mem_powerset] at hy ⊢ ** x y✝ z✝ : ZFSet h : IsTransitive x y : ZFSet hy : y ⊆ x z : ZFSet hz : z ∈ y ⊢ z ⊆ x ** exact h.subset_of_mem (hy hz) ** Qed | |
ZFSet.isTransitive_iff_sUnion_subset ** x y✝ z : ZFSet h : IsTransitive x y : ZFSet hy : y ∈ ⋃₀ x ⊢ y ∈ x ** rcases mem_sUnion.1 hy with ⟨z, hz, hz'⟩ ** case intro.intro x y✝ z✝ : ZFSet h : IsTransitive x y : ZFSet hy : y ∈ ⋃₀ x z : ZFSet hz : z ∈ x hz' : y ∈ z ⊢ y ∈ x ** exact h.mem_trans hz' hz ** Qed | |
Profinite.epi_iff_surjective ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Epi f ↔ Function.Surjective ↑f ** constructor ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Epi f → Function.Surjective ↑f ** dsimp [Function.Surjective] ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Epi f → ∀ (b : (forget Profinite).obj Y), ∃ a, ↑f a = b ** contrapose! ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ (∃ b, ∀ (a : (forget Profinite).obj X), ↑f a ≠ b) → ¬Epi f ** rintro ⟨y, hy⟩ hf ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f ⊢ False ** let C := Set.range f ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f ⊢ False ** have hC : IsClosed C := (isCompact_range f.continuous).isClosed ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C ⊢ False ** let U := Cᶜ ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ ⊢ False ** have hyU : y ∈ U := by
refine' Set.mem_compl _
rintro ⟨y', hy'⟩
exact hy y' hy' ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U ⊢ False ** have hUy : U ∈ 𝓝 y := hC.compl_mem_nhds hyU ** case mp.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y ⊢ False ** obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_clopen.mem_nhds_iff.mp hUy ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U ⊢ False ** classical
let Z := of (ULift.{u} <| Fin 2)
let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩
let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩
have H : h = g := by
rw [← cancel_epi f]
ext x
apply ULift.ext
dsimp [LocallyConstant.ofClopen]
erw [comp_apply, ContinuousMap.coe_mk, comp_apply, ContinuousMap.coe_mk,
Function.comp_apply, if_neg]
refine' mt (fun α => hVU α) _
simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff]
apply_fun fun e => (e y).down at H
dsimp [LocallyConstant.ofClopen] at H
erw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Function.comp_apply, if_pos hyV] at H
exact top_ne_bot H ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ ⊢ y ∈ U ** refine' Set.mem_compl _ ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ ⊢ ¬y ∈ C ** rintro ⟨y', hy'⟩ ** case intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ y' : (forget Profinite).obj X hy' : ↑f y' = y ⊢ False ** exact hy y' hy' ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U ⊢ False ** let Z := of (ULift.{u} <| Fin 2) ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) ⊢ False ** let g : Y ⟶ Z := ⟨(LocallyConstant.ofClopen hV).map ULift.up, LocallyConstant.continuous _⟩ ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) ⊢ False ** let h : Y ⟶ Z := ⟨fun _ => ⟨1⟩, continuous_const⟩ ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } ⊢ False ** have H : h = g := by
rw [← cancel_epi f]
ext x
apply ULift.ext
dsimp [LocallyConstant.ofClopen]
erw [comp_apply, ContinuousMap.coe_mk, comp_apply, ContinuousMap.coe_mk,
Function.comp_apply, if_neg]
refine' mt (fun α => hVU α) _
simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff] ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } H : h = g ⊢ False ** apply_fun fun e => (e y).down at H ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } H : (↑h y).down = (↑g y).down ⊢ False ** dsimp [LocallyConstant.ofClopen] at H ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } H : (↑(ContinuousMap.mk fun x => { down := 1 }) y).down = (↑(ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) y).down ⊢ False ** erw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Function.comp_apply, if_pos hyV] at H ** case mp.intro.intro.intro.intro X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } H : { down := 1 }.down = { down := 0 }.down ⊢ False ** exact top_ne_bot H ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } ⊢ h = g ** rw [← cancel_epi f] ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } ⊢ f ≫ h = f ≫ g ** ext x ** case w X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } x : (forget Profinite).obj X ⊢ ↑(f ≫ h) x = ↑(f ≫ g) x ** apply ULift.ext ** case w.h X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } x : (forget Profinite).obj X ⊢ (↑(f ≫ h) x).down = (↑(f ≫ g) x).down ** dsimp [LocallyConstant.ofClopen] ** case w.h X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } x : (forget Profinite).obj X ⊢ (↑(f ≫ ContinuousMap.mk fun x => { down := 1 }) x).down = (↑(f ≫ ContinuousMap.mk (ULift.up ∘ fun x => if x ∈ V then 0 else 1)) x).down ** erw [comp_apply, ContinuousMap.coe_mk, comp_apply, ContinuousMap.coe_mk,
Function.comp_apply, if_neg] ** case w.h.hnc X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } x : (forget Profinite).obj X ⊢ ¬↑f x ∈ V ** refine' mt (fun α => hVU α) _ ** case w.h.hnc X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y y : (forget Profinite).obj Y hy : ∀ (a : (forget Profinite).obj X), ↑f a ≠ y hf : Epi f C : Set ((forget Profinite).obj Y) := Set.range ↑f hC : IsClosed C U : Set ((forget Profinite).obj Y) := Cᶜ hyU : y ∈ U hUy : U ∈ 𝓝 y V : Set ((forget Profinite).obj Y) hV : V ∈ {s | IsClopen s} hyV : y ∈ V hVU : V ⊆ U Z : Profinite := of (ULift.{u, 0} (Fin 2)) g : Y ⟶ Z := ContinuousMap.mk ↑(LocallyConstant.map ULift.up (LocallyConstant.ofClopen hV)) h : Y ⟶ Z := ContinuousMap.mk fun x => { down := 1 } x : (forget Profinite).obj X ⊢ ¬↑f x ∈ U ** simp only [Set.mem_range_self, not_true, not_false_iff, Set.mem_compl_iff] ** case mpr X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Function.Surjective ↑f → Epi f ** rw [← CategoryTheory.epi_iff_surjective] ** case mpr X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Epi ↑f → Epi f ** apply (forget Profinite).epi_of_epi_map ** Qed | |
Profinite.mono_iff_injective ** X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Mono f ↔ Function.Injective ↑f ** constructor ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Mono f → Function.Injective ↑f ** intro h ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y h : Mono f ⊢ Function.Injective ↑f ** haveI : Limits.PreservesLimits profiniteToCompHaus := inferInstance ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y h : Mono f this : Limits.PreservesLimits profiniteToCompHaus ⊢ Function.Injective ↑f ** haveI : Mono (profiniteToCompHaus.map f) := inferInstance ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y h : Mono f this✝ : Limits.PreservesLimits profiniteToCompHaus this : Mono (profiniteToCompHaus.map f) ⊢ Function.Injective ↑f ** erw [← CompHaus.mono_iff_injective] ** case mp X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y h : Mono f this✝ : Limits.PreservesLimits profiniteToCompHaus this : Mono (profiniteToCompHaus.map f) ⊢ Mono f ** assumption ** case mpr X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Function.Injective ↑f → Mono f ** rw [← CategoryTheory.mono_iff_injective] ** case mpr X✝ Y✝ : Profinite f✝ : X✝ ⟶ Y✝ X Y : Profinite f : X ⟶ Y ⊢ Mono ↑f → Mono f ** apply (forget Profinite).mono_of_mono_map ** Qed | |
Function.Embedding.schroeder_bernstein ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g ⊢ ∃ h, Bijective h ** cases' isEmpty_or_nonempty β with hβ hβ ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β ⊢ ∃ h, Bijective h ** set F : Set α →o Set α :=
