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leandojo-lean4-formal-informal-strings-split

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formal stringlengths 41 427k	informal stringclasses 1 value
Cardinal.ord_univ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop ⊢ ord univ = Ordinal.univ refine' le_antisymm (ord_card_le _) <\| le_of_forall_lt fun o h => lt_ord.2 ?_ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ ⊢ card o < univ have := lift.principalSeg.{u, v}.down.1 (by simpa only [lift.principalSeg_coe] using h) α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ this : ∃ a, ↑lift.principalSeg.toRelEmbedding a = o ⊢ card o < univ rcases this with ⟨o, h'⟩ case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o✝ : Ordinal.{max (u + 1) v} h : o✝ < Ordinal.univ o : Ordinal.{u} h' : ↑lift.principalSeg.toRelEmbedding o = o✝ ⊢ card o✝ < univ rw [←h', lift.principalSeg_coe, ← lift_card] case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o✝ : Ordinal.{max (u + 1) v} h : o✝ < Ordinal.univ o : Ordinal.{u} h' : ↑lift.principalSeg.toRelEmbedding o = o✝ ⊢ lift.{max (u + 1) v, u} (card o) < univ apply lift_lt_univ' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ ⊢ ?m.226369 < lift.principalSeg.top simpa only [lift.principalSeg_coe] using h ** Qed
Cardinal.lt_univ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ ⊢ ∃ c', c = lift.{u + 1, u} c' have := ord_lt_ord.2 h α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < ord univ ⊢ ∃ c', c = lift.{u + 1, u} c' rw [ord_univ] at this α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ ⊢ ∃ c', c = lift.{u + 1, u} c' cases' lift.principalSeg.{u, u + 1}.down.1 (by simpa only [lift.principalSeg_top] ) with o e case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c ⊢ ∃ c', c = lift.{u + 1, u} c' have := card_ord c case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this✝ : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c this : card (ord c) = c ⊢ ∃ c', c = lift.{u + 1, u} c' rw [← e, lift.principalSeg_coe, ← lift_card] at this case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this✝ : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c this : lift.{u + 1, u} (card o) = c ⊢ ∃ c', c = lift.{u + 1, u} c' exact ⟨_, this.symm⟩ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ ⊢ ?m.226837 < lift.principalSeg.top simpa only [lift.principalSeg_top] ** Qed
Cardinal.lt_univ' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' let ⟨a, e, h'⟩ := lt_lift_iff.1 h α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ a : Cardinal.{u + 1} e : lift.{v, u + 1} a = c h' : a < #Ordinal.{u} ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' rw [← univ_id] at h' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ a : Cardinal.{u + 1} e : lift.{v, u + 1} a = c h' : a < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩ case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ c' : Cardinal.{u} e : lift.{v, u + 1} (lift.{u + 1, u} c') = c h' : lift.{u + 1, u} c' < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' exact ⟨c', by simp only [e.symm, lift_lift]⟩ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ c' : Cardinal.{u} e : lift.{v, u + 1} (lift.{u + 1, u} c') = c h' : lift.{u + 1, u} c' < univ ⊢ c = lift.{max (u + 1) v, u} c' simp only [e.symm, lift_lift] ** Qed
Cardinal.small_iff_lift_mk_lt_univ α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α ↔ lift.{v + 1, u} #α < univ rw [lt_univ'] α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α ↔ ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' constructor case mp α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α → ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' rintro ⟨β, e⟩ case mp.mk.intro α✝ : Type u β✝ : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β✝ → β✝ → Prop t : γ → γ → Prop α : Type u β : Type v e : Nonempty (α ≃ β) ⊢ ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩ case mpr α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ (∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c') → Small.{v, u} α rintro ⟨c, hc⟩ case mpr.intro α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u c : Cardinal.{v} hc : lift.{v + 1, u} #α = lift.{max u (v + 1), v} c ⊢ Small.{v, u} α exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩ ** Qed
Ordinal.nat_le_card α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑n ≤ card o ↔ ↑n ≤ o rw [← Cardinal.ord_le, Cardinal.ord_nat] ** Qed
