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Cardinal.isRegular_aleph_one ** α : Type u_1 r : α → α → Prop ⊢ IsRegular (aleph 1) ** rw [← succ_aleph0] ** α : Type u_1 r : α → α → Prop ⊢ IsRegular (succ ℵ₀) ** exact isRegular_succ le_rfl ** Qed | |
Cardinal.isRegular_aleph'_succ ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} h : ω ≤ o ⊢ IsRegular (aleph' (succ o)) ** rw [aleph'_succ] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} h : ω ≤ o ⊢ IsRegular (succ (aleph' o)) ** exact isRegular_succ (aleph0_le_aleph'.2 h) ** Qed | |
Cardinal.isRegular_aleph_succ ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ IsRegular (aleph (succ o)) ** rw [aleph_succ] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ IsRegular (succ (aleph o)) ** exact isRegular_succ (aleph0_le_aleph o) ** Qed | |
Cardinal.infinite_pigeonhole_card_lt ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α w : #α < #β w' : ℵ₀ ≤ #α ⊢ ∃ a, #α < #↑(f ⁻¹' {a}) ** simp_rw [← succ_le_iff] ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α w : #α < #β w' : ℵ₀ ≤ #α ⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a}) ** exact
Ordinal.infinite_pigeonhole_card f (succ #α) (succ_le_of_lt w) (w'.trans (lt_succ _).le)
((lt_succ _).trans_le (isRegular_succ w').2.ge) ** Qed | |
Cardinal.exists_infinite_fiber ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α w : #α < #β w' : Infinite α ⊢ ∃ a, Infinite ↑(f ⁻¹' {a}) ** simp_rw [Cardinal.infinite_iff] at w' ⊢ ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α w : #α < #β w' : ℵ₀ ≤ #α ⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a}) ** cases' infinite_pigeonhole_card_lt f w w' with a ha ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α w : #α < #β w' : ℵ₀ ≤ #α a : α ha : #α < #↑(f ⁻¹' {a}) ⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a}) ** exact ⟨a, w'.trans ha.le⟩ ** Qed | |
Cardinal.le_range_of_union_finset_eq_top ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ ⊢ #β ≤ #↑(range f) ** have k : _root_.Infinite (range f) := by
rw [infinite_coe_iff]
apply mt (union_finset_finite_of_range_finite f)
rw [w]
exact infinite_univ ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) ⊢ #β ≤ #↑(range f) ** by_contra h ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : ¬#β ≤ #↑(range f) ⊢ False ** simp only [not_le] at h ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β ⊢ False ** let u : ∀ b, ∃ a, b ∈ f a := fun b => by simpa using (w.ge : _) (Set.mem_univ b) ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) ⊢ False ** let u' : β → range f := fun b => ⟨f (u b).choose, by simp⟩ ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } ⊢ False ** have v' : ∀ a, u' ⁻¹' {⟨f a, by simp⟩} ≤ f a := by
rintro a p m
simp at m
rw [← m]
apply fun b => (u b).choose_spec ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } v' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a) ⊢ False ** obtain ⟨⟨-, ⟨a, rfl⟩⟩, p⟩ := exists_infinite_fiber u' h k ** case intro.mk.intro α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } v' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a) a : α p : Infinite ↑(u' ⁻¹' {{ val := f a, property := (_ : ∃ y, f y = f a) }}) ⊢ False ** exact (@Infinite.of_injective _ _ p (inclusion (v' a)) (inclusion_injective _)).false ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ ⊢ Infinite ↑(range f) ** rw [infinite_coe_iff] ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ ⊢ Set.Infinite (range f) ** apply mt (union_finset_finite_of_range_finite f) ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ ⊢ ¬Set.Finite (⋃ a, ↑(f a)) ** rw [w] ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ ⊢ ¬Set.Finite ⊤ ** exact infinite_univ ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β b : β ⊢ ∃ a, b ∈ f a ** simpa using (w.ge : _) (Set.mem_univ b) ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) b : β ⊢ f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f ** simp ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } a : α ⊢ f a ∈ range f ** simp ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } ⊢ ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a) ** rintro a p m ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } a : α p : β m : p ∈ u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ⊢ p ∈ ↑(f a) ** simp at m ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } a : α p : β m : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a ⊢ p ∈ ↑(f a) ** rw [← m] ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 β : Type u_3 inst✝ : Infinite β f : α → Finset β w : ⋃ a, ↑(f a) = ⊤ k : Infinite ↑(range f) h : #↑(range f) < #β u : ∀ (b : β), ∃ a, b ∈ f a := fun b => Eq.mp (Mathlib.Data.Set.Lattice._auxLemma.3.trans (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4))) (Eq.ge w (mem_univ b)) u' : β → ↑(range f) := fun b => { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) } a : α p : β m : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a ⊢ p ∈ ↑(f (Exists.choose (_ : ∃ a, p ∈ f a))) ** apply fun b => (u b).choose_spec ** Qed | |
Cardinal.lsub_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c ⊢ lift.{v, u} #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.lsub_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type (max u_2 u_3) f : ι → Ordinal.{max u_2 u_3} c : Cardinal.{max u_2 u_3} hc : IsRegular c hι : #ι < c ⊢ #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.sup_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c ⊢ lift.{v, u} #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.sup_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type (max u_2 u_3) f : ι → Ordinal.{max u_2 u_3} c : Cardinal.{max u_2 u_3} hc : IsRegular c hι : #ι < c ⊢ #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.blsub_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c ho : lift.{v, u} (card o) < c ⊢ lift.{v, u} (card o) < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.blsub_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{max u_2 u_3} f : (a : Ordinal.{max u_2 u_3}) → a < o → Ordinal.{max u_2 u_3} c : Cardinal.{max u_2 u_3} hc : IsRegular c ho : card o < c ⊢ card o < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.bsup_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} (card o) < c ⊢ lift.{v, u} (card o) < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.bsup_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{max u_2 u_3} f : (a : Ordinal.{max u_2 u_3}) → a < o → Ordinal.{max u_2 u_3} c : Cardinal.{max u_2 u_3} hc : IsRegular c hι : card o < c ⊢ card o < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.iSup_lt_lift_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c ⊢ lift.{v, u} #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.iSup_lt_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u_2 f : ι → Cardinal.{u_2} c : Cardinal.{u_2} hc : IsRegular c hι : #ι < c ⊢ #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** Qed | |
Cardinal.sum_lt_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{u} c : Cardinal.{u} hc : IsRegular c hι : #ι < c ⊢ lift.{u, u} #ι < c ** rwa [lift_id] ** Qed | |
Cardinal.nfpFamily_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ha : a < ord c ⊢ nfpFamily f a < ord c ** apply nfpFamily_lt_ord_lift.{u, v} _ _ hf ha <;> rw [hc.cof_eq] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ha : a < ord c ⊢ ℵ₀ < c α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ha : a < ord c ⊢ lift.{v, u} #ι < c ** exact lt_of_le_of_ne hc.1 hc'.symm ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ha : a < ord c ⊢ lift.{v, u} #ι < c ** exact hι ** Qed | |
Cardinal.nfpFamily_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} → Ordinal.{u} c : Cardinal.{u} hc : IsRegular c hι : #ι < c hc' : c ≠ ℵ₀ a : Ordinal.{u} hf : ∀ (i : ι) (b : Ordinal.{u}), b < ord c → f i b < ord c ⊢ lift.{u, u} #ι < c ** rwa [lift_id] ** Qed | |