{ toFun := fun s => (g '' (f '' s)ᶜ)ᶜ
monotone' := fun s t hst =>
compl_subset_compl.mpr <| image_subset _ <| compl_subset_compl.mpr <| image_subset _ hst } ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } ⊢ ∃ h, Bijective h ** set s : Set α := OrderHom.lfp F ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F ⊢ ∃ h, Bijective h ** have hs : (g '' (f '' s)ᶜ)ᶜ = s := F.map_lfp ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s ⊢ ∃ h, Bijective h ** have hns : g '' (f '' s)ᶜ = sᶜ := compl_injective (by simp [hs]) ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ ⊢ ∃ h, Bijective h ** set g' := invFun g ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g ⊢ ∃ h, Bijective h ** have g'g : LeftInverse g' g := leftInverse_invFun hg ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g ⊢ ∃ h, Bijective h ** have hg'ns : g' '' sᶜ = (f '' s)ᶜ := by rw [← hns, g'g.image_image] ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ ⊢ ∃ h, Bijective h ** set h : α → β := s.piecewise f g' ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' ⊢ ∃ h, Bijective h ** have : Surjective h := by rw [← range_iff_surjective, range_piecewise, hg'ns, union_compl_self] ** case inr α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this✝ : Surjective h this : Injective h ⊢ ∃ h, Bijective h ** exact ⟨h, ‹Injective h›, ‹Surjective h›⟩ ** case inl α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : IsEmpty β ⊢ ∃ h, Bijective h ** have : IsEmpty α := Function.isEmpty f ** case inl α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : IsEmpty β this : IsEmpty α ⊢ ∃ h, Bijective h ** exact ⟨_, ((Equiv.equivEmpty α).trans (Equiv.equivEmpty β).symm).bijective⟩ ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s ⊢ (g '' (f '' s)ᶜ)ᶜ = sᶜᶜ ** simp [hs] ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g ⊢ g' '' sᶜ = (f '' s)ᶜ ** rw [← hns, g'g.image_image] ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' ⊢ Surjective h ** rw [← range_iff_surjective, range_piecewise, hg'ns, union_compl_self] ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h ⊢ Injective h ** refine' (injective_piecewise_iff _).2 ⟨hf.injOn _, _, _⟩ ** case refine'_1 α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h ⊢ InjOn g' sᶜ ** intro x hx y hy hxy ** case refine'_1 α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ sᶜ y : α hy : y ∈ sᶜ hxy : g' x = g' y ⊢ x = y ** obtain ⟨x', _, rfl⟩ : x ∈ g '' (f '' s)ᶜ := by rwa [hns] ** case refine'_1.intro.intro α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h y : α hy : y ∈ sᶜ x' : β left✝ : x' ∈ (f '' s)ᶜ hx : g x' ∈ sᶜ hxy : g' (g x') = g' y ⊢ g x' = y ** obtain ⟨y', _, rfl⟩ : y ∈ g '' (f '' s)ᶜ := by rwa [hns] ** case refine'_1.intro.intro.intro.intro α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x' : β left✝¹ : x' ∈ (f '' s)ᶜ hx : g x' ∈ sᶜ y' : β left✝ : y' ∈ (f '' s)ᶜ hy : g y' ∈ sᶜ hxy : g' (g x') = g' (g y') ⊢ g x' = g y' ** rw [g'g _, g'g _] at hxy ** case refine'_1.intro.intro.intro.intro α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x' : β left✝¹ : x' ∈ (f '' s)ᶜ hx : g x' ∈ sᶜ y' : β left✝ : y' ∈ (f '' s)ᶜ hy : g y' ∈ sᶜ hxy : x' = y' ⊢ g x' = g y' ** rw [hxy] ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ sᶜ y : α hy : y ∈ sᶜ hxy : g' x = g' y ⊢ x ∈ g '' (f '' s)ᶜ ** rwa [hns] ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h y : α hy : y ∈ sᶜ x' : β left✝ : x' ∈ (f '' s)ᶜ hx : g x' ∈ sᶜ hxy : g' (g x') = g' y ⊢ y ∈ g '' (f '' s)ᶜ ** rwa [hns] ** case refine'_2 α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h ⊢ ∀ (x : α), x ∈ s → ∀ (y : α), ¬y ∈ s → f x ≠ g' y ** intro x hx y hy hxy ** case refine'_2 α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ s y : α hy : ¬y ∈ s hxy : f x = g' y ⊢ False ** obtain ⟨y', hy', rfl⟩ : y ∈ g '' (f '' s)ᶜ := by rwa [hns] ** case refine'_2.intro.intro α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ s y' : β hy' : y' ∈ (f '' s)ᶜ hy : ¬g y' ∈ s hxy : f x = g' (g y') ⊢ False ** rw [g'g _] at hxy ** case refine'_2.intro.intro α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ s y' : β hy' : y' ∈ (f '' s)ᶜ hy : ¬g y' ∈ s hxy : f x = y' ⊢ False ** exact hy' ⟨x, hx, hxy⟩ ** α : Type u β : Type v f : α → β g : β → α hf : Injective f hg : Injective g hβ : Nonempty β F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := (_ : ∀ (s t : Set α), s ≤ t → (g '' (f '' s)ᶜ)ᶜ ⊆ (g '' (f '' t)ᶜ)ᶜ) } s : Set α := ↑OrderHom.lfp F hs : (g '' (f '' s)ᶜ)ᶜ = s hns : g '' (f '' s)ᶜ = sᶜ g' : α → β := invFun g g'g : LeftInverse g' g hg'ns : g' '' sᶜ = (f '' s)ᶜ h : α → β := piecewise s f g' this : Surjective h x : α hx : x ∈ s y : α hy : ¬y ∈ s hxy : f x = g' y ⊢ y ∈ g '' (f '' s)ᶜ ** rwa [hns] ** Qed | |
Function.Embedding.min_injective ** ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y ⊢ ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y ** simpa only [ne_eq, not_exists, not_forall, not_and] using h ** ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x : (i : ι) → β i hx : x ∈ insert f s y : (i : ι) → β i hy : y ∈ insert f s ⊢ ∀ (i : ι), x i = y i → x = y ** cases' hx with hx hx <;> cases' hy with hy hy ** case inl.inl ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x y : (i : ι) → β i hx : x = f hy : y = f ⊢ ∀ (i : ι), x i = y i → x = y ** simp [hx, hy] ** case inl.inr ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x y : (i : ι) → β i hx : x = f hy : y ∈ s ⊢ ∀ (i : ι), x i = y i → x = y ** subst x ** case inl.inr ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x y : (i : ι) → β i hy : y ∈ s ⊢ ∀ (i : ι), f i = y i → f = y ** exact fun i e => (hf i y hy e.symm).elim ** case inr.inl ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x y : (i : ι) → β i hx : x ∈ s hy : y = f ⊢ ∀ (i : ι), x i = y i → x = y ** subst y ** case inr.inl ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x : (i : ι) → β i hx : x ∈ s ⊢ ∀ (i : ι), x i = f i → x = f ** exact fun i e => (hf i x hx e).elim ** case inr.inr ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s h✝ : ¬∃ i, ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y h : ∀ (i : ι), ∃ y, ∀ (x : (i : ι) → β i), x ∈ s → x i ≠ y f : (x : ι) → β x hf : ∀ (x : ι) (x_1 : (i : ι) → β i), x_1 ∈ s → x_1 x ≠ f x x y : (i : ι) → β i hx : x ∈ s hy : y ∈ s ⊢ ∀ (i : ι), x i = y i → x = y ** exact hs x hx y hy ** ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s i : ι e : ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y f : β i → (i : ι) → β i hf : ∀ (x : β i), f x ∈ s ∧ f x i = x j : ι a b : β i e' : (fun a => f a j) a = (fun a => f a j) b ⊢ a = b ** let ⟨sa, ea⟩ := hf a ** ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s i : ι e : ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y f : β i → (i : ι) → β i hf : ∀ (x : β i), f x ∈ s ∧ f x i = x j : ι a b : β i e' : (fun a => f a j) a = (fun a => f a j) b sa : f a ∈ s ea : f a i = a ⊢ a = b ** let ⟨sb, eb⟩ := hf b ** ι : Type u β : ι → Type v I : Nonempty ι s : Set ((i : ι) → β i) hs : s ∈ Function.Embedding.sets β ms : ∀ (a : Set ((i : ι) → β i)), a ∈ Function.Embedding.sets β → s ⊆ a → a = s i : ι e : ∀ (y : β i), ∃ x, x ∈ s ∧ x i = y f : β i → (i : ι) → β i hf : ∀ (x : β i), f x ∈ s ∧ f x i = x j : ι a b : β i e' : (fun a => f a j) a = (fun a => f a j) b sa : f a ∈ s ea : f a i = a sb : f b ∈ s eb : f b i = b ⊢ a = b ** rw [← ea, ← eb, hs _ sa _ sb _ e'] ** Qed | |