Ordinal.nat_lt_card α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑n < card o ↔ ↑n < o rw [← succ_le_iff, ← succ_le_iff, ← nat_succ, nat_le_card] α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑(Nat.succ n) ≤ o ↔ succ ↑n ≤ o rfl ** Qed
Ordinal.card_eq_nat α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ card o = ↑n ↔ o = ↑n simp only [le_antisymm_iff, card_le_nat, nat_le_card] ** Qed
Ordinal.type_fintype α : Type u β : Type u_1 γ : Type u_2 r✝ : α → α → Prop s : β → β → Prop t : γ → γ → Prop r : α → α → Prop inst✝¹ : IsWellOrder α r inst✝ : Fintype α ⊢ type r = ↑(Fintype.card α) rw [← card_eq_nat, card_type, mk_fintype] ** Qed
Ordinal.type_fin α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop n : ℕ ⊢ (type fun x x_1 => x < x_1) = ↑n simp ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheaf F ↔ IsSheafPairwiseIntersections F rw [isSheaf_iff_isSheafOpensLeCover, isSheafOpensLeCover_iff_isSheafPairwiseIntersections] ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheaf F ↔ IsSheafPreservesLimitPairwiseIntersections F rw [isSheaf_iff_isSheafPairwiseIntersections] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPairwiseIntersections F ↔ IsSheafPreservesLimitPairwiseIntersections F constructor case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPairwiseIntersections F → IsSheafPreservesLimitPairwiseIntersections F intro h ι U case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPairwiseIntersections F ι : Type w U : ι → Opens ↑X ⊢ Nonempty (PreservesLimit (Pairwise.diagram U).op F) exact ⟨preservesLimitOfPreservesLimitCone (Pairwise.coconeIsColimit U).op (h U).some⟩ case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPreservesLimitPairwiseIntersections F → IsSheafPairwiseIntersections F intro h ι U case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPreservesLimitPairwiseIntersections F ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U)))) haveI := (h U).some case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPreservesLimitPairwiseIntersections F ι : Type w U : ι → Opens ↑X this : PreservesLimit (Pairwise.diagram U).op F ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U)))) exact ⟨PreservesLimit.preserves (Pairwise.coconeIsColimit U).op⟩ ** Qed
TopCat.Sheaf.interUnionPullbackConeLift_left C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ interUnionPullbackConeLift F U V s ≫ F.val.map (homOfLE (_ : U ≤ U ⊔ V)).op = PullbackCone.fst s erw [Category.assoc] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Opposite.casesOn X_1 fun unop => Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s)) fun a a_1 => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ V)).op) } ≫ F.val.map (eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ F.val.map (homOfLE (_ : U ≤ U ⊔ V)).op = PullbackCone.fst s simp_rw [← F.1.map_comp] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down) (fun a a_1 => WalkingPair.rec (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ WalkingPair.rec U V a_1.down ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ WalkingPair.rec U V a_1.down ≤ V)).op) a.down) X_1.unop } ≫ F.val.map ((eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ (homOfLE (_ : U ≤ U ⊔ V)).op) = PullbackCone.fst s exact (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <\| op <\| Pairwise.single <\| ULift.up WalkingPair.left ** Qed
TopCat.Sheaf.interUnionPullbackConeLift_right C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ interUnionPullbackConeLift F U V s ≫ F.val.map (homOfLE (_ : V ≤ U ⊔ V)).op = PullbackCone.snd s erw [Category.assoc] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Opposite.casesOn X_1 fun unop => Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s)) fun a a_1 => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ V)).op) } ≫ F.val.map (eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ F.val.map (homOfLE (_ : V ≤ U ⊔ V)).op = PullbackCone.snd s simp_rw [← F.1.map_comp] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down) (fun a a_1 => WalkingPair.rec (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ WalkingPair.rec U V a_1.down ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ WalkingPair.rec U V a_1.down ≤ V)).op) a.down) X_1.unop } ≫ F.val.map ((eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ (homOfLE (_ : V ≤ U ⊔ V)).op) = PullbackCone.snd s exact (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <\| op <\| Pairwise.single <\| ULift.up WalkingPair.right ** Qed