Cardinal.nfpBFamily_lt_ord_lift_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c ho : lift.{v, u} (card o) < c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{max u v}), b < ord c → f i hi b < ord c a : Ordinal.{max u v} ⊢ lift.{v, u} #(Quotient.out o).α < c ** rwa [mk_ordinal_out] ** Qed | |
Cardinal.nfpBFamily_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} → Ordinal.{u} c : Cardinal.{u} hc : IsRegular c ho : card o < c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{u}), b < ord c → f i hi b < ord c a : Ordinal.{u} ⊢ lift.{u, u} (card o) < c ** rwa [lift_id] ** Qed | |
Cardinal.nfp_lt_ord_of_isRegular ** α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} c : Cardinal.{u_2} hc : IsRegular c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u_2}), i < ord c → f i < ord c a : Ordinal.{u_2} ⊢ ℵ₀ < Ordinal.cof (ord c) ** rw [hc.cof_eq] ** α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} c : Cardinal.{u_2} hc : IsRegular c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u_2}), i < ord c → f i < ord c a : Ordinal.{u_2} ⊢ ℵ₀ < c ** exact lt_of_le_of_ne hc.1 hc'.symm ** Qed | |
Cardinal.derivFamily_lt_ord_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ⊢ a < ord c → derivFamily f a < ord c ** have hω : ℵ₀ < c.ord.cof := by
rw [hc.cof_eq]
exact lt_of_le_of_ne hc.1 hc'.symm ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} hω : ℵ₀ < Ordinal.cof (ord c) ⊢ a < ord c → derivFamily f a < ord c ** induction a using limitRecOn with
| H₁ =>
rw [derivFamily_zero]
exact nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf
| H₂ b hb =>
intro hb'
rw [derivFamily_succ]
exact
nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf
((ord_isLimit hc.1).2 _ (hb ((lt_succ b).trans hb')))
| H₃ b hb H =>
intro hb'
rw [derivFamily_limit f hb]
exact
bsup_lt_ord_of_isRegular.{u, v} hc (ord_lt_ord.1 ((ord_card_le b).trans_lt hb')) fun o' ho' =>
H o' ho' (ho'.trans hb') ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ⊢ ℵ₀ < Ordinal.cof (ord c) ** rw [hc.cof_eq] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c a : Ordinal.{max u v} ⊢ ℵ₀ < c ** exact lt_of_le_of_ne hc.1 hc'.symm ** case H₁ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) ⊢ 0 < ord c → derivFamily f 0 < ord c ** rw [derivFamily_zero] ** case H₁ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) ⊢ 0 < ord c → nfpFamily f 0 < ord c ** exact nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) ⊢ lift.{v, u} #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** case H₂ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : b < ord c → derivFamily f b < ord c ⊢ succ b < ord c → derivFamily f (succ b) < ord c ** intro hb' ** case H₂ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : b < ord c → derivFamily f b < ord c hb' : succ b < ord c ⊢ derivFamily f (succ b) < ord c ** rw [derivFamily_succ] ** case H₂ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : b < ord c → derivFamily f b < ord c hb' : succ b < ord c ⊢ nfpFamily f (succ (derivFamily f b)) < ord c ** exact
nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf
((ord_isLimit hc.1).2 _ (hb ((lt_succ b).trans hb'))) ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : b < ord c → derivFamily f b < ord c hb' : succ b < ord c ⊢ lift.{v, u} #ι < Ordinal.cof (ord c) ** rwa [hc.cof_eq] ** case H₃ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : Ordinal.IsLimit b H : ∀ (o' : Ordinal.{max u v}), o' < b → o' < ord c → derivFamily f o' < ord c ⊢ b < ord c → derivFamily f b < ord c ** intro hb' ** case H₃ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : Ordinal.IsLimit b H : ∀ (o' : Ordinal.{max u v}), o' < b → o' < ord c → derivFamily f o' < ord c hb' : b < ord c ⊢ derivFamily f b < ord c ** rw [derivFamily_limit f hb] ** case H₃ α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < ord c → f i b < ord c hω : ℵ₀ < Ordinal.cof (ord c) b : Ordinal.{max u v} hb : Ordinal.IsLimit b H : ∀ (o' : Ordinal.{max u v}), o' < b → o' < ord c → derivFamily f o' < ord c hb' : b < ord c ⊢ (bsup b fun a x => derivFamily f a) < ord c ** exact
bsup_lt_ord_of_isRegular.{u, v} hc (ord_lt_ord.1 ((ord_card_le b).trans_lt hb')) fun o' ho' =>
H o' ho' (ho'.trans hb') ** Qed | |
Cardinal.derivFamily_lt_ord ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} → Ordinal.{u} c : Cardinal.{u} hc : IsRegular c hι : #ι < c hc' : c ≠ ℵ₀ hf : ∀ (i : ι) (b : Ordinal.{u}), b < ord c → f i b < ord c a : Ordinal.{u} ⊢ lift.{u, u} #ι < c ** rwa [lift_id] ** Qed | |
Cardinal.derivBFamily_lt_ord_lift ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} → Ordinal.{max u v} c : Cardinal.{max u v} hc : IsRegular c hι : lift.{v, u} (card o) < c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{max u v}), b < ord c → f i hi b < ord c a : Ordinal.{max u v} ⊢ lift.{v, u} #(Quotient.out o).α < c ** rwa [mk_ordinal_out] ** Qed | |
Cardinal.derivBFamily_lt_ord ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} → Ordinal.{u} c : Cardinal.{u} hc : IsRegular c hι : card o < c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{u}), b < ord c → f i hi b < ord c a : Ordinal.{u} ⊢ lift.{u, u} (card o) < c ** rwa [lift_id] ** Qed | |
Cardinal.deriv_lt_ord ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} c : Cardinal.{u} hc : IsRegular c hc' : c ≠ ℵ₀ hf : ∀ (i : Ordinal.{u}), i < ord c → f i < ord c a : Ordinal.{u} ⊢ lift.{u, 0} #Unit < c ** simpa using Cardinal.one_lt_aleph0.trans (lt_of_le_of_ne hc.1 hc'.symm) ** Qed | |
Cardinal.univ_inaccessible ** α : Type u_1 r : α → α → Prop ⊢ ℵ₀ < univ ** simpa using lift_lt_univ' ℵ₀ ** α : Type u_1 r : α → α → Prop ⊢ univ ≤ Ordinal.cof (ord univ) ** simp ** α : Type u_1 r : α → α → Prop c : Cardinal.{max (u + 1) v} h : c < univ ⊢ 2 ^ c < univ ** rcases lt_univ'.1 h with ⟨c, rfl⟩ ** case intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h : lift.{max (u + 1) v, u} c < univ ⊢ 2 ^ lift.{max (u + 1) v, u} c < univ ** rw [← lift_two_power.{u, max (u + 1) v}] ** case intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h : lift.{max (u + 1) v, u} c < univ ⊢ lift.{max (u + 1) v, u} (2 ^ c) < univ ** apply lift_lt_univ' ** Qed | |
Cardinal.lt_power_cof ** α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α ⊢ Quotient.mk isEquivalent α < Quotient.mk isEquivalent α ^ Ordinal.cof (ord (Quotient.mk isEquivalent α)) ** rcases ord_eq α with ⟨r, wo, re⟩ ** case intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r ⊢ Quotient.mk isEquivalent α < Quotient.mk isEquivalent α ^ Ordinal.cof (ord (Quotient.mk isEquivalent α)) ** have := ord_isLimit h ** case intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (ord (Quotient.mk isEquivalent α)) ⊢ Quotient.mk isEquivalent α < Quotient.mk isEquivalent α ^ Ordinal.cof (ord (Quotient.mk isEquivalent α)) ** rw [mk'_def, re] at this ⊢ ** case intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) ⊢ #α < #α ^ Ordinal.cof (type r) ** rcases cof_eq' r this with ⟨S, H, Se⟩ ** case intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ #α < #α ^ Ordinal.cof (type r) ** have := sum_lt_prod (fun a : S => #{ x // r x a }) (fun _ => #α) fun i => ?_ ** case intro.intro.intro.intro.refine_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) this : (sum fun a => #{ x // r x ↑a }) < prod fun x => #α ⊢ #α < #α ^ Ordinal.cof (type r) ** simp only [Cardinal.prod_const, Cardinal.lift_id, ← Se, ← mk_sigma, power_def] at this ⊢ ** case intro.intro.intro.intro.refine_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) this : #((i : ↑S) × { x // r x ↑i }) < #(↑S → α) ⊢ #α < #(↑S → α) ** refine' lt_of_le_of_lt _ this ** case intro.intro.intro.intro.refine_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) this : #((i : ↑S) × { x // r x ↑i }) < #(↑S → α) ⊢ #α ≤ #((i : ↑S) × { x // r x ↑i }) ** refine' ⟨Embedding.ofSurjective _ _⟩ ** case intro.intro.intro.intro.refine_2.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) this : #((i : ↑S) × { x // r x ↑i }) < #(↑S → α) ⊢ (i : ↑S) × { x // r x ↑i } → α ** exact fun x => x.2.1 ** case intro.intro.intro.intro.refine_2.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) this : #((i : ↑S) × { x // r x ↑i }) < #(↑S → α) ⊢ Surjective fun x => ↑x.snd ** exact fun a =>