Ordinal.nfpFamily_le_apply ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} inst✝ : Nonempty ι H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} ⊢ (∃ i, nfpFamily f a ≤ f i b) ↔ nfpFamily f a ≤ b ** rw [← not_iff_not] ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} inst✝ : Nonempty ι H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} ⊢ (¬∃ i, nfpFamily f a ≤ f i b) ↔ ¬nfpFamily f a ≤ b ** push_neg ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} inst✝ : Nonempty ι H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} ⊢ (∀ (i : ι), f i b < nfpFamily f a) ↔ b < nfpFamily f a ** exact apply_lt_nfpFamily_iff H ** Qed | |
Ordinal.nfpFamily_le_fp ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), Monotone (f i) a b : Ordinal.{max u v} ab : a ≤ b h : ∀ (i : ι), f i b ≤ b l : List ι ⊢ List.foldr f a l ≤ b ** by_cases hι : IsEmpty ι ** case pos ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), Monotone (f i) a b : Ordinal.{max u v} ab : a ≤ b h : ∀ (i : ι), f i b ≤ b l : List ι hι : IsEmpty ι ⊢ List.foldr f a l ≤ b ** rwa [Unique.eq_default l] ** case neg ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), Monotone (f i) a b : Ordinal.{max u v} ab : a ≤ b h : ∀ (i : ι), f i b ≤ b l : List ι hι : ¬IsEmpty ι ⊢ List.foldr f a l ≤ b ** induction' l with i l IH generalizing a ** case neg.cons ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), Monotone (f i) a✝ b : Ordinal.{max u v} ab✝ : a✝ ≤ b h : ∀ (i : ι), f i b ≤ b hι : ¬IsEmpty ι i : ι l : List ι IH : ∀ {a : Ordinal.{max u v}}, a ≤ b → List.foldr f a l ≤ b a : Ordinal.{max u v} ab : a ≤ b ⊢ List.foldr f a (i :: l) ≤ b ** exact (H i (IH ab)).trans (h i) ** case neg.nil ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), Monotone (f i) a✝ b : Ordinal.{max u v} ab✝ : a✝ ≤ b h : ∀ (i : ι), f i b ≤ b hι : ¬IsEmpty ι a : Ordinal.{max u v} ab : a ≤ b ⊢ List.foldr f a [] ≤ b ** exact ab ** Qed | |
Ordinal.nfpFamily_fp ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) a : Ordinal.{max u v} ⊢ f i (nfpFamily f a) = nfpFamily f a ** unfold nfpFamily ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) a : Ordinal.{max u v} ⊢ f i (sup (List.foldr f a)) = sup (List.foldr f a) ** rw [@IsNormal.sup.{u, v, v} _ H _ _ ⟨[]⟩] ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) a : Ordinal.{max u v} ⊢ sup (f i ∘ List.foldr f a) = sup (List.foldr f a) ** apply le_antisymm <;> refine' Ordinal.sup_le fun l => _ ** case a ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) a : Ordinal.{max u v} l : List ι ⊢ (f i ∘ List.foldr f a) l ≤ sup (List.foldr f a) ** exact le_sup _ (i::l) ** case a ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) a : Ordinal.{max u v} l : List ι ⊢ List.foldr f a l ≤ sup (f i ∘ List.foldr f a) ** exact (H.self_le _).trans (le_sup _ _) ** Qed | |
Ordinal.apply_le_nfpFamily ** ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} hι : Nonempty ι f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} ⊢ (∀ (i : ι), f i b ≤ nfpFamily f a) ↔ b ≤ nfpFamily f a ** refine' ⟨fun h => _, fun h i => _⟩ ** case refine'_2 ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} hι : Nonempty ι f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} h : b ≤ nfpFamily f a i : ι ⊢ f i b ≤ nfpFamily f a ** rw [← nfpFamily_fp (H i)] ** case refine'_2 ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} hι : Nonempty ι f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} h : b ≤ nfpFamily f a i : ι ⊢ f i b ≤ f i (nfpFamily f a) ** exact (H i).monotone h ** case refine'_1 ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} hι : Nonempty ι f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} h : ∀ (i : ι), f i b ≤ nfpFamily f a ⊢ b ≤ nfpFamily f a ** cases' hι with i ** case refine'_1.intro ι : Type u f✝ f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a b : Ordinal.{max u v} h : ∀ (i : ι), f i b ≤ nfpFamily f a i : ι ⊢ b ≤ nfpFamily f a ** exact ((H i).self_le b).trans (h i) ** Qed | |
Ordinal.nfpFamily_eq_self ** ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} f : ι → Ordinal.{max u u_1} → Ordinal.{max u u_1} a : Ordinal.{max u u_1} h : ∀ (i : ι), f i a = a l : List ι ⊢ List.foldr f a l ≤ a ** rw [List.foldr_fixed' h l] ** Qed | |
Ordinal.fp_family_unbounded ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} s : Set Ordinal.{max u v} x✝ : s ∈ Set.range fun i => fixedPoints (f i) i : ι hi : (fun i => fixedPoints (f i)) i = s ⊢ nfpFamily f a ∈ s ** rw [← hi, mem_fixedPoints_iff] ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} s : Set Ordinal.{max u v} x✝ : s ∈ Set.range fun i => fixedPoints (f i) i : ι hi : (fun i => fixedPoints (f i)) i = s ⊢ f i (nfpFamily f a) = nfpFamily f a ** exact nfpFamily_fp.{u, v} (H i) a ** Qed | |
Ordinal.derivFamily_isNormal ** ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} f : ι → Ordinal.{max u u_1} → Ordinal.{max u u_1} o : Ordinal.{max u_1 u} ⊢ derivFamily f o < derivFamily f (succ o) ** rw [derivFamily_succ, ← succ_le_iff] ** ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} f : ι → Ordinal.{max u u_1} → Ordinal.{max u u_1} o : Ordinal.{max u_1 u} ⊢ succ (derivFamily f o) ≤ nfpFamily f (succ (derivFamily f o)) ** apply le_nfpFamily ** ι : Type u f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v} f : ι → Ordinal.{max u u_1} → Ordinal.{max u u_1} o : Ordinal.{max u_1 u} l : IsLimit o a : Ordinal.{max u_1 u} ⊢ derivFamily f o ≤ a ↔ ∀ (b : Ordinal.{max u_1 u}), b < o → derivFamily f b ≤ a ** rw [derivFamily_limit _ l, bsup_le_iff] ** Qed | |
Ordinal.derivFamily_fp ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} ⊢ f i (derivFamily f o) = derivFamily f o ** induction' o using limitRecOn with o _ o l IH ** case H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) ⊢ f i (derivFamily f 0) = derivFamily f 0 ** rw [derivFamily_zero] ** case H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) ⊢ f i (nfpFamily f 0) = nfpFamily f 0 ** exact nfpFamily_fp H 0 ** case H₂ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} a✝ : f i (derivFamily f o) = derivFamily f o ⊢ f i (derivFamily f (succ o)) = derivFamily f (succ o) ** rw [derivFamily_succ] ** case H₂ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} a✝ : f i (derivFamily f o) = derivFamily f o ⊢ f i (nfpFamily f (succ (derivFamily f o))) = nfpFamily f (succ (derivFamily f o)) ** exact nfpFamily_fp H _ ** case H₃ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → f i (derivFamily f o') = derivFamily f o' ⊢ f i (derivFamily f o) = derivFamily f o ** rw [derivFamily_limit _ l,
IsNormal.bsup.{max u v, u, max u v} H (fun a _ => derivFamily f a) l.1] ** case H₃ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → f i (derivFamily f o') = derivFamily f o' ⊢ (bsup o fun a h => f i (derivFamily f a)) = bsup o fun a x => derivFamily f a ** refine' eq_of_forall_ge_iff fun c => _ ** case H₃ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} i : ι H : IsNormal (f i) o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → f i (derivFamily f o') = derivFamily f o' c : Ordinal.{max u v} ⊢ (bsup o fun a h => f i (derivFamily f a)) ≤ c ↔ (bsup o fun a x => derivFamily f a) ≤ c ** simp (config := { contextual := true }) only [bsup_le_iff, IH] ** Qed | |