Cardinal.out_embedding α β : Type u c c' : Cardinal.{u_1} ⊢ c ≤ c' ↔ Nonempty (Quotient.out c ↪ Quotient.out c') trans α β : Type u c c' : Cardinal.{u_1} ⊢ c ≤ c' ↔ ?m.14684 rw [← Quotient.out_eq c, ← Quotient.out_eq c'] α β : Type u c c' : Cardinal.{u_1} ⊢ Quotient.mk isEquivalent (Quotient.out c) ≤ Quotient.mk isEquivalent (Quotient.out c') ↔ Nonempty (Quotient.out c ↪ Quotient.out c') rw [mk'_def, mk'_def, le_def] ** Qed
Cardinal.lift_le α✝ β✝ : Type u a b : Cardinal.{v} α β : Type v ⊢ lift.{u, v} #α ≤ lift.{u, v} #β ↔ #α ≤ #β rw [← lift_umax] α✝ β✝ : Type u a b : Cardinal.{v} α β : Type v ⊢ lift.{max v u, v} #α ≤ lift.{max v u, v} #β ↔ #α ≤ #β exact lift_mk_le.{u} ** Qed
Cardinal.mk_option α✝ β α : Type u ⊢ #(Option α) = #α + 1 rw [(Equiv.optionEquivSumPUnit.{u, u} α).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id] ** Qed
Cardinal.mk_fintype α✝ β α : Type u h : Fintype α ⊢ Fintype.card α = Fintype.card (ULift.{u, 0} (Fin (Fintype.card α))) simp ** Qed
Cardinal.cast_succ α β : Type u n : ℕ ⊢ ↑(n + 1) = ↑n + 1 change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1 α β : Type u n : ℕ ⊢ #(ULift.{u, 0} (Fin (n + 1))) = #(ULift.{u, 0} (Fin n)) + 1 rw [← mk_option, mk_fintype, mk_fintype] α β : Type u n : ℕ ⊢ ↑(Fintype.card (ULift.{u, 0} (Fin (n + 1)))) = ↑(Fintype.card (Option (ULift.{u, 0} (Fin n)))) simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option] ** Qed
Cardinal.power_bit1 ** α β : Type u a b : Cardinal.{u_1} ⊢ a ^ bit1 b = a ^ b * a ^ b * a rw [bit1, ← power_bit0, power_add, power_one] Qed
Cardinal.mk_bool α β : Type u ⊢ #Bool = 2 simp ** Qed
Cardinal.mk_Prop α β : Type u ⊢ #Prop = 2 simp ** Qed
Cardinal.power_mul ** α β : Type u a b c : Cardinal.{u_1} ⊢ a ^ (b * c) = (a ^ b) ^ c rw [mul_comm b c] α β : Type u a b c : Cardinal.{u_1} ⊢ a ^ (c * b) = (a ^ b) ^ c exact inductionOn₃ a b c fun α β γ => mk_congr <\| Equiv.curry γ β α Qed
Cardinal.lift_bit1 α β : Type u a : Cardinal.{u_1} ⊢ lift.{v, u_1} (bit1 a) = bit1 (lift.{v, u_1} a) simp [bit1] ** Qed
Cardinal.lift_two α β : Type u ⊢ lift.{u, v} 2 = 2 simp [←one_add_one_eq_two] ** Qed
Cardinal.mk_set α✝ β α : Type u ⊢ #(Set α) = 2 ^ #α simp [←one_add_one_eq_two, Set, mk_arrow] ** Qed
Cardinal.lift_two_power α β : Type u a : Cardinal.{u_1} ⊢ lift.{v, u_1} (2 ^ a) = 2 ^ lift.{v, u_1} a simp [←one_add_one_eq_two] ** Qed
Cardinal.zero_le α β : Type u ⊢ ∀ (a : Cardinal.{u_1}), 0 ≤ a rintro ⟨α⟩ case mk α✝ β : Type u a✝ : Cardinal.{u_1} α : Type u_1 ⊢ 0 ≤ Quot.mk Setoid.r α exact ⟨Embedding.ofIsEmpty⟩ ** Qed
Cardinal.add_le_add' α β : Type u ⊢ ∀ {a b c d : Cardinal.{u_1}}, a ≤ b → c ≤ d → a + c ≤ b + d rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩ case mk.mk.mk.mk.intro.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 d✝ : Cardinal.{u_1} δ : Type u_1 e₁ : α ↪ β e₂ : γ ↪ δ ⊢ Quot.mk Setoid.r α + Quot.mk Setoid.r γ ≤ Quot.mk Setoid.r β + Quot.mk Setoid.r δ exact ⟨e₁.sumMap e₂⟩ ** Qed
Cardinal.zero_power_le α β : Type u c : Cardinal.{u} ⊢ 0 ^ c ≤ 1 by_cases h : c = 0 case pos α β : Type u c : Cardinal.{u} h : c = 0 ⊢ 0 ^ c ≤ 1 rw [h, power_zero] case neg α β : Type u c : Cardinal.{u} h : ¬c = 0 ⊢ 0 ^ c ≤ 1 rw [zero_power h] case neg α β : Type u c : Cardinal.{u} h : ¬c = 0 ⊢ 0 ≤ 1 apply zero_le ** Qed
Cardinal.power_le_power_left α β : Type u ⊢ ∀ {a b c : Cardinal.{u_1}}, a ≠ 0 → b ≤ c → a ^ b ≤ a ^ c rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ hα ⟨e⟩ case mk.mk.mk.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 hα : Quot.mk Setoid.r α ≠ 0 e : β ↪ γ ⊢ Quot.mk Setoid.r α ^ Quot.mk Setoid.r β ≤ Quot.mk Setoid.r α ^ Quot.mk Setoid.r γ let ⟨a⟩ := mk_ne_zero_iff.1 hα case mk.mk.mk.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 hα : Quot.mk Setoid.r α ≠ 0 e : β ↪ γ a : α ⊢ Quot.mk Setoid.r α ^ Quot.mk Setoid.r β ≤ Quot.mk Setoid.r α ^ Quot.mk Setoid.r γ exact ⟨@Function.Embedding.arrowCongrLeft _ _ _ ⟨a⟩ e⟩ ** Qed
Cardinal.self_le_power α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ⊢ a ≤ a ^ b rcases eq_or_ne a 0 with (rfl \| ha) case inl α β : Type u b : Cardinal.{u_1} hb : 1 ≤ b ⊢ 0 ≤ 0 ^ b exact zero_le _ case inr α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ha : a ≠ 0 ⊢ a ≤ a ^ b convert power_le_power_left ha hb case h.e'_3 α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ha : a ≠ 0 ⊢ a = a ^ 1 exact power_one.symm ** Qed