let ⟨b, h, ab⟩ := H a
⟨⟨⟨_, h⟩, _, ab⟩, rfl⟩ ** case intro.intro.intro.intro.refine_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) i : ↑S ⊢ (fun a => #{ x // r x ↑a }) i < (fun x => #α) i ** have := typein_lt_type r i ** case intro.intro.intro.intro.refine_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} α : Type u h : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this✝ : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) i : ↑S this : typein r ↑i < type r ⊢ (fun a => #{ x // r x ↑a }) i < (fun x => #α) i ** rwa [← re, lt_ord] at this ** Qed | |
Cardinal.lt_cof_power ** α : Type u_1 r : α → α → Prop a b : Cardinal.{u_2} ha : ℵ₀ ≤ a b1 : 1 < b ⊢ a < Ordinal.cof (ord (b ^ a)) ** have b0 : b ≠ 0 := (zero_lt_one.trans b1).ne' ** α : Type u_1 r : α → α → Prop a b : Cardinal.{u_2} ha : ℵ₀ ≤ a b1 : 1 < b b0 : b ≠ 0 ⊢ a < Ordinal.cof (ord (b ^ a)) ** apply lt_imp_lt_of_le_imp_le (power_le_power_left <| power_ne_zero a b0) ** α : Type u_1 r : α → α → Prop a b : Cardinal.{u_2} ha : ℵ₀ ≤ a b1 : 1 < b b0 : b ≠ 0 ⊢ (b ^ a) ^ a < (b ^ a) ^ Ordinal.cof (ord (b ^ a)) ** rw [← power_mul, mul_eq_self ha] ** α : Type u_1 r : α → α → Prop a b : Cardinal.{u_2} ha : ℵ₀ ≤ a b1 : 1 < b b0 : b ≠ 0 ⊢ b ^ a < (b ^ a) ^ Ordinal.cof (ord (b ^ a)) ** exact lt_power_cof (ha.trans <| (cantor' _ b1).le) ** Qed | |
Cardinal.ord_isLimit ** c : Cardinal.{u_1} co : ℵ₀ ≤ c ⊢ Ordinal.IsLimit (ord c) ** refine' ⟨fun h => aleph0_ne_zero _, fun a => lt_imp_lt_of_le_imp_le fun h => _⟩ ** case refine'_1 c : Cardinal.{u_1} co : ℵ₀ ≤ c h : ord c = 0 ⊢ ℵ₀ = 0 ** rw [← Ordinal.le_zero, ord_le] at h ** case refine'_1 c : Cardinal.{u_1} co : ℵ₀ ≤ c h : c ≤ card 0 ⊢ ℵ₀ = 0 ** simpa only [card_zero, nonpos_iff_eq_zero] using co.trans h ** case refine'_2 c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : ord c ≤ succ a ⊢ ord c ≤ a ** rw [ord_le] at h ⊢ ** case refine'_2 c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : c ≤ card (succ a) ⊢ c ≤ card a ** rwa [← @add_one_of_aleph0_le (card a), ← card_succ] ** case refine'_2 c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : c ≤ card (succ a) ⊢ ℵ₀ ≤ card a ** rw [← ord_le, ← le_succ_of_isLimit, ord_le] ** case refine'_2 c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : c ≤ card (succ a) ⊢ ℵ₀ ≤ card (succ a) ** exact co.trans h ** case refine'_2.h c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : c ≤ card (succ a) ⊢ Ordinal.IsLimit (ord ℵ₀) ** rw [ord_aleph0] ** case refine'_2.h c : Cardinal.{u_1} co : ℵ₀ ≤ c a : Ordinal.{u_1} h : c ≤ card (succ a) ⊢ Ordinal.IsLimit ω ** exact omega_isLimit ** Qed | |
Cardinal.alephIdx_le ** a b : Cardinal.{u_1} ⊢ alephIdx a ≤ alephIdx b ↔ a ≤ b ** rw [← not_lt, ← not_lt, alephIdx_lt] ** Qed | |
Cardinal.type_cardinal ** ⊢ (type fun x x_1 => x < x_1) = Ordinal.univ ** rw [Ordinal.univ_id] ** ⊢ (type fun x x_1 => x < x_1) = type fun x x_1 => x < x_1 ** exact Quotient.sound ⟨alephIdx.relIso⟩ ** Qed | |
Cardinal.mk_cardinal ** ⊢ #Cardinal.{u} = univ ** simpa only [card_type, card_univ] using congr_arg card type_cardinal ** Qed | |
Cardinal.aleph'_zero ** ⊢ aleph' 0 = 0 ** rw [← nonpos_iff_eq_zero, ← aleph'_alephIdx 0, aleph'_le] ** ⊢ 0 ≤ alephIdx 0 ** apply Ordinal.zero_le ** Qed | |
Cardinal.aleph'_succ ** o : Ordinal.{u_1} ⊢ aleph' (succ o) = succ (aleph' o) ** apply (succ_le_of_lt <| aleph'_lt.2 <| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <| _) ** o : Ordinal.{u_1} ⊢ alephIdx (aleph' (succ o)) ≤ alephIdx (succ (aleph' o)) ** rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx] ** o : Ordinal.{u_1} ⊢ aleph' o < succ (aleph' o) ** apply lt_succ ** Qed | |
Cardinal.aleph'_nat ** n : ℕ ⊢ aleph' (succ ↑n) = ↑(Nat.succ n) ** rw [aleph'_succ, aleph'_nat n, nat_succ] ** Qed | |
Cardinal.aleph'_le_of_limit ** o : Ordinal.{u_1} l : Ordinal.IsLimit o c : Cardinal.{u_1} h : ∀ (o' : Ordinal.{u_1}), o' < o → aleph' o' ≤ c ⊢ aleph' o ≤ c ** rw [← aleph'_alephIdx c, aleph'_le, limit_le l] ** o : Ordinal.{u_1} l : Ordinal.IsLimit o c : Cardinal.{u_1} h : ∀ (o' : Ordinal.{u_1}), o' < o → aleph' o' ≤ c ⊢ ∀ (x : Ordinal.{u_1}), x < o → x ≤ alephIdx c ** intro x h' ** o : Ordinal.{u_1} l : Ordinal.IsLimit o c : Cardinal.{u_1} h : ∀ (o' : Ordinal.{u_1}), o' < o → aleph' o' ≤ c x : Ordinal.{u_1} h' : x < o ⊢ x ≤ alephIdx c ** rw [← aleph'_le, aleph'_alephIdx] ** o : Ordinal.{u_1} l : Ordinal.IsLimit o c : Cardinal.{u_1} h : ∀ (o' : Ordinal.{u_1}), o' < o → aleph' o' ≤ c x : Ordinal.{u_1} h' : x < o ⊢ aleph' x ≤ c ** exact h _ h' ** Qed | |
Cardinal.aleph'_limit ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ aleph' o = ⨆ a, aleph' ↑a ** refine' le_antisymm _ (ciSup_le' fun i => aleph'_le.2 (le_of_lt i.2)) ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ aleph' o ≤ ⨆ a, aleph' ↑a ** rw [aleph'_le_of_limit ho] ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ ∀ (o' : Ordinal.{u_1}), o' < o → aleph' o' ≤ ⨆ a, aleph' ↑a ** exact fun a ha => le_ciSup (bddAbove_of_small _) (⟨a, ha⟩ : Iio o) ** Qed | |
Cardinal.aleph'_omega ** c : Cardinal.{u_1} ⊢ aleph' ω ≤ c ↔ ℵ₀ ≤ c ** simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le] ** c : Cardinal.{u_1} ⊢ (∀ (o' : Ordinal.{u_1}) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c ** exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat]) ** c : Cardinal.{u_1} n : ℕ ⊢ (∀ (x : Ordinal.{u_1}), x = ↑n → aleph' x ≤ c) ↔ ↑n ≤ c ** simp only [forall_eq, aleph'_nat] ** Qed | |
Cardinal.max_aleph_eq ** o₁ o₂ : Ordinal.{u_1} ⊢ max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) ** cases' le_total (aleph o₁) (aleph o₂) with h h ** case inl o₁ o₂ : Ordinal.{u_1} h : aleph o₁ ≤ aleph o₂ ⊢ max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) ** rw [max_eq_right h, max_eq_right (aleph_le.1 h)] ** case inr o₁ o₂ : Ordinal.{u_1} h : aleph o₂ ≤ aleph o₁ ⊢ max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) ** rw [max_eq_left h, max_eq_left (aleph_le.1 h)] ** Qed | |
Cardinal.aleph_succ ** o : Ordinal.{u_1} ⊢ aleph (succ o) = succ (aleph o) ** rw [aleph, add_succ, aleph'_succ, aleph] ** Qed | |
Cardinal.aleph_zero ** ⊢ aleph 0 = ℵ₀ ** rw [aleph, add_zero, aleph'_omega] ** Qed | |
Cardinal.aleph_limit ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ aleph o = ⨆ a, aleph ↑a ** apply le_antisymm _ (ciSup_le' _) ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ aleph o ≤ ⨆ i, aleph ↑i ** rw [aleph, aleph'_limit (ho.add _)] ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ ⨆ a, aleph' ↑a ≤ ⨆ i, aleph ↑i ** refine' ciSup_mono' (bddAbove_of_small _) _ ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ ∀ (i : ↑(Iio (ω + o))), ∃ i', aleph' ↑i ≤ aleph ↑i' ** rintro ⟨i, hi⟩ ** case mk o : Ordinal.{u_1} ho : Ordinal.IsLimit o i : Ordinal.{u_1} hi : i ∈ Iio (ω + o) ⊢ ∃ i', aleph' ↑{ val := i, property := hi } ≤ aleph ↑i' ** cases' lt_or_le i ω with h h ** case mk.inl o : Ordinal.{u_1} ho : Ordinal.IsLimit o i : Ordinal.{u_1} hi : i ∈ Iio (ω + o) h : i < ω ⊢ ∃ i', aleph' ↑{ val := i, property := hi } ≤ aleph ↑i' ** rcases lt_omega.1 h with ⟨n, rfl⟩ ** case mk.inl.intro o : Ordinal.{u_1} ho : Ordinal.IsLimit o n : ℕ hi : ↑n ∈ Iio (ω + o) h : ↑n < ω ⊢ ∃ i', aleph' ↑{ val := ↑n, property := hi } ≤ aleph ↑i' ** use ⟨0, ho.pos⟩ ** case h o : Ordinal.{u_1} ho : Ordinal.IsLimit o n : ℕ hi : ↑n ∈ Iio (ω + o) h : ↑n < ω ⊢ aleph' ↑{ val := ↑n, property := hi } ≤ aleph ↑{ val := 0, property := (_ : 0 < o) } ** simpa using (nat_lt_aleph0 n).le ** case mk.inr o : Ordinal.{u_1} ho : Ordinal.IsLimit o i : Ordinal.{u_1} hi : i ∈ Iio (ω + o) h : ω ≤ i ⊢ ∃ i', aleph' ↑{ val := i, property := hi } ≤ aleph ↑i' ** exact ⟨⟨_, (sub_lt_of_le h).2 hi⟩, aleph'_le.2 (le_add_sub _ _)⟩ ** o : Ordinal.{u_1} ho : Ordinal.IsLimit o ⊢ ∀ (i : ↑(Iio o)), aleph ↑i ≤ aleph o ** exact fun i => aleph_le.2 (le_of_lt i.2) ** Qed | |