Ordinal.le_iff_derivFamily ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a ⊢ ∃ o, derivFamily f o = a ** suffices : ∀ (o) (_ : a ≤ derivFamily.{u, v} f o), ∃ o, derivFamily.{u, v} f o = a ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a this : ∀ (o : Ordinal.{max u v}), a ≤ derivFamily f o → ∃ o, derivFamily f o = a ⊢ ∃ o, derivFamily f o = a case this ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a ⊢ ∀ (o : Ordinal.{max u v}), a ≤ derivFamily f o → ∃ o, derivFamily f o = a ** exact this a ((derivFamily_isNormal _).self_le _) ** case this ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a ⊢ ∀ (o : Ordinal.{max u v}), a ≤ derivFamily f o → ∃ o, derivFamily f o = a ** intro o ** case this ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} ⊢ a ≤ derivFamily f o → ∃ o, derivFamily f o = a ** induction' o using limitRecOn with o IH o l IH ** case this.H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a ⊢ a ≤ derivFamily f 0 → ∃ o, derivFamily f o = a ** intro h₁ ** case this.H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a h₁ : a ≤ derivFamily f 0 ⊢ ∃ o, derivFamily f o = a ** refine' ⟨0, le_antisymm _ h₁⟩ ** case this.H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a h₁ : a ≤ derivFamily f 0 ⊢ derivFamily f 0 ≤ a ** rw [derivFamily_zero] ** case this.H₁ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a h₁ : a ≤ derivFamily f 0 ⊢ nfpFamily f 0 ≤ a ** exact nfpFamily_le_fp (fun i => (H i).monotone) (Ordinal.zero_le _) ha ** case this.H₂ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a ⊢ a ≤ derivFamily f (succ o) → ∃ o, derivFamily f o = a ** intro h₁ ** case this.H₂ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f (succ o) ⊢ ∃ o, derivFamily f o = a ** cases' le_or_lt a (derivFamily.{u, v} f o) with h h ** case this.H₂.inr ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f (succ o) h : derivFamily f o < a ⊢ ∃ o, derivFamily f o = a ** refine' ⟨succ o, le_antisymm _ h₁⟩ ** case this.H₂.inr ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f (succ o) h : derivFamily f o < a ⊢ derivFamily f (succ o) ≤ a ** rw [derivFamily_succ] ** case this.H₂.inr ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f (succ o) h : derivFamily f o < a ⊢ nfpFamily f (succ (derivFamily f o)) ≤ a ** exact nfpFamily_le_fp (fun i => (H i).monotone) (succ_le_of_lt h) ha ** case this.H₂.inl ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} IH : a ≤ derivFamily f o → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f (succ o) h : a ≤ derivFamily f o ⊢ ∃ o, derivFamily f o = a ** exact IH h ** case this.H₃ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → a ≤ derivFamily f o' → ∃ o, derivFamily f o = a ⊢ a ≤ derivFamily f o → ∃ o, derivFamily f o = a ** intro h₁ ** case this.H₃ ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → a ≤ derivFamily f o' → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f o ⊢ ∃ o, derivFamily f o = a ** cases' eq_or_lt_of_le h₁ with h h ** case this.H₃.inr ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → a ≤ derivFamily f o' → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f o h : a < derivFamily f o ⊢ ∃ o, derivFamily f o = a ** rw [derivFamily_limit _ l, ← not_le, bsup_le_iff, not_ball] at h ** case this.H₃.inr ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → a ≤ derivFamily f o' → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f o h : ∃ x h, ¬derivFamily f x ≤ a ⊢ ∃ o, derivFamily f o = a ** exact
let ⟨o', h, hl⟩ := h
IH o' h (le_of_not_le hl) ** case this.H₃.inl ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), f i a ≤ a o : Ordinal.{max u v} l : IsLimit o IH : ∀ (o' : Ordinal.{max u v}), o' < o → a ≤ derivFamily f o' → ∃ o, derivFamily f o = a h₁ : a ≤ derivFamily f o h : a = derivFamily f o ⊢ ∃ o, derivFamily f o = a ** exact ⟨_, h.symm⟩ ** Qed | |
Ordinal.derivFamily_eq_enumOrd ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) ⊢ derivFamily f = enumOrd (⋂ i, fixedPoints (f i)) ** rw [← eq_enumOrd _ (fp_family_unbounded.{u, v} H)] ** ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) ⊢ StrictMono (derivFamily f) ∧ Set.range (derivFamily f) = ⋂ i, fixedPoints (f i) ** use (derivFamily_isNormal f).strictMono ** case right ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) ⊢ Set.range (derivFamily f) = ⋂ i, fixedPoints (f i) ** rw [Set.range_eq_iff] ** case right ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) ⊢ (∀ (a : Ordinal.{max u v}), derivFamily f a ∈ ⋂ i, fixedPoints (f i)) ∧ ∀ (b : Ordinal.{max u v}), b ∈ ⋂ i, fixedPoints (f i) → ∃ a, derivFamily f a = b ** refine' ⟨_, fun a ha => _⟩ ** case right.refine'_2 ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : a ∈ ⋂ i, fixedPoints (f i) ⊢ ∃ a_1, derivFamily f a_1 = a ** rw [Set.mem_iInter] at ha ** case right.refine'_2 ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} ha : ∀ (i : ι), a ∈ fixedPoints (f i) ⊢ ∃ a_1, derivFamily f a_1 = a ** rwa [← fp_iff_derivFamily H] ** case right.refine'_1 ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) ⊢ ∀ (a : Ordinal.{max u v}), derivFamily f a ∈ ⋂ i, fixedPoints (f i) ** rintro a S ⟨i, hi⟩ ** case right.refine'_1.intro ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} S : Set Ordinal.{max u v} i : ι hi : (fun i => fixedPoints (f i)) i = S ⊢ derivFamily f a ∈ S ** rw [← hi] ** case right.refine'_1.intro ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : ι), IsNormal (f i) a : Ordinal.{max u v} S : Set Ordinal.{max u v} i : ι hi : (fun i => fixedPoints (f i)) i = S ⊢ derivFamily f a ∈ (fun i => fixedPoints (f i)) i ** exact derivFamily_fp (H i) a ** Qed | |
Ordinal.apply_lt_nfpBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a b : Ordinal.{max v u} hb : b < nfpBFamily o f a i : Ordinal.{u} hi : i < o ⊢ f i hi b < nfpBFamily o f a ** rw [←familyOfBFamily_enum o f] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a b : Ordinal.{max v u} hb : b < nfpBFamily o f a i : Ordinal.{u} hi : i < o ⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) b < nfpBFamily o f a ** apply apply_lt_nfpFamily (fun _ => H _ _) hb ** Qed | |
Ordinal.apply_lt_nfpBFamily_iff ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b < nfpBFamily o f a ⊢ b < nfpBFamily o f a ** haveI := out_nonempty_iff_ne_zero.2 ho ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b < nfpBFamily o f a this : Nonempty (Quotient.out o).α ⊢ b < nfpBFamily o f a ** refine' (apply_lt_nfpFamily_iff.{u, v} _).1 fun _ => h _ _ ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b < nfpBFamily o f a this : Nonempty (Quotient.out o).α ⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i) ** exact fun _ => H _ _ ** Qed | |
Ordinal.nfpBFamily_le_apply ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} ⊢ (∃ i hi, nfpBFamily o f a ≤ f i hi b) ↔ nfpBFamily o f a ≤ b ** rw [← not_iff_not] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} ⊢ (¬∃ i hi, nfpBFamily o f a ≤ f i hi b) ↔ ¬nfpBFamily o f a ≤ b ** push_neg ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi b < nfpBFamily o f a) ↔ b < nfpBFamily o f a ** exact apply_lt_nfpBFamily_iff.{u, v} ho H ** Qed | |
Ordinal.nfpBFamily_fp ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max v u} ⊢ f i hi (nfpBFamily o f a) = nfpBFamily o f a ** rw [← familyOfBFamily_enum o f] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max v u} ⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) (nfpBFamily o f a) = nfpBFamily o f a ** apply nfpFamily_fp ** case H o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max v u} ⊢ IsNormal (familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1))) ** rw [familyOfBFamily_enum] ** case H o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max v u} ⊢ IsNormal (f i ?H.hi) case H.hi o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max v u} ⊢ i < o ** exact H ** Qed | |