Cardinal.aleph0_le_aleph' ** o : Ordinal.{u_1} ⊢ ℵ₀ ≤ aleph' o ↔ ω ≤ o ** rw [← aleph'_omega, aleph'_le] ** Qed | |
Cardinal.aleph0_le_aleph ** o : Ordinal.{u_1} ⊢ ℵ₀ ≤ aleph o ** rw [aleph, aleph0_le_aleph'] ** o : Ordinal.{u_1} ⊢ ω ≤ ω + o ** apply Ordinal.le_add_right ** Qed | |
Cardinal.aleph'_pos ** o : Ordinal.{u_1} ho : 0 < o ⊢ 0 < aleph' o ** rwa [← aleph'_zero, aleph'_lt] ** Qed | |
Cardinal.exists_aleph ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c = aleph (alephIdx c - ω) ** rw [aleph, Ordinal.add_sub_cancel_of_le, aleph'_alephIdx] ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ ω ≤ alephIdx c ** rwa [← aleph0_le_aleph', aleph'_alephIdx] ** Qed | |
Cardinal.aleph'_isNormal ** o : Ordinal.{u_1} l : Ordinal.IsLimit o a : Ordinal.{u_1} ⊢ (ord ∘ aleph') o ≤ a ↔ ∀ (b : Ordinal.{u_1}), b < o → (ord ∘ aleph') b ≤ a ** simp [ord_le, aleph'_le_of_limit l] ** Qed | |
Cardinal.succ_aleph0 ** ⊢ succ ℵ₀ = aleph 1 ** rw [← aleph_zero, ← aleph_succ, Ordinal.succ_zero] ** Qed | |
Cardinal.aleph0_lt_aleph_one ** ⊢ ℵ₀ < aleph 1 ** rw [← succ_aleph0] ** ⊢ ℵ₀ < succ ℵ₀ ** apply lt_succ ** Qed | |
Cardinal.countable_iff_lt_aleph_one ** α : Type u_1 s : Set α ⊢ Set.Countable s ↔ #↑s < aleph 1 ** rw [← succ_aleph0, lt_succ_iff, le_aleph0_iff_set_countable] ** Qed | |
Cardinal.ord_card_unbounded ** a : Ordinal.{u_1} ⊢ ord (succ (card a)) ∈ {b | ord (card b) = b} ** dsimp ** a : Ordinal.{u_1} ⊢ ord (card (ord (succ (card a)))) = ord (succ (card a)) ** rw [card_ord] ** Qed | |
Cardinal.eq_aleph'_of_eq_card_ord ** o : Ordinal.{u_1} ho : ord (card o) = o ⊢ ord (aleph' (↑alephIdx.relIso (card o))) = o ** simpa using ho ** Qed | |
Cardinal.ord_aleph'_eq_enum_card ** ⊢ ord ∘ aleph' = enumOrd {b | ord (card b) = b} ** rw [← eq_enumOrd _ ord_card_unbounded, range_eq_iff] ** ⊢ StrictMono (ord ∘ aleph') ∧ (∀ (a : Ordinal.{u_1}), (ord ∘ aleph') a ∈ {b | ord (card b) = b}) ∧ ∀ (b : Ordinal.{u_1}), b ∈ {b | ord (card b) = b} → ∃ a, (ord ∘ aleph') a = b ** exact
⟨aleph'_isNormal.strictMono,
⟨fun a => by
dsimp
rw [card_ord], fun b hb => eq_aleph'_of_eq_card_ord hb⟩⟩ ** a : Ordinal.{u_1} ⊢ (ord ∘ aleph') a ∈ {b | ord (card b) = b} ** dsimp ** a : Ordinal.{u_1} ⊢ ord (card (ord (aleph' a))) = ord (aleph' a) ** rw [card_ord] ** Qed | |
Cardinal.eq_aleph_of_eq_card_ord ** o : Ordinal.{u_1} ho : ord (card o) = o ho' : ω ≤ o ⊢ ∃ a, ord (aleph a) = o ** cases' eq_aleph'_of_eq_card_ord ho with a ha ** case intro o : Ordinal.{u_1} ho : ord (card o) = o ho' : ω ≤ o a : Ordinal.{u_1} ha : ord (aleph' a) = o ⊢ ∃ a, ord (aleph a) = o ** use a - ω ** case h o : Ordinal.{u_1} ho : ord (card o) = o ho' : ω ≤ o a : Ordinal.{u_1} ha : ord (aleph' a) = o ⊢ ord (aleph (a - ω)) = o ** unfold aleph ** case h o : Ordinal.{u_1} ho : ord (card o) = o ho' : ω ≤ o a : Ordinal.{u_1} ha : ord (aleph' a) = o ⊢ ord (aleph' (ω + (a - ω))) = o ** rwa [Ordinal.add_sub_cancel_of_le] ** case h o : Ordinal.{u_1} ho : ord (card o) = o ho' : ω ≤ o a : Ordinal.{u_1} ha : ord (aleph' a) = o ⊢ ω ≤ a ** rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0] ** Qed | |
Cardinal.ord_aleph_eq_enum_card ** ⊢ ord ∘ aleph = enumOrd {b | ord (card b) = b ∧ ω ≤ b} ** rw [← eq_enumOrd _ ord_card_unbounded'] ** ⊢ StrictMono (ord ∘ aleph) ∧ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b} ** use aleph_isNormal.strictMono ** case right ⊢ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b} ** rw [range_eq_iff] ** case right ⊢ (∀ (a : Ordinal.{u_1}), (ord ∘ aleph) a ∈ {b | ord (card b) = b ∧ ω ≤ b}) ∧ ∀ (b : Ordinal.{u_1}), b ∈ {b | ord (card b) = b ∧ ω ≤ b} → ∃ a, (ord ∘ aleph) a = b ** refine' ⟨fun a => ⟨_, _⟩, fun b hb => eq_aleph_of_eq_card_ord hb.1 hb.2⟩ ** case right.refine'_1 a : Ordinal.{u_1} ⊢ ord (card ((ord ∘ aleph) a)) = (ord ∘ aleph) a ** rw [Function.comp_apply, card_ord] ** case right.refine'_2 a : Ordinal.{u_1} ⊢ ω ≤ (ord ∘ aleph) a ** rw [← ord_aleph0, Function.comp_apply, ord_le_ord] ** case right.refine'_2 a : Ordinal.{u_1} ⊢ ℵ₀ ≤ aleph a ** exact aleph0_le_aleph _ ** Qed | |
Cardinal.beth_strictMono ** ⊢ StrictMono beth ** intro a b ** a b : Ordinal.{u_1} ⊢ a < b → beth a < beth b ** induction' b using Ordinal.induction with b IH generalizing a ** case h a✝ b : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < b → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k a : Ordinal.{u_1} ⊢ a < b → beth a < beth b ** intro h ** case h a✝ b : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < b → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k a : Ordinal.{u_1} h : a < b ⊢ beth a < beth b ** rcases zero_or_succ_or_limit b with (rfl | ⟨c, rfl⟩ | hb) ** case h.inl a✝ a : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < 0 → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a < 0 ⊢ beth a < beth 0 ** exact (Ordinal.not_lt_zero a h).elim ** case h.inr.inl.intro a✝ a c : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ c → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a < succ c ⊢ beth a < beth (succ c) ** rw [lt_succ_iff] at h ** case h.inr.inl.intro a✝ a c : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ c → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a ≤ c ⊢ beth a < beth (succ c) ** rw [beth_succ] ** case h.inr.inl.intro a✝ a c : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ c → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a ≤ c ⊢ beth a < 2 ^ beth c ** apply lt_of_le_of_lt _ (cantor _) ** a✝ a c : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ c → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a ≤ c ⊢ beth a ≤ beth c ** rcases eq_or_lt_of_le h with (rfl | h) ** case inr a✝ a c : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ c → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h✝ : a ≤ c h : a < c ⊢ beth a ≤ beth c ** exact (IH c (lt_succ c) h).le ** case inl a✝ a : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < succ a → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k h : a ≤ a ⊢ beth a ≤ beth a ** rfl ** case h.inr.inr a✝ b : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < b → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k a : Ordinal.{u_1} h : a < b hb : Ordinal.IsLimit b ⊢ beth a < beth b ** apply (cantor _).trans_le ** case h.inr.inr a✝ b : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < b → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k a : Ordinal.{u_1} h : a < b hb : Ordinal.IsLimit b ⊢ 2 ^ beth a ≤ beth b ** rw [beth_limit hb, ← beth_succ] ** case h.inr.inr a✝ b : Ordinal.{u_1} IH : ∀ (k : Ordinal.{u_1}), k < b → ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k a : Ordinal.{u_1} h : a < b hb : Ordinal.IsLimit b ⊢ beth (succ a) ≤ ⨆ a, beth ↑a ** exact le_ciSup (bddAbove_of_small _) (⟨_, hb.succ_lt h⟩ : Iio b) ** Qed | |
Cardinal.aleph_le_beth ** o : Ordinal.{u_1} ⊢ aleph o ≤ beth o ** induction o using limitRecOn with
| H₁ => simp
| H₂ o h =>
rw [aleph_succ, beth_succ, succ_le_iff]
exact (cantor _).trans_le (power_le_power_left two_ne_zero h)
| H₃ o ho IH =>
rw [aleph_limit ho, beth_limit ho]
exact ciSup_mono (bddAbove_of_small _) fun x => IH x.1 x.2 ** case H₁ ⊢ aleph 0 ≤ beth 0 ** simp ** case H₂ o : Ordinal.{u_1} h : aleph o ≤ beth o ⊢ aleph (succ o) ≤ beth (succ o) ** rw [aleph_succ, beth_succ, succ_le_iff] ** case H₂ o : Ordinal.{u_1} h : aleph o ≤ beth o ⊢ aleph o < 2 ^ beth o ** exact (cantor _).trans_le (power_le_power_left two_ne_zero h) ** case H₃ o : Ordinal.{u_1} ho : Ordinal.IsLimit o IH : ∀ (o' : Ordinal.{u_1}), o' < o → aleph o' ≤ beth o' ⊢ aleph o ≤ beth o ** rw [aleph_limit ho, beth_limit ho] ** case H₃ o : Ordinal.{u_1} ho : Ordinal.IsLimit o IH : ∀ (o' : Ordinal.{u_1}), o' < o → aleph o' ≤ beth o' ⊢ ⨆ a, aleph ↑a ≤ ⨆ a, beth ↑a ** exact ciSup_mono (bddAbove_of_small _) fun x => IH x.1 x.2 ** Qed | |
Cardinal.beth_normal ** o : Ordinal.{u} ho : Ordinal.IsLimit o a : Ordinal.{u} ha : ∀ (b : Ordinal.{u}), b < o → ord (beth b) ≤ a ⊢ ord (beth o) ≤ a ** rw [beth_limit ho, ord_le] ** o : Ordinal.{u} ho : Ordinal.IsLimit o a : Ordinal.{u} ha : ∀ (b : Ordinal.{u}), b < o → ord (beth b) ≤ a ⊢ ⨆ a, beth ↑a ≤ card a ** exact ciSup_le' fun b => ord_le.1 (ha _ b.2) ** Qed | |
Cardinal.mul_eq_self ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c * c = c ** refine' le_antisymm _ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c) ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c * c ≤ c ** refine' Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => _) h ** c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** rcases ord_eq α with ⟨r, wo, e⟩ ** case intro.intro c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** letI := linearOrderOfSTO r ** case intro.intro c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝ : LinearOrder α := linearOrderOfSTO r this : IsWellOrder α fun x x_1 => x < x_1 ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** let g : α × α → α := fun p => max p.1 p.2 ** case intro.intro c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝ : LinearOrder α := linearOrderOfSTO r this : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** haveI : IsWellOrder _ s := (RelEmbedding.preimage _ _).isWellOrder ** case intro.intro c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** suffices type s ≤ type r by exact card_le_card this ** case intro.intro c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s ⊢ type s ≤ type r ** refine' le_of_forall_lt fun o h => _ ** case intro.intro c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s o : Ordinal.{u_1} h : o < type s ⊢ o < type r ** rcases typein_surj s h with ⟨p, rfl⟩ ** case intro.intro.intro c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s ⊢ typein s p < type r ** rw [← e, lt_ord] ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ≤ c * c ** simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c ** c✝ : Cardinal.{u_1} h : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s this : type s ≤ type r ⊢ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α ** exact card_le_card this ** case intro.intro.intro.refine'_1 c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s ⊢ card (typein s p) ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) ** have : { q | s q p } ⊆ insert (g p) { x | x < g p } ×ˢ insert (g p) { x | x < g p } := by
intro q h
simp only [Preimage, ge_iff_le, Embedding.coeFn_mk, Prod.lex_def, typein_lt_typein,
typein_inj, mem_setOf_eq] at h
exact max_le_iff.1 (le_iff_lt_or_eq.2 <| h.imp_right And.left) ** case intro.intro.intro.refine'_1 c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s p : α × α h : typein s p < type s this : {q | s q p} ⊆ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ⊢ card (typein s p) ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) ** suffices H : (insert (g p) { x | r x (g p) } : Set α) ≃ Sum { x | r x (g p) } PUnit ** case H c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s p : α × α h : typein s p < type s this : {q | s q p} ⊆ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ⊢ ↑(insert (g p) {x | r x (g p)}) ≃ ↑{x | r x (g p)} ⊕ PUnit.{u_1 + 1} ** refine' (Equiv.Set.insert _).trans ((Equiv.refl _).sumCongr punitEquivPUnit) ** case H c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s p : α × α h : typein s p < type s this : {q | s q p} ⊆ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ⊢ ¬g p ∈ {x | r x (g p)} ** apply @irrefl _ r ** c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s ⊢ {q | s q p} ⊆ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ** intro q h ** c✝ : Cardinal.{u_1} h✝¹ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h✝ : typein s p < type s q : α × α h : q ∈ {q | s q p} ⊢ q ∈ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ** simp only [Preimage, ge_iff_le, Embedding.coeFn_mk, Prod.lex_def, typein_lt_typein,
typein_inj, mem_setOf_eq] at h ** c✝ : Cardinal.{u_1} h✝¹ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h✝ : typein s p < type s q : α × α h : max q.1 q.2 < max p.1 p.2 ∨ max q.1 q.2 = max p.1 p.2 ∧ (q.1 < p.1 ∨ q.1 = p.1 ∧ q.2 < p.2) ⊢ q ∈ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} ** exact max_le_iff.1 (le_iff_lt_or_eq.2 <| h.imp_right And.left) ** case intro.intro.intro.refine'_1 c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s p : α × α h : typein s p < type s this : {q | s q p} ⊆ insert (g p) {x | x < g p} ×ˢ insert (g p) {x | x < g p} H : ↑(insert (g p) {x | r x (g p)}) ≃ ↑{x | r x (g p)} ⊕ PUnit.{?u.110308 + 1} ⊢ card (typein s p) ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) ** exact
⟨(Set.embeddingOfSubset _ _ this).trans
((Equiv.Set.prod _ _).trans (H.prodCongr H)).toEmbedding⟩ ** case intro.intro.intro.refine'_2.inl c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : card (succ (typein (fun x x_1 => x < x_1) (g p))) < ℵ₀ ⊢ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) < #α ** exact (mul_lt_aleph0 qo qo).trans_le ol ** case intro.intro.intro.refine'_2.inr c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) ⊢ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) < #α ** suffices : (succ (typein LT.lt (g p))).card < ⟦α⟧ ** case this c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) ⊢ card (succ (typein LT.lt (g p))) < Quotient.mk isEquivalent α ** rw [← lt_ord] ** case this c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) ⊢ succ (typein LT.lt (g p)) < ord (Quotient.mk isEquivalent α) ** apply (ord_isLimit ol).2 ** case this.a c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) ⊢ typein LT.lt (g p) < ord (Quotient.mk isEquivalent α) ** rw [mk'_def, e] ** case this.a c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝¹ : LinearOrder α := linearOrderOfSTO r this✝ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) ⊢ typein LT.lt (g p) < type r ** apply typein_lt_type ** case intro.intro.intro.refine'_2.inr c✝ : Cardinal.{u_1} h✝ : ℵ₀ ≤ c✝ c : Cardinal.{u_1} x✝ : ∀ (y : Cardinal.{u_1}), y < c → Acc (fun x x_1 => x < x_1) y α : Type u_1 IH : ∀ (y : Cardinal.{u_1}), y < Quotient.mk isEquivalent α → ℵ₀ ≤ y → y * y ≤ y ol : ℵ₀ ≤ Quotient.mk isEquivalent α r : α → α → Prop wo : IsWellOrder α r e : ord #α = type r this✝² : LinearOrder α := linearOrderOfSTO r this✝¹ : IsWellOrder α fun x x_1 => x < x_1 g : α × α → α := fun p => max p.1 p.2 f : α × α ↪ Ordinal.{u_1} × α × α := { toFun := fun p => (typein (fun x x_1 => x < x_1) (g p), p), inj' := (_ : ∀ (p q : α × α), (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p = (fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q → ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) p).2 = ((fun p => (typein (fun x x_1 => x < x_1) (g p), p)) q).2) } s : α × α → α × α → Prop := ↑f ⁻¹'o Prod.Lex (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1) this✝ : IsWellOrder (α × α) s p : α × α h : typein s p < type s qo : ℵ₀ ≤ card (succ (typein (fun x x_1 => x < x_1) (g p))) this : card (succ (typein LT.lt (g p))) < Quotient.mk isEquivalent α ⊢ card (succ (typein (fun x x_1 => x < x_1) (g p))) * card (succ (typein (fun x x_1 => x < x_1) (g p))) < #α ** exact (IH _ this qo).trans_lt this ** Qed | |
Cardinal.mul_eq_max ** a b : Cardinal.{u_1} ha : ℵ₀ ≤ a hb : ℵ₀ ≤ b ⊢ a ≤ a * b ** simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans hb) a ** a b : Cardinal.{u_1} ha : ℵ₀ ≤ a hb : ℵ₀ ≤ b ⊢ b ≤ a * b ** simpa only [one_mul] using mul_le_mul_right' (one_le_aleph0.trans ha) b ** Qed | |