Ordinal.apply_le_nfpBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi b ≤ nfpBFamily o f a) ↔ b ≤ nfpBFamily o f a ** refine' ⟨fun h => _, fun h i hi => _⟩ ** case refine'_1 o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b ≤ nfpBFamily o f a ⊢ b ≤ nfpBFamily o f a ** have ho' : 0 < o := Ordinal.pos_iff_ne_zero.2 ho ** case refine'_1 o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b ≤ nfpBFamily o f a ho' : 0 < o ⊢ b ≤ nfpBFamily o f a ** exact ((H 0 ho').self_le b).trans (h 0 ho') ** case refine'_2 o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : b ≤ nfpBFamily o f a i : Ordinal.{u} hi : i < o ⊢ f i hi b ≤ nfpBFamily o f a ** rw [← nfpBFamily_fp (H i hi)] ** case refine'_2 o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} ho : o ≠ 0 H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max v u} b : Ordinal.{max u v} h : b ≤ nfpBFamily o f a i : Ordinal.{u} hi : i < o ⊢ f i hi b ≤ f i hi (nfpBFamily o f a) ** exact (H i hi).monotone h ** Qed | |
Ordinal.fp_bfamily_unbounded ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ nfpBFamily o f a ∈ ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi) ** rw [Set.mem_iInter₂] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ ∀ (i : Ordinal.{u}) (j : i < o), nfpBFamily o f a ∈ fixedPoints (f i j) ** exact fun i hi => nfpBFamily_fp (H i hi) _ ** Qed | |
Ordinal.derivBFamily_fp ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max u v} ⊢ f i hi (derivBFamily o f a) = derivBFamily o f a ** rw [← familyOfBFamily_enum o f] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max u v} ⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) (derivBFamily o f a) = derivBFamily o f a ** apply derivFamily_fp ** case H o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max u v} ⊢ IsNormal (familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1))) ** rw [familyOfBFamily_enum] ** case H o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max u v} ⊢ IsNormal (f i ?H.hi) case H.hi o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} i : Ordinal.{u} hi : i < o H : IsNormal (f i hi) a : Ordinal.{max u v} ⊢ i < o ** exact H ** Qed | |
Ordinal.le_iff_derivBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a) ↔ ∃ b, derivBFamily o f b = a ** unfold derivBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a) ↔ ∃ b, derivFamily (familyOfBFamily o f) b = a ** rw [← le_iff_derivFamily] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a) ↔ ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a ** refine' ⟨fun h i => h _ _, fun h i hi => _⟩ ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a i : Ordinal.{u} hi : i < o ⊢ f i hi a ≤ a ** rw [← familyOfBFamily_enum o f] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i a ≤ a i : Ordinal.{u} hi : i < o ⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) a ≤ a ** apply h ** case H o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i) ** exact fun _ => H _ _ ** Qed | |
Ordinal.fp_iff_derivBFamily ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi a = a) ↔ ∃ b, derivBFamily o f b = a ** rw [← le_iff_derivBFamily H] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ⊢ (∀ (i : Ordinal.{u}) (hi : i < o), f i hi a = a) ↔ ∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a ** refine' ⟨fun h i hi => le_of_eq (h i hi), fun h i hi => _⟩ ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a i : Ordinal.{u} hi : i < o ⊢ f i hi a = a ** rw [← (H i hi).le_iff_eq] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a i : Ordinal.{u} hi : i < o ⊢ f i hi a ≤ a ** exact h i hi ** Qed | |
Ordinal.derivBFamily_eq_enumOrd ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) ⊢ derivBFamily o f = enumOrd (⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi)) ** rw [← eq_enumOrd _ (fp_bfamily_unbounded.{u, v} H)] ** o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) ⊢ StrictMono (derivBFamily o f) ∧ Set.range (derivBFamily o f) = ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi) ** use (derivBFamily_isNormal f).strictMono ** case right o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) ⊢ Set.range (derivBFamily o f) = ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi) ** rw [Set.range_eq_iff] ** case right o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) ⊢ (∀ (a : Ordinal.{max u v}), derivBFamily o f a ∈ ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi)) ∧ ∀ (b : Ordinal.{max u v}), b ∈ ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi) → ∃ a, derivBFamily o f a = b ** refine' ⟨fun a => Set.mem_iInter₂.2 fun i hi => derivBFamily_fp (H i hi) a, fun a ha => _⟩ ** case right o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ha : a ∈ ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi) ⊢ ∃ a_1, derivBFamily o f a_1 = a ** rw [Set.mem_iInter₂] at ha ** case right o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi) a : Ordinal.{max u v} ha : ∀ (i : Ordinal.{u}) (j : i < o), a ∈ fixedPoints (f i j) ⊢ ∃ a_1, derivBFamily o f a_1 = a ** rwa [← fp_iff_derivBFamily H] ** Qed | |
Ordinal.sup_iterate_eq_nfp ** f✝ f : Ordinal.{u} → Ordinal.{u} ⊢ (fun a => sup fun n => f^[n] a) = nfp f ** refine' funext fun a => le_antisymm _ (sup_le fun l => _) ** case refine'_1 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} ⊢ (sup fun n => f^[n] a) ≤ nfp f a ** rw [sup_le_iff] ** case refine'_1 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} ⊢ ∀ (i : ℕ), f^[i] a ≤ nfp f a ** intro n ** case refine'_1 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} n : ℕ ⊢ f^[n] a ≤ nfp f a ** rw [← List.length_replicate n Unit.unit, ← List.foldr_const f a] ** case refine'_1 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} n : ℕ ⊢ List.foldr (fun x => f) a (List.replicate n ()) ≤ nfp f a ** apply le_sup ** case refine'_2 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} l : List Unit ⊢ List.foldr (fun x => f) a l ≤ sup fun n => f^[n] a ** rw [List.foldr_const f a l] ** case refine'_2 f✝ f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u} l : List Unit ⊢ f^[List.length l] a ≤ sup fun n => f^[n] a ** exact le_sup _ _ ** Qed | |
Ordinal.iterate_le_nfp ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} a : Ordinal.{u_1} n : ℕ ⊢ f^[n] a ≤ nfp f a ** rw [← sup_iterate_eq_nfp] ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} a : Ordinal.{u_1} n : ℕ ⊢ f^[n] a ≤ (fun a => sup fun n => f^[n] a) a ** exact le_sup _ n ** Qed | |
Ordinal.lt_nfp ** f : Ordinal.{u} → Ordinal.{u} a b : Ordinal.{u} ⊢ a < nfp f b ↔ ∃ n, a < f^[n] b ** rw [← sup_iterate_eq_nfp] ** f : Ordinal.{u} → Ordinal.{u} a b : Ordinal.{u} ⊢ a < (fun a => sup fun n => f^[n] a) b ↔ ∃ n, a < f^[n] b ** exact lt_sup ** Qed | |
Ordinal.nfp_le_iff ** f : Ordinal.{u} → Ordinal.{u} a b : Ordinal.{u} ⊢ nfp f a ≤ b ↔ ∀ (n : ℕ), f^[n] a ≤ b ** rw [← sup_iterate_eq_nfp] ** f : Ordinal.{u} → Ordinal.{u} a b : Ordinal.{u} ⊢ (fun a => sup fun n => f^[n] a) a ≤ b ↔ ∀ (n : ℕ), f^[n] a ≤ b ** exact sup_le_iff ** Qed | |
Ordinal.nfp_id ** f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u_1} ⊢ nfp id a = id a ** simp_rw [← sup_iterate_eq_nfp, iterate_id] ** f : Ordinal.{u} → Ordinal.{u} a : Ordinal.{u_1} ⊢ (sup fun n => id a) = id a ** exact sup_const a ** Qed | |