Cardinal.aleph_mul_aleph ** o₁ o₂ : Ordinal.{u_1} ⊢ aleph o₁ * aleph o₂ = aleph (max o₁ o₂) ** rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq] ** Qed | |
Cardinal.mul_lt_of_lt ** a b c : Cardinal.{u_1} hc : ℵ₀ ≤ c h1 : a < c h2 : b < c h : ℵ₀ ≤ max a b ⊢ max a b * max a b < c ** rw [mul_eq_self h] ** a b c : Cardinal.{u_1} hc : ℵ₀ ≤ c h1 : a < c h2 : b < c h : ℵ₀ ≤ max a b ⊢ max a b < c ** exact max_lt h1 h2 ** Qed | |
Cardinal.mul_le_max_of_aleph0_le_left ** a b : Cardinal.{u_1} h : ℵ₀ ≤ a ⊢ a * b ≤ max a b ** convert mul_le_mul' (le_max_left a b) (le_max_right a b) using 1 ** case h.e'_4 a b : Cardinal.{u_1} h : ℵ₀ ≤ a ⊢ max a b = max a b * max a b ** rw [mul_eq_self] ** case h.e'_4 a b : Cardinal.{u_1} h : ℵ₀ ≤ a ⊢ ℵ₀ ≤ max a b ** refine' h.trans (le_max_left a b) ** Qed | |
Cardinal.mul_eq_max_of_aleph0_le_left ** a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 ⊢ a * b = max a b ** cases' le_or_lt ℵ₀ b with hb hb ** case inr a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : b < ℵ₀ ⊢ a * b = max a b ** refine' (mul_le_max_of_aleph0_le_left h).antisymm _ ** case inr a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : b < ℵ₀ ⊢ max a b ≤ a * b ** have : b ≤ a := hb.le.trans h ** case inr a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : b < ℵ₀ this : b ≤ a ⊢ max a b ≤ a * b ** rw [max_eq_left this] ** case inr a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : b < ℵ₀ this : b ≤ a ⊢ a ≤ a * b ** convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') a ** case h.e'_3 a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : b < ℵ₀ this : b ≤ a ⊢ a = a * 1 ** rw [mul_one] ** case inl a b : Cardinal.{u_1} h : ℵ₀ ≤ a h' : b ≠ 0 hb : ℵ₀ ≤ b ⊢ a * b = max a b ** exact mul_eq_max h hb ** Qed | |
Cardinal.mul_le_max_of_aleph0_le_right ** a b : Cardinal.{u_1} h : ℵ₀ ≤ b ⊢ a * b ≤ max a b ** simpa only [mul_comm b, max_comm b] using mul_le_max_of_aleph0_le_left h ** Qed | |
Cardinal.mul_eq_max_of_aleph0_le_right ** a b : Cardinal.{u_1} h' : a ≠ 0 h : ℵ₀ ≤ b ⊢ a * b = max a b ** rw [mul_comm, max_comm] ** a b : Cardinal.{u_1} h' : a ≠ 0 h : ℵ₀ ≤ b ⊢ b * a = max b a ** exact mul_eq_max_of_aleph0_le_left h h' ** Qed | |
Cardinal.mul_eq_max' ** a b : Cardinal.{u_1} h : ℵ₀ ≤ a * b ⊢ a * b = max a b ** rcases aleph0_le_mul_iff.mp h with ⟨ha, hb, ha' | hb'⟩ ** case intro.intro.inl a b : Cardinal.{u_1} h : ℵ₀ ≤ a * b ha : a ≠ 0 hb : b ≠ 0 ha' : ℵ₀ ≤ a ⊢ a * b = max a b ** exact mul_eq_max_of_aleph0_le_left ha' hb ** case intro.intro.inr a b : Cardinal.{u_1} h : ℵ₀ ≤ a * b ha : a ≠ 0 hb : b ≠ 0 hb' : ℵ₀ ≤ b ⊢ a * b = max a b ** exact mul_eq_max_of_aleph0_le_right ha hb' ** Qed | |
Cardinal.mul_le_max ** a b : Cardinal.{u_1} ⊢ a * b ≤ max (max a b) ℵ₀ ** rcases eq_or_ne a 0 with (rfl | ha0) ** case inr a b : Cardinal.{u_1} ha0 : a ≠ 0 ⊢ a * b ≤ max (max a b) ℵ₀ ** rcases eq_or_ne b 0 with (rfl | hb0) ** case inr.inr a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ⊢ a * b ≤ max (max a b) ℵ₀ ** cases' le_or_lt ℵ₀ a with ha ha ** case inl b : Cardinal.{u_1} ⊢ 0 * b ≤ max (max 0 b) ℵ₀ ** simp ** case inr.inl a : Cardinal.{u_1} ha0 : a ≠ 0 ⊢ a * 0 ≤ max (max a 0) ℵ₀ ** simp ** case inr.inr.inl a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : ℵ₀ ≤ a ⊢ a * b ≤ max (max a b) ℵ₀ ** rw [mul_eq_max_of_aleph0_le_left ha hb0] ** case inr.inr.inl a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : ℵ₀ ≤ a ⊢ max a b ≤ max (max a b) ℵ₀ ** exact le_max_left _ _ ** case inr.inr.inr a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : a < ℵ₀ ⊢ a * b ≤ max (max a b) ℵ₀ ** cases' le_or_lt ℵ₀ b with hb hb ** case inr.inr.inr.inl a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : a < ℵ₀ hb : ℵ₀ ≤ b ⊢ a * b ≤ max (max a b) ℵ₀ ** rw [mul_comm, mul_eq_max_of_aleph0_le_left hb ha0, max_comm] ** case inr.inr.inr.inl a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : a < ℵ₀ hb : ℵ₀ ≤ b ⊢ max a b ≤ max (max a b) ℵ₀ ** exact le_max_left _ _ ** case inr.inr.inr.inr a b : Cardinal.{u_1} ha0 : a ≠ 0 hb0 : b ≠ 0 ha : a < ℵ₀ hb : b < ℵ₀ ⊢ a * b ≤ max (max a b) ℵ₀ ** exact le_max_of_le_right (mul_lt_aleph0 ha hb).le ** Qed | |
Cardinal.mul_eq_left ** a b : Cardinal.{u_1} ha : ℵ₀ ≤ a hb : b ≤ a hb' : b ≠ 0 ⊢ a * b = a ** rw [mul_eq_max_of_aleph0_le_left ha hb', max_eq_left hb] ** Qed | |
Cardinal.mul_eq_right ** a b : Cardinal.{u_1} hb : ℵ₀ ≤ b ha : a ≤ b ha' : a ≠ 0 ⊢ a * b = b ** rw [mul_comm, mul_eq_left hb ha ha'] ** Qed | |
Cardinal.le_mul_left ** a b : Cardinal.{u_1} h : b ≠ 0 ⊢ a ≤ b * a ** convert mul_le_mul_right' (one_le_iff_ne_zero.mpr h) a ** case h.e'_3 a b : Cardinal.{u_1} h : b ≠ 0 ⊢ a = 1 * a ** rw [one_mul] ** Qed | |
Cardinal.le_mul_right ** a b : Cardinal.{u_1} h : b ≠ 0 ⊢ a ≤ a * b ** rw [mul_comm] ** a b : Cardinal.{u_1} h : b ≠ 0 ⊢ a ≤ b * a ** exact le_mul_left h ** Qed | |
Cardinal.mul_eq_left_iff ** a b : Cardinal.{u_1} ⊢ a * b = a ↔ max ℵ₀ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** rw [max_le_iff] ** a b : Cardinal.{u_1} ⊢ a * b = a ↔ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** refine' ⟨fun h => _, _⟩ ** case refine'_1 a b : Cardinal.{u_1} h : a * b = a ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** cases' le_or_lt ℵ₀ a with ha ha ** case refine'_1.inr a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** right ** case refine'_1.inr.h a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ ⊢ b = 1 ∨ a = 0 ** by_cases h2a : a = 0 ** case neg a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ h2a : ¬a = 0 ⊢ b = 1 ∨ a = 0 ** have hb : b ≠ 0 := by
rintro rfl
apply h2a
rw [mul_zero] at h
exact h.symm ** case neg a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ h2a : ¬a = 0 hb : b ≠ 0 ⊢ b = 1 ∨ a = 0 ** left ** case neg.h a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ h2a : ¬a = 0 hb : b ≠ 0 ⊢ b = 1 ** rw [← h, mul_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha ** case neg.h a b : Cardinal.{u_1} h : a * b = a ha : a = 0 ∨ b = 0 ∨ (∃ n, a = ↑n) ∧ ∃ n, b = ↑n h2a : ¬a = 0 hb : b ≠ 0 ⊢ b = 1 ** rcases ha with (rfl | rfl | ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩) ** case neg.h.inl b : Cardinal.{u_1} hb : b ≠ 0 h : 0 * b = 0 h2a : ¬0 = 0 ⊢ b = 1 case neg.h.inr.inl a : Cardinal.{u_1} h2a : ¬a = 0 h : a * 0 = a hb : 0 ≠ 0 ⊢ 0 = 1 case neg.h.inr.inr.intro.intro.intro n : ℕ h2a : ¬↑n = 0 m : ℕ hb : ↑m ≠ 0 h : ↑n * ↑m = ↑n ⊢ ↑m = 1 ** contradiction ** case neg.h.inr.inl a : Cardinal.{u_1} h2a : ¬a = 0 h : a * 0 = a hb : 0 ≠ 0 ⊢ 0 = 1 case neg.h.inr.inr.intro.intro.intro n : ℕ h2a : ¬↑n = 0 m : ℕ hb : ↑m ≠ 0 h : ↑n * ↑m = ↑n ⊢ ↑m = 1 ** contradiction ** case neg.h.inr.inr.intro.intro.intro n : ℕ h2a : ¬↑n = 0 m : ℕ hb : ↑m ≠ 0 h : ↑n * ↑m = ↑n ⊢ ↑m = 1 ** rw [← Ne] at h2a ** case neg.h.inr.inr.intro.intro.intro n : ℕ h2a : ↑n ≠ 0 m : ℕ hb : ↑m ≠ 0 h : ↑n * ↑m = ↑n ⊢ ↑m = 1 ** rw [← one_le_iff_ne_zero] at h2a hb ** case neg.h.inr.inr.intro.intro.intro n : ℕ h2a : 1 ≤ ↑n m : ℕ hb : 1 ≤ ↑m h : ↑n * ↑m = ↑n ⊢ ↑m = 1 ** norm_cast at h2a hb h ⊢ ** case neg.h.inr.inr.intro.intro.intro n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ m = 1 ** apply le_antisymm _ hb ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ m ≤ 1 ** rw [← not_lt] ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ ¬1 < m ** apply fun h2b => ne_of_gt _ h ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ 1 < m → n < n * m ** conv_rhs => left; rw [← mul_one n] ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ 1 < m → n * 1 < n * m ** rw [mul_lt_mul_left] ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ 1 < m → 1 < m n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ 0 < n ** exact id ** n m : ℕ h2a : 1 ≤ n hb : 1 ≤ m h : n * m = n ⊢ 0 < n ** apply Nat.lt_of_succ_le h2a ** case refine'_1.inl a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** have : a ≠ 0 := by
rintro rfl
exact ha.not_lt aleph0_pos ** case refine'_1.inl a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 ** left ** case refine'_1.inl.h a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ** rw [and_assoc] ** case refine'_1.inl.h a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ ℵ₀ ≤ a ∧ b ≤ a ∧ b ≠ 0 ** use ha ** case right a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ b ≤ a ∧ b ≠ 0 ** constructor ** a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a ⊢ a ≠ 0 ** rintro rfl ** b : Cardinal.{u_1} h : 0 * b = 0 ha : ℵ₀ ≤ 0 ⊢ False ** exact ha.not_lt aleph0_pos ** case right.left a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ b ≤ a ** rw [← not_lt] ** case right.left a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ ¬a < b ** exact fun hb => ne_of_gt (hb.trans_le (le_mul_left this)) h ** case right.right a b : Cardinal.{u_1} h : a * b = a ha : ℵ₀ ≤ a this : a ≠ 0 ⊢ b ≠ 0 ** rintro rfl ** case right.right a : Cardinal.{u_1} ha : ℵ₀ ≤ a this : a ≠ 0 h : a * 0 = a ⊢ False ** apply this ** case right.right a : Cardinal.{u_1} ha : ℵ₀ ≤ a this : a ≠ 0 h : a * 0 = a ⊢ a = 0 ** rw [mul_zero] at h ** case right.right a : Cardinal.{u_1} ha : ℵ₀ ≤ a this : a ≠ 0 h : 0 = a ⊢ a = 0 ** exact h.symm ** case pos a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ h2a : a = 0 ⊢ b = 1 ∨ a = 0 ** exact Or.inr h2a ** a b : Cardinal.{u_1} h : a * b = a ha : a < ℵ₀ h2a : ¬a = 0 ⊢ b ≠ 0 ** rintro rfl ** a : Cardinal.{u_1} ha : a < ℵ₀ h2a : ¬a = 0 h : a * 0 = a ⊢ False ** apply h2a ** a : Cardinal.{u_1} ha : a < ℵ₀ h2a : ¬a = 0 h : a * 0 = a ⊢ a = 0 ** rw [mul_zero] at h ** a : Cardinal.{u_1} ha : a < ℵ₀ h2a : ¬a = 0 h : 0 = a ⊢ a = 0 ** exact h.symm ** case refine'_2 a b : Cardinal.{u_1} ⊢ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 → a * b = a ** rintro (⟨⟨ha, hab⟩, hb⟩ | rfl | rfl) ** case refine'_2.inr.inl a : Cardinal.{u_1} ⊢ a * 1 = a case refine'_2.inr.inr b : Cardinal.{u_1} ⊢ 0 * b = 0 ** all_goals simp ** case refine'_2.inl.intro.intro a b : Cardinal.{u_1} hb : b ≠ 0 ha : ℵ₀ ≤ a hab : b ≤ a ⊢ a * b = a ** rw [mul_eq_max_of_aleph0_le_left ha hb, max_eq_left hab] ** case refine'_2.inr.inr b : Cardinal.{u_1} ⊢ 0 * b = 0 ** simp ** Qed | |
Cardinal.add_eq_self ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c + c ≤ c ** convert mul_le_mul_right' ((nat_lt_aleph0 2).le.trans h) c using 1
<;> simp [two_mul, mul_eq_self h] ** Qed | |
Cardinal.add_eq_max' ** a b : Cardinal.{u_1} ha : ℵ₀ ≤ b ⊢ a + b = max a b ** rw [add_comm, max_comm, add_eq_max ha] ** Qed | |
Cardinal.add_le_max ** a b : Cardinal.{u_1} ⊢ a + b ≤ max (max a b) ℵ₀ ** cases' le_or_lt ℵ₀ a with ha ha ** case inl a b : Cardinal.{u_1} ha : ℵ₀ ≤ a ⊢ a + b ≤ max (max a b) ℵ₀ ** rw [add_eq_max ha] ** case inl a b : Cardinal.{u_1} ha : ℵ₀ ≤ a ⊢ max a b ≤ max (max a b) ℵ₀ ** exact le_max_left _ _ ** case inr a b : Cardinal.{u_1} ha : a < ℵ₀ ⊢ a + b ≤ max (max a b) ℵ₀ ** cases' le_or_lt ℵ₀ b with hb hb ** case inr.inl a b : Cardinal.{u_1} ha : a < ℵ₀ hb : ℵ₀ ≤ b ⊢ a + b ≤ max (max a b) ℵ₀ ** rw [add_comm, add_eq_max hb, max_comm] ** case inr.inl a b : Cardinal.{u_1} ha : a < ℵ₀ hb : ℵ₀ ≤ b ⊢ max a b ≤ max (max a b) ℵ₀ ** exact le_max_left _ _ ** case inr.inr a b : Cardinal.{u_1} ha : a < ℵ₀ hb : b < ℵ₀ ⊢ a + b ≤ max (max a b) ℵ₀ ** exact le_max_of_le_right (add_lt_aleph0 ha hb).le ** Qed | |
Cardinal.add_lt_of_lt ** a b c : Cardinal.{u_1} hc : ℵ₀ ≤ c h1 : a < c h2 : b < c h : ℵ₀ ≤ max a b ⊢ max a b + max a b < c ** rw [add_eq_self h] ** a b c : Cardinal.{u_1} hc : ℵ₀ ≤ c h1 : a < c h2 : b < c h : ℵ₀ ≤ max a b ⊢ max a b < c ** exact max_lt h1 h2 ** Qed | |
Cardinal.eq_of_add_eq_of_aleph0_le ** a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c ⊢ b = c ** apply le_antisymm ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c ⊢ c ≤ b ** rw [← not_lt] ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c ⊢ ¬b < c ** intro hb ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c hb : b < c ⊢ False ** have : a + b < c := add_lt_of_lt hc ha hb ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c hb : b < c this : a + b < c ⊢ False ** simp [h, lt_irrefl] at this ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c ⊢ b ≤ c ** rw [← h] ** case a a b c : Cardinal.{u_1} h : a + b = c ha : a < c hc : ℵ₀ ≤ c ⊢ b ≤ a + b ** apply self_le_add_left ** Qed | |
Cardinal.add_eq_left ** a b : Cardinal.{u_1} ha : ℵ₀ ≤ a hb : b ≤ a ⊢ a + b = a ** rw [add_eq_max ha, max_eq_left hb] ** Qed | |
Cardinal.add_eq_right ** a b : Cardinal.{u_1} hb : ℵ₀ ≤ b ha : a ≤ b ⊢ a + b = b ** rw [add_comm, add_eq_left hb ha] ** Qed | |
Cardinal.add_eq_left_iff ** a b : Cardinal.{u_1} ⊢ a + b = a ↔ max ℵ₀ b ≤ a ∨ b = 0 ** rw [max_le_iff] ** a b : Cardinal.{u_1} ⊢ a + b = a ↔ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 ** refine' ⟨fun h => _, _⟩ ** case refine'_1 a b : Cardinal.{u_1} h : a + b = a ⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 ** cases' le_or_lt ℵ₀ a with ha ha ** case refine'_1.inr a b : Cardinal.{u_1} h : a + b = a ha : a < ℵ₀ ⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 ** right ** case refine'_1.inr.h a b : Cardinal.{u_1} h : a + b = a ha : a < ℵ₀ ⊢ b = 0 ** rw [← h, add_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha ** case refine'_1.inr.h a b : Cardinal.{u_1} h : a + b = a ha : (∃ n, a = ↑n) ∧ ∃ n, b = ↑n ⊢ b = 0 ** rcases ha with ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩ ** case refine'_1.inr.h.intro.intro.intro n m : ℕ h : ↑n + ↑m = ↑n ⊢ ↑m = 0 ** norm_cast at h ⊢ ** case refine'_1.inr.h.intro.intro.intro n m : ℕ h : n + m = n ⊢ m = 0 ** rw [← add_right_inj, h, add_zero] ** case refine'_1.inl a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a ⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 ** left ** case refine'_1.inl.h a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a ⊢ ℵ₀ ≤ a ∧ b ≤ a ** use ha ** case right a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a ⊢ b ≤ a ** rw [← not_lt] ** case right a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a ⊢ ¬a < b ** apply fun hb => ne_of_gt _ h ** a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a ⊢ a < b → a < a + b ** intro hb ** a b : Cardinal.{u_1} h : a + b = a ha : ℵ₀ ≤ a hb : a < b ⊢ a < a + b ** exact hb.trans_le (self_le_add_left b a) ** case refine'_2 a b : Cardinal.{u_1} ⊢ ℵ₀ ≤ a ∧ b ≤ a ∨ b = 0 → a + b = a ** rintro (⟨h1, h2⟩ | h3) ** case refine'_2.inl.intro a b : Cardinal.{u_1} h1 : ℵ₀ ≤ a h2 : b ≤ a ⊢ a + b = a ** rw [add_eq_max h1, max_eq_left h2] ** case refine'_2.inr a b : Cardinal.{u_1} h3 : b = 0 ⊢ a + b = a ** rw [h3, add_zero] ** Qed | |
Cardinal.add_eq_right_iff ** a b : Cardinal.{u_1} ⊢ a + b = b ↔ max ℵ₀ a ≤ b ∨ a = 0 ** rw [add_comm, add_eq_left_iff] ** Qed | |
Cardinal.nat_add_eq ** a : Cardinal.{u_1} n : ℕ ha : ℵ₀ ≤ a ⊢ ↑n + a = a ** rw [add_comm, add_nat_eq n ha] ** Qed | |
Cardinal.eq_of_add_eq_add_left ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ ⊢ b = c ** cases' le_or_lt ℵ₀ b with hb hb ** case inl a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : ℵ₀ ≤ b ⊢ b = c ** have : a < b := ha.trans_le hb ** case inl a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : ℵ₀ ≤ b this : a < b ⊢ b = c ** rw [add_eq_right hb this.le, eq_comm] at h ** case inl a b c : Cardinal.{u_1} h : a + c = b ha : a < ℵ₀ hb : ℵ₀ ≤ b this : a < b ⊢ b = c ** rw [eq_of_add_eq_of_aleph0_le h this hb] ** case inr a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ ⊢ b = c ** have hc : c < ℵ₀ := by
rw [← not_le]
intro hc
apply lt_irrefl ℵ₀
apply (hc.trans (self_le_add_left _ a)).trans_lt
rw [← h]