Ordinal.IsNormal.apply_lt_nfp ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a b : Ordinal.{u_1} ⊢ f b < nfp f a ↔ b < nfp f a ** unfold nfp ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a b : Ordinal.{u_1} ⊢ f b < nfpFamily (fun x => f) a ↔ b < nfpFamily (fun x => f) a ** rw [← @apply_lt_nfpFamily_iff Unit (fun _ => f) _ (fun _ => H) a b] ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a b : Ordinal.{u_1} ⊢ f b < nfpFamily (fun x => f) a ↔ Unit → f b < nfpFamily (fun x => f) a ** exact ⟨fun h _ => h, fun h => h Unit.unit⟩ ** Qed | |
Ordinal.IsNormal.apply_le_nfp ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a b : Ordinal.{u_1} h : b ≤ nfp f a ⊢ f b ≤ nfp f a ** simpa only [H.nfp_fp] using H.le_iff.2 h ** Qed | |
Ordinal.fp_unbounded ** f : Ordinal.{u} → Ordinal.{u} H : IsNormal f ⊢ Set.Unbounded (fun x x_1 => x < x_1) (fixedPoints f) ** convert fp_family_unbounded fun _ : Unit => H ** case h.e'_3 f : Ordinal.{u} → Ordinal.{u} H : IsNormal f ⊢ fixedPoints f = ⋂ i, fixedPoints f ** exact (Set.iInter_const _).symm ** Qed | |
Ordinal.deriv_id_of_nfp_id ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} h : nfp f = id ⊢ deriv f 0 = id 0 ∧ ∀ (a : Ordinal.{u_1}), deriv f a = id a → deriv f (succ a) = id (succ a) ** simp [h] ** Qed | |
Ordinal.IsNormal.le_iff_deriv ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a : Ordinal.{u_1} ⊢ f a ≤ a ↔ ∃ o, deriv f o = a ** unfold deriv ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a : Ordinal.{u_1} ⊢ f a ≤ a ↔ ∃ o, derivFamily (fun x => f) o = a ** rw [← le_iff_derivFamily fun _ : Unit => H] ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a : Ordinal.{u_1} ⊢ f a ≤ a ↔ Unit → f a ≤ a ** exact ⟨fun h _ => h, fun h => h Unit.unit⟩ ** Qed | |
Ordinal.IsNormal.fp_iff_deriv ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} H : IsNormal f a : Ordinal.{u_1} ⊢ f a = a ↔ ∃ o, deriv f o = a ** rw [← H.le_iff_eq, H.le_iff_deriv] ** Qed | |
Ordinal.deriv_eq_enumOrd ** f : Ordinal.{u} → Ordinal.{u} H : IsNormal f ⊢ deriv f = enumOrd (fixedPoints f) ** convert derivFamily_eq_enumOrd fun _ : Unit => H ** case h.e'_3.h.e'_1 f : Ordinal.{u} → Ordinal.{u} H : IsNormal f ⊢ fixedPoints f = ⋂ i, fixedPoints f ** exact (Set.iInter_const _).symm ** Qed | |
Ordinal.deriv_eq_id_of_nfp_eq_id ** f✝ : Ordinal.{u} → Ordinal.{u} f : Ordinal.{u_1} → Ordinal.{u_1} h : nfp f = id ⊢ deriv f 0 = id 0 ∧ ∀ (a : Ordinal.{u_1}), deriv f a = id a → deriv f (succ a) = id (succ a) ** simp [h] ** Qed | |
Ordinal.nfp_add_zero ** a : Ordinal.{u_1} ⊢ nfp (fun x => a + x) 0 = a * ω ** simp_rw [← sup_iterate_eq_nfp, ← sup_mul_nat] ** a : Ordinal.{u_1} ⊢ (sup fun n => (fun x => a + x)^[n] 0) = sup fun n => a * ↑n ** congr ** case e_f a : Ordinal.{u_1} ⊢ (fun n => (fun x => a + x)^[n] 0) = fun n => a * ↑n ** funext n ** case e_f.h a : Ordinal.{u_1} n : ℕ ⊢ (fun x => a + x)^[n] 0 = a * ↑n ** induction' n with n hn ** case e_f.h.zero a : Ordinal.{u_1} ⊢ (fun x => a + x)^[Nat.zero] 0 = a * ↑Nat.zero ** rw [Nat.cast_zero, mul_zero, iterate_zero_apply] ** case e_f.h.succ a : Ordinal.{u_1} n : ℕ hn : (fun x => a + x)^[n] 0 = a * ↑n ⊢ (fun x => a + x)^[Nat.succ n] 0 = a * ↑(Nat.succ n) ** nth_rw 2 [Nat.succ_eq_one_add] ** case e_f.h.succ a : Ordinal.{u_1} n : ℕ hn : (fun x => a + x)^[n] 0 = a * ↑n ⊢ (fun x => a + x)^[Nat.succ n] 0 = a * ↑(1 + n) ** rw [Nat.cast_add, Nat.cast_one, mul_one_add, iterate_succ_apply', hn] ** Qed | |
Ordinal.nfp_add_eq_mul_omega ** a b : Ordinal.{u_1} hba : b ≤ a * ω ⊢ nfp (fun x => a + x) b = a * ω ** apply le_antisymm (nfp_le_fp (add_isNormal a).monotone hba _) ** a b : Ordinal.{u_1} hba : b ≤ a * ω ⊢ a * ω ≤ nfp ((fun x x_1 => x + x_1) a) b ** rw [← nfp_add_zero] ** a b : Ordinal.{u_1} hba : b ≤ a * ω ⊢ nfp (fun x => a + x) 0 ≤ nfp ((fun x x_1 => x + x_1) a) b ** exact nfp_monotone (add_isNormal a).monotone (Ordinal.zero_le b) ** a b : Ordinal.{u_1} hba : b ≤ a * ω ⊢ (fun x x_1 => x + x_1) a (a * ω) ≤ a * ω ** dsimp ** a b : Ordinal.{u_1} hba : b ≤ a * ω ⊢ a + a * ω ≤ a * ω ** rw [← mul_one_add, one_add_omega] ** Qed | |
Ordinal.add_eq_right_iff_mul_omega_le ** a b : Ordinal.{u_1} ⊢ a + b = b ↔ a * ω ≤ b ** refine' ⟨fun h => _, fun h => _⟩ ** case refine'_1 a b : Ordinal.{u_1} h : a + b = b ⊢ a * ω ≤ b ** rw [← nfp_add_zero a, ← deriv_zero] ** case refine'_1 a b : Ordinal.{u_1} h : a + b = b ⊢ deriv (fun x => a + x) 0 ≤ b ** cases' (add_isNormal a).fp_iff_deriv.1 h with c hc ** case refine'_1.intro a b : Ordinal.{u_1} h : a + b = b c : Ordinal.{u_1} hc : deriv ((fun x x_1 => x + x_1) a) c = b ⊢ deriv (fun x => a + x) 0 ≤ b ** rw [← hc] ** case refine'_1.intro a b : Ordinal.{u_1} h : a + b = b c : Ordinal.{u_1} hc : deriv ((fun x x_1 => x + x_1) a) c = b ⊢ deriv (fun x => a + x) 0 ≤ deriv ((fun x x_1 => x + x_1) a) c ** exact (deriv_isNormal _).monotone (Ordinal.zero_le _) ** case refine'_2 a b : Ordinal.{u_1} h : a * ω ≤ b ⊢ a + b = b ** have := Ordinal.add_sub_cancel_of_le h ** case refine'_2 a b : Ordinal.{u_1} h : a * ω ≤ b this : a * ω + (b - a * ω) = b ⊢ a + b = b ** nth_rw 1 [← this] ** case refine'_2 a b : Ordinal.{u_1} h : a * ω ≤ b this : a * ω + (b - a * ω) = b ⊢ a + (a * ω + (b - a * ω)) = b ** rwa [← add_assoc, ← mul_one_add, one_add_omega] ** Qed | |
Ordinal.add_le_right_iff_mul_omega_le ** a b : Ordinal.{u_1} ⊢ a + b ≤ b ↔ a * ω ≤ b ** rw [← add_eq_right_iff_mul_omega_le] ** a b : Ordinal.{u_1} ⊢ a + b ≤ b ↔ a + b = b ** exact (add_isNormal a).le_iff_eq ** Qed | |
Ordinal.deriv_add_eq_mul_omega_add ** a b : Ordinal.{u} ⊢ deriv (fun x => a + x) b = a * ω + b ** revert b ** a : Ordinal.{u} ⊢ ∀ (b : Ordinal.{u}), deriv (fun x => a + x) b = a * ω + b ** rw [← funext_iff, IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (add_isNormal _)] ** a : Ordinal.{u} ⊢ deriv (fun x => a + x) 0 = (fun x x_1 => x + x_1) (a * ω) 0 ∧ ∀ (a_1 : Ordinal.{u}), deriv (fun x => a + x) a_1 = (fun x x_1 => x + x_1) (a * ω) a_1 → deriv (fun x => a + x) (succ a_1) = (fun x x_1 => x + x_1) (a * ω) (succ a_1) ** refine' ⟨_, fun a h => _⟩ ** case refine'_1 a : Ordinal.{u} ⊢ deriv (fun x => a + x) 0 = (fun x x_1 => x + x_1) (a * ω) 0 ** dsimp ** case refine'_1 a : Ordinal.{u} ⊢ deriv (fun x => a + x) 0 = a * ω + 0 ** rw [deriv_zero, add_zero] ** case refine'_1 a : Ordinal.{u} ⊢ nfp (fun x => a + x) 0 = a * ω ** exact nfp_add_zero a ** case refine'_2 a✝ a : Ordinal.{u} h : deriv (fun x => a✝ + x) a = (fun x x_1 => x + x_1) (a✝ * ω) a ⊢ deriv (fun x => a✝ + x) (succ a) = (fun x x_1 => x + x_1) (a✝ * ω) (succ a) ** dsimp ** case refine'_2 a✝ a : Ordinal.{u} h : deriv (fun x => a✝ + x) a = (fun x x_1 => x + x_1) (a✝ * ω) a ⊢ deriv (fun x => a✝ + x) (succ a) = a✝ * ω + succ a ** rw [deriv_succ, h, add_succ] ** case refine'_2 a✝ a : Ordinal.{u} h : deriv (fun x => a✝ + x) a = (fun x x_1 => x + x_1) (a✝ * ω) a ⊢ nfp (fun x => a✝ + x) (succ ((fun x x_1 => x + x_1) (a✝ * ω) a)) = succ (a✝ * ω + a) ** exact nfp_eq_self (add_eq_right_iff_mul_omega_le.2 ((le_add_right _ _).trans (le_succ _))) ** Qed | |
Ordinal.nfp_mul_one ** a : Ordinal.{u_1} ha : 0 < a ⊢ nfp (fun x => a * x) 1 = a ^ ω ** rw [← sup_iterate_eq_nfp, ← sup_opow_nat] ** a : Ordinal.{u_1} ha : 0 < a ⊢ (fun a_1 => sup fun n => (fun x => a * x)^[n] a_1) 1 = sup fun n => a ^ ↑n ** dsimp ** a : Ordinal.{u_1} ha : 0 < a ⊢ (sup fun n => (fun x => a * x)^[n] 1) = sup fun n => a ^ ↑n ** congr ** case e_f a : Ordinal.{u_1} ha : 0 < a ⊢ (fun n => (fun x => a * x)^[n] 1) = fun n => a ^ ↑n ** funext n ** case e_f.h a : Ordinal.{u_1} ha : 0 < a n : ℕ ⊢ (fun x => a * x)^[n] 1 = a ^ ↑n ** induction' n with n hn ** case e_f.h.succ a : Ordinal.{u_1} ha : 0 < a n : ℕ hn : (fun x => a * x)^[n] 1 = a ^ ↑n ⊢ (fun x => a * x)^[Nat.succ n] 1 = a ^ ↑(Nat.succ n) ** nth_rw 2 [Nat.succ_eq_one_add] ** case e_f.h.succ a : Ordinal.{u_1} ha : 0 < a n : ℕ hn : (fun x => a * x)^[n] 1 = a ^ ↑n ⊢ (fun x => a * x)^[Nat.succ n] 1 = a ^ ↑(1 + n) ** rw [Nat.cast_add, Nat.cast_one, opow_add, opow_one, iterate_succ_apply', hn] ** case e_f.h.zero a : Ordinal.{u_1} ha : 0 < a ⊢ (fun x => a * x)^[Nat.zero] 1 = a ^ ↑Nat.zero ** rw [Nat.cast_zero, opow_zero, iterate_zero_apply] ** a : Ordinal.{u_1} ha : 0 < a ⊢ 0 < a ** exact ha ** Qed | |