apply add_lt_aleph0 ha hb ** case inr a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ hc : c < ℵ₀ ⊢ b = c ** rw [lt_aleph0] at * ** case inr a b c : Cardinal.{u_1} h : a + b = a + c ha : ∃ n, a = ↑n hb : ∃ n, b = ↑n hc : ∃ n, c = ↑n ⊢ b = c ** rcases ha with ⟨n, rfl⟩ ** case inr.intro b c : Cardinal.{u_1} hb : ∃ n, b = ↑n hc : ∃ n, c = ↑n n : ℕ h : ↑n + b = ↑n + c ⊢ b = c ** rcases hb with ⟨m, rfl⟩ ** case inr.intro.intro c : Cardinal.{u_1} hc : ∃ n, c = ↑n n m : ℕ h : ↑n + ↑m = ↑n + c ⊢ ↑m = c ** rcases hc with ⟨k, rfl⟩ ** case inr.intro.intro.intro n m k : ℕ h : ↑n + ↑m = ↑n + ↑k ⊢ ↑m = ↑k ** norm_cast at h ⊢ ** case inr.intro.intro.intro n m k : ℕ h : n + m = n + k ⊢ m = k ** apply add_left_cancel h ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ ⊢ c < ℵ₀ ** rw [← not_le] ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ ⊢ ¬ℵ₀ ≤ c ** intro hc ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ hc : ℵ₀ ≤ c ⊢ False ** apply lt_irrefl ℵ₀ ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ hc : ℵ₀ ≤ c ⊢ ℵ₀ < ℵ₀ ** apply (hc.trans (self_le_add_left _ a)).trans_lt ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ hc : ℵ₀ ≤ c ⊢ a + c < ℵ₀ ** rw [← h] ** a b c : Cardinal.{u_1} h : a + b = a + c ha : a < ℵ₀ hb : b < ℵ₀ hc : ℵ₀ ≤ c ⊢ a + b < ℵ₀ ** apply add_lt_aleph0 ha hb ** Qed | |
Cardinal.eq_of_add_eq_add_right ** a b c : Cardinal.{u_1} h : a + b = c + b hb : b < ℵ₀ ⊢ a = c ** rw [add_comm a b, add_comm c b] at h ** a b c : Cardinal.{u_1} h : b + a = b + c hb : b < ℵ₀ ⊢ a = c ** exact Cardinal.eq_of_add_eq_add_left h hb ** Qed | |
Cardinal.aleph_add_aleph ** o₁ o₂ : Ordinal.{u_1} ⊢ aleph o₁ + aleph o₂ = aleph (max o₁ o₂) ** rw [Cardinal.add_eq_max (aleph0_le_aleph o₁), max_aleph_eq] ** Qed | |
Cardinal.principal_add_ord ** c : Cardinal.{u_1} hc : ℵ₀ ≤ c a b : Ordinal.{u_1} ha : a < ord c hb : b < ord c ⊢ (fun x x_1 => x + x_1) a b < ord c ** rw [lt_ord, Ordinal.card_add] at * ** c : Cardinal.{u_1} hc : ℵ₀ ≤ c a b : Ordinal.{u_1} ha : card a < c hb : card b < c ⊢ card a + card b < c ** exact add_lt_of_lt hc ha hb ** Qed | |
Cardinal.add_le_add_iff_of_lt_aleph0 ** α β γ : Cardinal.{u_1} γ₀ : γ < ℵ₀ ⊢ α + γ ≤ β + γ ↔ α ≤ β ** refine' ⟨fun h => _, fun h => add_le_add_right h γ⟩ ** α β γ : Cardinal.{u_1} γ₀ : γ < ℵ₀ h : α + γ ≤ β + γ ⊢ α ≤ β ** contrapose h ** α β γ : Cardinal.{u_1} γ₀ : γ < ℵ₀ h : ¬α ≤ β ⊢ ¬α + γ ≤ β + γ ** rw [not_le, lt_iff_le_and_ne, Ne] at h ⊢ ** α β γ : Cardinal.{u_1} γ₀ : γ < ℵ₀ h : β ≤ α ∧ ¬β = α ⊢ β + γ ≤ α + γ ∧ ¬β + γ = α + γ ** exact ⟨add_le_add_right h.1 γ, mt (add_right_inj_of_lt_aleph0 γ₀).1 h.2⟩ ** Qed | |
Cardinal.pow_le ** κ μ : Cardinal.{u} H1✝ : ℵ₀ ≤ κ H2 : μ < ℵ₀ n : ℕ H3 : μ = ↑n α : Type u H1 : ℵ₀ ≤ Quotient.mk isEquivalent α ⊢ Quotient.mk isEquivalent α ^ ↑Nat.zero < ℵ₀ ** rw [Nat.cast_zero, power_zero] ** κ μ : Cardinal.{u} H1✝ : ℵ₀ ≤ κ H2 : μ < ℵ₀ n : ℕ H3 : μ = ↑n α : Type u H1 : ℵ₀ ≤ Quotient.mk isEquivalent α ⊢ 1 < ℵ₀ ** exact one_lt_aleph0 ** κ μ : Cardinal.{u} H1✝ : ℵ₀ ≤ κ H2 : μ < ℵ₀ n✝ : ℕ H3 : μ = ↑n✝ α : Type u H1 : ℵ₀ ≤ Quotient.mk isEquivalent α n : ℕ ih : Quotient.mk isEquivalent α ^ ↑n ≤ Quotient.mk isEquivalent α ⊢ Quotient.mk isEquivalent α ^ ↑(Nat.succ n) ≤ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ** rw [Nat.cast_succ, power_add, power_one] ** κ μ : Cardinal.{u} H1✝ : ℵ₀ ≤ κ H2 : μ < ℵ₀ n✝ : ℕ H3 : μ = ↑n✝ α : Type u H1 : ℵ₀ ≤ Quotient.mk isEquivalent α n : ℕ ih : Quotient.mk isEquivalent α ^ ↑n ≤ Quotient.mk isEquivalent α ⊢ Quotient.mk isEquivalent α ^ ↑n * Quotient.mk isEquivalent α ≤ Quotient.mk isEquivalent α * Quotient.mk isEquivalent α ** exact mul_le_mul_right' ih _ ** Qed | |
Cardinal.power_self_eq ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ^ c = 2 ^ c ** apply ((power_le_power_right <| (cantor c).le).trans _).antisymm ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ 2 ^ c ≤ c ^ c ** exact power_le_power_right ((nat_lt_aleph0 2).le.trans h) ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ (2 ^ c) ^ c ≤ 2 ^ c ** rw [← power_mul, mul_eq_self h] ** Qed | |
Cardinal.prod_eq_two_power ** ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ prod c = 2 ^ lift.{v, u} #ι ** rw [← lift_id'.{u, v} (prod.{u, v} c), lift_prod, ← lift_two_power] ** ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ (prod fun i => lift.{u, v} (c i)) = lift.{v, u} (2 ^ #ι) ** apply le_antisymm ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ (prod fun i => lift.{u, v} (c i)) ≤ lift.{v, u} (2 ^ #ι) ** refine' (prod_le_prod _ _ h₂).trans_eq _ ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ (prod fun i => lift.{v, u} #ι) = lift.{v, u} (2 ^ #ι) ** rw [prod_const, lift_lift, ← lift_power, power_self_eq (aleph0_le_mk ι), lift_umax.{u, v}] ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ lift.{v, u} (2 ^ #ι) ≤ prod fun i => lift.{u, v} (c i) ** rw [← prod_const', lift_prod] ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι ⊢ (prod fun i => lift.{v, u} 2) ≤ prod fun i => lift.{u, v} (c i) ** refine' prod_le_prod _ _ fun i => _ ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι i : ι ⊢ lift.{v, u} 2 ≤ lift.{u, v} (c i) ** rw [lift_two, ← lift_two.{u, v}, lift_le] ** case a ι : Type u inst✝ : Infinite ι c : ι → Cardinal.{v} h₁ : ∀ (i : ι), 2 ≤ c i h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι i : ι ⊢ 2 ≤ c i ** exact h₁ i ** Qed | |
Cardinal.nat_power_eq ** c : Cardinal.{u} h : ℵ₀ ≤ c n : ℕ hn : 2 ≤ n ⊢ 2 ≤ ↑n ** assumption_mod_cast ** Qed | |
Cardinal.power_nat_eq ** c : Cardinal.{u} n : ℕ h1 : ℵ₀ ≤ c h2 : 1 ≤ n ⊢ 1 ≤ ↑n ** exact_mod_cast h2 ** Qed | |
Cardinal.power_nat_le_max ** c : Cardinal.{u} n : ℕ ⊢ c ^ ↑n ≤ max c ℵ₀ ** cases' le_or_lt ℵ₀ c with hc hc ** case inl c : Cardinal.{u} n : ℕ hc : ℵ₀ ≤ c ⊢ c ^ ↑n ≤ max c ℵ₀ ** exact le_max_of_le_left (power_nat_le hc) ** case inr c : Cardinal.{u} n : ℕ hc : c < ℵ₀ ⊢ c ^ ↑n ≤ max c ℵ₀ ** exact le_max_of_le_right (power_lt_aleph0 hc (nat_lt_aleph0 _)).le ** Qed | |
Cardinal.powerlt_aleph0 ** c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ^< ℵ₀ = c ** apply le_antisymm ** case a c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ≤ c ^< ℵ₀ ** convert le_powerlt c one_lt_aleph0 ** case h.e'_3 c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c = c ^ 1 ** rw [power_one] ** case a c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ^< ℵ₀ ≤ c ** rw [powerlt_le] ** case a c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ ∀ (x : Cardinal.{u_1}), x < ℵ₀ → c ^ x ≤ c ** intro c' ** case a c : Cardinal.{u_1} h : ℵ₀ ≤ c c' : Cardinal.{u_1} ⊢ c' < ℵ₀ → c ^ c' ≤ c ** rw [lt_aleph0] ** case a c : Cardinal.{u_1} h : ℵ₀ ≤ c c' : Cardinal.{u_1} ⊢ (∃ n, c' = ↑n) → c ^ c' ≤ c ** rintro ⟨n, rfl⟩ ** case a.intro c : Cardinal.{u_1} h : ℵ₀ ≤ c n : ℕ ⊢ c ^ ↑n ≤ c ** apply power_nat_le h ** Qed | |
Cardinal.powerlt_aleph0_le ** c : Cardinal.{u_1} ⊢ c ^< ℵ₀ ≤ max c ℵ₀ ** cases' le_or_lt ℵ₀ c with h h ** case inr c : Cardinal.{u_1} h : c < ℵ₀ ⊢ c ^< ℵ₀ ≤ max c ℵ₀ ** rw [powerlt_le] ** case inr c : Cardinal.{u_1} h : c < ℵ₀ ⊢ ∀ (x : Cardinal.{u_1}), x < ℵ₀ → c ^ x ≤ max c ℵ₀ ** exact fun c' hc' => (power_lt_aleph0 h hc').le.trans (le_max_right _ _) ** case inl c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ^< ℵ₀ ≤ max c ℵ₀ ** rw [powerlt_aleph0 h] ** case inl c : Cardinal.{u_1} h : ℵ₀ ≤ c ⊢ c ≤ max c ℵ₀ ** apply le_max_left ** Qed | |
Cardinal.mk_list_eq_mk ** α : Type u inst✝ : Infinite α H1 : ℵ₀ ≤ #α x✝ : α ⊢ ∀ ⦃a₂ : α⦄, (fun a => [a]) x✝ = (fun a => [a]) a₂ → x✝ = a₂ ** simp ** α : Type u inst✝ : Infinite α H1 : ℵ₀ ≤ #α ⊢ (sum fun x => #α) = #α ** simp [H1] ** Qed | |
Cardinal.mk_list_eq_max_mk_aleph0 ** α : Type u inst✝ : Nonempty α ⊢ #(List α) = max #α ℵ₀ ** cases finite_or_infinite α ** case inl α : Type u inst✝ : Nonempty α h✝ : Finite α ⊢ #(List α) = max #α ℵ₀ ** rw [mk_list_eq_aleph0, eq_comm, max_eq_right] ** case inl α : Type u inst✝ : Nonempty α h✝ : Finite α ⊢ #α ≤ ℵ₀ ** exact mk_le_aleph0 ** case inr α : Type u inst✝ : Nonempty α h✝ : Infinite α ⊢ #(List α) = max #α ℵ₀ ** rw [mk_list_eq_mk, eq_comm, max_eq_left] ** case inr α : Type u inst✝ : Nonempty α h✝ : Infinite α ⊢ ℵ₀ ≤ #α ** exact aleph0_le_mk α ** Qed |
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