Ordinal.nfp_mul_zero ** a : Ordinal.{u_1} ⊢ nfp (fun x => a * x) 0 = 0 ** rw [← Ordinal.le_zero, nfp_le_iff] ** a : Ordinal.{u_1} ⊢ ∀ (n : ℕ), (fun x => a * x)^[n] 0 ≤ 0 ** intro n ** a : Ordinal.{u_1} n : ℕ ⊢ (fun x => a * x)^[n] 0 ≤ 0 ** induction' n with n hn ** case succ a : Ordinal.{u_1} n : ℕ hn : (fun x => a * x)^[n] 0 ≤ 0 ⊢ (fun x => a * x)^[Nat.succ n] 0 ≤ 0 ** rwa [iterate_succ_apply, mul_zero] ** case zero a : Ordinal.{u_1} ⊢ (fun x => a * x)^[Nat.zero] 0 ≤ 0 ** rfl ** Qed | |
Ordinal.nfp_zero_mul ** ⊢ nfp (HMul.hMul 0) = id ** rw [← sup_iterate_eq_nfp] ** a : Ordinal.{u_1} n : ℕ ⊢ (HMul.hMul 0)^[n] a ≤ id a ** induction' n with n _ ** case succ a : Ordinal.{u_1} n : ℕ n_ih✝ : (HMul.hMul 0)^[n] a ≤ id a ⊢ (HMul.hMul 0)^[Nat.succ n] a ≤ id a ** rw [Function.iterate_succ'] ** case succ a : Ordinal.{u_1} n : ℕ n_ih✝ : (HMul.hMul 0)^[n] a ≤ id a ⊢ (HMul.hMul 0 ∘ (HMul.hMul 0)^[n]) a ≤ id a ** change 0 * _ ≤ a ** case succ a : Ordinal.{u_1} n : ℕ n_ih✝ : (HMul.hMul 0)^[n] a ≤ id a ⊢ 0 * (HMul.hMul 0)^[n] a ≤ a ** rw [zero_mul] ** case succ a : Ordinal.{u_1} n : ℕ n_ih✝ : (HMul.hMul 0)^[n] a ≤ id a ⊢ 0 ≤ a ** exact Ordinal.zero_le a ** case zero a : Ordinal.{u_1} ⊢ (HMul.hMul 0)^[Nat.zero] a ≤ id a ** rfl ** Qed | |
Ordinal.nfp_mul_eq_opow_omega ** a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ⊢ nfp (fun x => a * x) b = a ^ ω ** cases' eq_zero_or_pos a with ha ha ** case inr a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ nfp (fun x => a * x) b = a ^ ω ** apply le_antisymm ** case inr.a a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ a ^ ω ≤ nfp (fun x => a * x) b ** rw [← nfp_mul_one ha] ** case inr.a a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ nfp (fun x => a * x) 1 ≤ nfp (fun x => a * x) b ** exact nfp_monotone (mul_isNormal ha).monotone (one_le_iff_pos.2 hb) ** case inl a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : a = 0 ⊢ nfp (fun x => a * x) b = a ^ ω ** rw [ha, zero_opow omega_ne_zero] at hba ⊢ ** case inl a b : Ordinal.{u} hb : 0 < b hba : b ≤ 0 ha : a = 0 ⊢ nfp (fun x => 0 * x) b = 0 ** rw [Ordinal.le_zero.1 hba, nfp_zero_mul] ** case inl a b : Ordinal.{u} hb : 0 < b hba : b ≤ 0 ha : a = 0 ⊢ id 0 = 0 ** rfl ** case inr.a a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ nfp (fun x => a * x) b ≤ a ^ ω ** apply nfp_le_fp (mul_isNormal ha).monotone hba ** case inr.a a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ (fun x x_1 => x * x_1) a (a ^ ω) ≤ a ^ ω ** dsimp only ** case inr.a a b : Ordinal.{u} hb : 0 < b hba : b ≤ a ^ ω ha : 0 < a ⊢ a * a ^ ω ≤ a ^ ω ** rw [← opow_one_add, one_add_omega] ** Qed | |
Ordinal.eq_zero_or_opow_omega_le_of_mul_eq_right ** a b : Ordinal.{u} hab : a * b = b ⊢ b = 0 ∨ a ^ ω ≤ b ** cases' eq_zero_or_pos a with ha ha ** case inr a b : Ordinal.{u} hab : a * b = b ha : 0 < a ⊢ b = 0 ∨ a ^ ω ≤ b ** rw [or_iff_not_imp_left] ** case inr a b : Ordinal.{u} hab : a * b = b ha : 0 < a ⊢ ¬b = 0 → a ^ ω ≤ b ** intro hb ** case inr a b : Ordinal.{u} hab : a * b = b ha : 0 < a hb : ¬b = 0 ⊢ a ^ ω ≤ b ** rw [← nfp_mul_one ha] ** case inr a b : Ordinal.{u} hab : a * b = b ha : 0 < a hb : ¬b = 0 ⊢ nfp (fun x => a * x) 1 ≤ b ** rw [← Ne.def, ← one_le_iff_ne_zero] at hb ** case inr a b : Ordinal.{u} hab : a * b = b ha : 0 < a hb : 1 ≤ b ⊢ nfp (fun x => a * x) 1 ≤ b ** exact nfp_le_fp (mul_isNormal ha).monotone hb (le_of_eq hab) ** case inl a b : Ordinal.{u} hab : a * b = b ha : a = 0 ⊢ b = 0 ∨ a ^ ω ≤ b ** rw [ha, zero_opow omega_ne_zero] ** case inl a b : Ordinal.{u} hab : a * b = b ha : a = 0 ⊢ b = 0 ∨ 0 ≤ b ** exact Or.inr (Ordinal.zero_le b) ** Qed | |
Ordinal.mul_eq_right_iff_opow_omega_dvd ** a b : Ordinal.{u_1} ⊢ a * b = b ↔ a ^ ω ∣ b ** cases' eq_zero_or_pos a with ha ha ** case inr a b : Ordinal.{u_1} ha : 0 < a ⊢ a * b = b ↔ a ^ ω ∣ b ** refine' ⟨fun hab => _, fun h => _⟩ ** case inr.refine'_2 a b : Ordinal.{u_1} ha : 0 < a h : a ^ ω ∣ b ⊢ a * b = b ** cases' h with c hc ** case inr.refine'_2.intro a b : Ordinal.{u_1} ha : 0 < a c : Ordinal.{u_1} hc : b = a ^ ω * c ⊢ a * b = b ** rw [hc, ← mul_assoc, ← opow_one_add, one_add_omega] ** case inl a b : Ordinal.{u_1} ha : a = 0 ⊢ a * b = b ↔ a ^ ω ∣ b ** rw [ha, zero_mul, zero_opow omega_ne_zero, zero_dvd_iff] ** case inl a b : Ordinal.{u_1} ha : a = 0 ⊢ 0 = b ↔ b = 0 ** exact eq_comm ** case inr.refine'_1 a b : Ordinal.{u_1} ha : 0 < a hab : a * b = b ⊢ a ^ ω ∣ b ** rw [dvd_iff_mod_eq_zero] ** case inr.refine'_1 a b : Ordinal.{u_1} ha : 0 < a hab : a * b = b ⊢ b % a ^ ω = 0 ** rw [← div_add_mod b (a^omega), mul_add, ← mul_assoc, ← opow_one_add, one_add_omega,
add_left_cancel] at hab ** case inr.refine'_1 a b : Ordinal.{u_1} ha : 0 < a hab : a * (b % a ^ ω) = b % a ^ ω ⊢ b % a ^ ω = 0 ** cases' eq_zero_or_opow_omega_le_of_mul_eq_right hab with hab hab ** case inr.refine'_1.inr a b : Ordinal.{u_1} ha : 0 < a hab✝ : a * (b % a ^ ω) = b % a ^ ω hab : a ^ ω ≤ b % a ^ ω ⊢ b % a ^ ω = 0 ** refine' (not_lt_of_le hab (mod_lt b (opow_ne_zero omega _))).elim ** case inr.refine'_1.inr a b : Ordinal.{u_1} ha : 0 < a hab✝ : a * (b % a ^ ω) = b % a ^ ω hab : a ^ ω ≤ b % a ^ ω ⊢ a ≠ 0 ** rwa [← Ordinal.pos_iff_ne_zero] ** case inr.refine'_1.inl a b : Ordinal.{u_1} ha : 0 < a hab✝ : a * (b % a ^ ω) = b % a ^ ω hab : b % a ^ ω = 0 ⊢ b % a ^ ω = 0 ** exact hab ** Qed | |
Ordinal.mul_le_right_iff_opow_omega_dvd ** a b : Ordinal.{u_1} ha : 0 < a ⊢ a * b ≤ b ↔ a ^ ω ∣ b ** rw [← mul_eq_right_iff_opow_omega_dvd] ** a b : Ordinal.{u_1} ha : 0 < a ⊢ a * b ≤ b ↔ a * b = b ** exact (mul_isNormal ha).le_iff_eq ** Qed | |
Ordinal.nfp_mul_opow_omega_add ** a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ nfp (fun x => a * x) (a ^ ω * b + c) = a ^ ω * succ b ** apply le_antisymm ** case a a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ nfp (fun x => a * x) (a ^ ω * b + c) ≤ a ^ ω * succ b ** apply nfp_le_fp (mul_isNormal ha).monotone ** case a.ab a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ a ^ ω * b + c ≤ a ^ ω * succ b ** rw [mul_succ] ** case a.ab a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ a ^ ω * b + c ≤ a ^ ω * b + a ^ ω ** apply add_le_add_left hca ** case a.h a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ (fun x x_1 => x * x_1) a (a ^ ω * succ b) ≤ a ^ ω * succ b ** dsimp only ** case a.h a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ a * (a ^ ω * succ b) ≤ a ^ ω * succ b ** rw [← mul_assoc, ← opow_one_add, one_add_omega] ** case a a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω ⊢ a ^ ω * succ b ≤ nfp (fun x => a * x) (a ^ ω * b + c) ** cases' mul_eq_right_iff_opow_omega_dvd.1 ((mul_isNormal ha).nfp_fp ((a^omega) * b + c)) with
d hd ** case a.intro a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d ⊢ a ^ ω * succ b ≤ nfp (fun x => a * x) (a ^ ω * b + c) ** rw [hd] ** case a.intro a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d ⊢ a ^ ω * succ b ≤ a ^ ω * d ** apply mul_le_mul_left' ** case a.intro.bc a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d ⊢ succ b ≤ d ** have := le_nfp (Mul.mul a) ((a^omega) * b + c) ** case a.intro.bc a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d this : a ^ ω * b + c ≤ nfp (Mul.mul a) (a ^ ω * b + c) ⊢ succ b ≤ d ** erw [hd] at this ** case a.intro.bc a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d this : a ^ ω * b + c ≤ a ^ ω * d ⊢ succ b ≤ d ** have := (add_lt_add_left hc ((a^omega) * b)).trans_le this ** case a.intro.bc a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d this✝ : a ^ ω * b + c ≤ a ^ ω * d this : a ^ ω * b + 0 < a ^ ω * d ⊢ succ b ≤ d ** rw [add_zero, mul_lt_mul_iff_left (opow_pos omega ha)] at this ** case a.intro.bc a c b : Ordinal.{u} ha : 0 < a hc : 0 < c hca : c ≤ a ^ ω d : Ordinal.{u} hd : nfp ((fun x x_1 => x * x_1) a) (a ^ ω * b + c) = a ^ ω * d this✝ : a ^ ω * b + c ≤ a ^ ω * d this : b < d ⊢ succ b ≤ d ** rwa [succ_le_iff] ** Qed | |
Ordinal.deriv_mul_eq_opow_omega_mul ** a : Ordinal.{u} ha : 0 < a b : Ordinal.{u} ⊢ deriv (fun x => a * x) b = a ^ ω * b ** revert b ** a : Ordinal.{u} ha : 0 < a ⊢ ∀ (b : Ordinal.{u}), deriv (fun x => a * x) b = a ^ ω * b ** rw [← funext_iff,
IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (mul_isNormal (opow_pos omega ha))] ** a : Ordinal.{u} ha : 0 < a ⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0 ∧ ∀ (a_1 : Ordinal.{u}), deriv (fun x => a * x) a_1 = (fun x x_1 => x * x_1) (a ^ ω) a_1 → deriv (fun x => a * x) (succ a_1) = (fun x x_1 => x * x_1) (a ^ ω) (succ a_1) ** refine' ⟨_, fun c h => _⟩ ** case refine'_1 a : Ordinal.{u} ha : 0 < a ⊢ deriv (fun x => a * x) 0 = (fun x x_1 => x * x_1) (a ^ ω) 0 ** dsimp only ** case refine'_1 a : Ordinal.{u} ha : 0 < a ⊢ deriv (fun x => a * x) 0 = a ^ ω * 0 ** rw [deriv_zero, nfp_mul_zero, mul_zero] ** case refine'_2 a : Ordinal.{u} ha : 0 < a c : Ordinal.{u} h : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c ⊢ deriv (fun x => a * x) (succ c) = (fun x x_1 => x * x_1) (a ^ ω) (succ c) ** rw [deriv_succ, h] ** case refine'_2 a : Ordinal.{u} ha : 0 < a c : Ordinal.{u} h : deriv (fun x => a * x) c = (fun x x_1 => x * x_1) (a ^ ω) c ⊢ nfp (fun x => a * x) (succ ((fun x x_1 => x * x_1) (a ^ ω) c)) = (fun x x_1 => x * x_1) (a ^ ω) (succ c) ** exact nfp_mul_opow_omega_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha)) ** Qed | |
SetTheory.PGame.nim_def ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); nim o = mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) ** rw [nim] ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => let_fun x := (_ : typein (fun x x_1 => x < x_1) o₂ < o); nim (typein (Quotient.out o).r o₂)) fun o₂ => let_fun x := (_ : typein (fun x x_1 => x < x_1) o₂ < o); nim (typein (Quotient.out o).r o₂)) = mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) ** rfl ** Qed | |
SetTheory.PGame.leftMoves_nim ** o : Ordinal.{u_1} ⊢ LeftMoves (nim o) = (Quotient.out o).α ** rw [nim_def] ** o : Ordinal.{u_1} ⊢ LeftMoves (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) = (Quotient.out o).α ** rfl ** Qed | |
SetTheory.PGame.rightMoves_nim ** o : Ordinal.{u_1} ⊢ RightMoves (nim o) = (Quotient.out o).α ** rw [nim_def] ** o : Ordinal.{u_1} ⊢ RightMoves (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) = (Quotient.out o).α ** rfl ** Qed | |
SetTheory.PGame.moveLeft_nim_hEq ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); HEq (moveLeft (nim o)) fun i => nim (typein (fun x x_1 => x < x_1) i) ** rw [nim_def] ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); HEq (moveLeft (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))) fun i => nim (typein (fun x x_1 => x < x_1) i) ** rfl ** Qed | |
SetTheory.PGame.moveRight_nim_hEq ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); HEq (moveRight (nim o)) fun i => nim (typein (fun x x_1 => x < x_1) i) ** rw [nim_def] ** o : Ordinal.{u_1} ⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1); HEq (moveRight (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))) fun i => nim (typein (fun x x_1 => x < x_1) i) ** rfl ** Qed | |
SetTheory.PGame.moveLeft_nim ** o : Ordinal.{u_1} i : ↑(Set.Iio o) ⊢ moveLeft (nim o) (↑toLeftMovesNim i) = nim ↑i ** simp ** Qed | |
SetTheory.PGame.moveRight_nim ** o : Ordinal.{u_1} i : ↑(Set.Iio o) ⊢ moveRight (nim o) (↑toRightMovesNim i) = nim ↑i ** simp ** Qed | |
SetTheory.PGame.toLeftMovesNim_one_symm ** i : LeftMoves (nim 1) ⊢ ↑toLeftMovesNim.symm i = { val := 0, property := (_ : 0 ∈ Set.Iio 1) } ** simp ** Qed | |
SetTheory.PGame.toRightMovesNim_one_symm ** i : RightMoves (nim 1) ⊢ ↑toRightMovesNim.symm i = { val := 0, property := (_ : 0 ∈ Set.Iio 1) } ** simp ** Qed | |
SetTheory.PGame.nim_one_moveLeft ** x : LeftMoves (nim 1) ⊢ moveLeft (nim 1) x = nim 0 ** simp ** Qed | |
SetTheory.PGame.nim_one_moveRight ** x : RightMoves (nim 1) ⊢ moveRight (nim 1) x = nim 0 ** simp ** Qed | |
SetTheory.PGame.nim_birthday ** o : Ordinal.{u_1} ⊢ birthday (nim o) = o ** induction' o using Ordinal.induction with o IH ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (nim k) = k ⊢ birthday (nim o) = o ** rw [nim_def, birthday_def] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (nim k) = k ⊢ max (lsub fun i => birthday (moveLeft (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) i)) (lsub fun i => birthday (moveRight (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) i)) = o ** dsimp ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (nim k) = k ⊢ max (lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i))) (lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i))) = o ** rw [max_eq_right le_rfl] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (nim k) = k ⊢ (lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i))) = o ** convert lsub_typein o with i ** case h.e'_2.h.e'_2.h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → birthday (nim k) = k i : (Quotient.out o).α ⊢ birthday (nim (typein (fun x x_1 => x < x_1) i)) = typein (fun x x_1 => x < x_1) i ** exact IH _ (typein_lt_self i) ** Qed | |
SetTheory.PGame.neg_nim ** o : Ordinal.{u_1} ⊢ -nim o = nim o ** induction' o using Ordinal.induction with o IH ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → -nim k = nim k ⊢ -nim o = nim o ** rw [nim_def] ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → -nim k = nim k ⊢ (-mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) = mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) ** dsimp ** case h o : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < o → -nim k = nim k ⊢ (mk (Quotient.out o).α (Quotient.out o).α (fun j => -nim (typein (fun x x_1 => x < x_1) j)) fun i => -nim (typein (fun x x_1 => x < x_1) i)) = mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) ** congr <;> funext i <;> exact IH _ (Ordinal.typein_lt_self i) ** Qed | |
SetTheory.PGame.nim_fuzzy_zero_of_ne_zero ** o : Ordinal.{u_1} ho : o ≠ 0 ⊢ nim o ‖ 0 ** rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le] ** o : Ordinal.{u_1} ho : o ≠ 0 ⊢ ∃ j, moveRight (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) j ≤ 0 ** rw [← Ordinal.pos_iff_ne_zero] at ho ** o : Ordinal.{u_1} ho : 0 < o ⊢ ∃ j, moveRight (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) j ≤ 0 ** exact ⟨(Ordinal.principalSegOut ho).top, by simp⟩ ** o : Ordinal.{u_1} ho : 0 < o ⊢ moveRight (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) (principalSegOut ho).top ≤ 0 ** simp ** Qed |
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