Cardinal.cantor α β : Type u a : Cardinal.{u} ⊢ a < 2 ^ a induction' a using Cardinal.inductionOn with α case h α✝ β α : Type u ⊢ #α < 2 ^ #α rw [← mk_set] case h α✝ β α : Type u ⊢ #α < #(Set α) refine' ⟨⟨⟨singleton, fun a b => singleton_eq_singleton_iff.1⟩⟩, _⟩ case h α✝ β α : Type u ⊢ ¬Quotient.liftOn₂ (#(Set α)) (#α) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 rintro ⟨⟨f, hf⟩⟩ case h.intro.mk α✝ β α : Type u f : Set α → α hf : Injective f ⊢ False exact cantor_injective f hf ** Qed
Cardinal.one_lt_iff_nontrivial α✝ β α : Type u ⊢ 1 < #α ↔ Nontrivial α rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not] ** Qed
Cardinal.power_le_max_power_one α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ⊢ a ^ b ≤ max (a ^ c) 1 by_cases ha : a = 0 case pos α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ha : a = 0 ⊢ a ^ b ≤ max (a ^ c) 1 simp [ha, zero_power_le] case neg α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ha : ¬a = 0 ⊢ a ^ b ≤ max (a ^ c) 1 exact (power_le_power_left ha h).trans (le_max_left _ _) ** Qed
Cardinal.lt_wf α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (?u.61137 + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } ⊢ False let f : ι → Cardinal := Subtype.val α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (?u.61137 + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{?u.61137} := Subtype.val ⊢ False haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩ case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) ⊢ False refine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_) case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) j : Cardinal.{u} h' : j < c hj : ¬Acc (fun x x_1 => x < x_1) j ⊢ Quotient.liftOn₂ c j (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 have : #_ ≤ #_ := ⟨h_1 ⟨j, hj⟩⟩ case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) j : Cardinal.{u} h' : j < c hj : ¬Acc (fun x x_1 => x < x_1) j this : #(Quotient.out (f { val := c, property := hc })) ≤ #(Quotient.out (f { val := j, property := hj })) ⊢ Quotient.liftOn₂ c j (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 simpa only [mk_out] using this ** Qed
Cardinal.add_one_le_succ α β : Type u c : Cardinal.{u} ⊢ c + 1 ≤ succ c have : Set.Nonempty { c' \| c < c' } := exists_gt c α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} ⊢ c + 1 ≤ succ c simp_rw [succ_def, le_csInf_iff'' this, mem_setOf] α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} ⊢ ∀ (b : Cardinal.{u}), c < b → c + 1 ≤ b intro b hlt α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} b : Cardinal.{u} hlt : c < b ⊢ c + 1 ≤ b rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩ case mk.mk α β✝ : Type u c b : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β cases' le_of_lt hlt with f case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β have : ¬Surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn) case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this✝ : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β this : ¬Surjective ↑f ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β simp only [Surjective, not_forall] at this case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this✝ : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β this : ∃ x, ¬∃ a, ↑f a = x ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β rcases this with ⟨b, hb⟩ case mk.mk.intro.intro α β✝ : Type u c b✝ : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β b : β hb : ¬∃ a, ↑f a = b ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β calc #γ + 1 = #(Option γ) := mk_option.symm _ ≤ #β := (f.optionElim b hb).cardinal_le ** Qed
Cardinal.le_sum α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι ⊢ f i ≤ sum f rw [← Quotient.out_eq (f i)] α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι ⊢ Quotient.mk isEquivalent (Quotient.out (f i)) ≤ sum f exact ⟨⟨fun a => ⟨i, a⟩, fun a b h => by injection h⟩⟩ α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι a b : Quotient.out (f i) h : (fun a => { fst := i, snd := a }) a = (fun a => { fst := i, snd := a }) b ⊢ a = b injection h ** Qed
Cardinal.sum_const' ** α β ι : Type u a : Cardinal.{u} ⊢ (sum fun x => a) = #ι * a simp Qed
Cardinal.sum_add_distrib α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} ⊢ sum (f + g) = sum f + sum g have := mk_congr (Equiv.sigmaSumDistrib (Quotient.out ∘ f) (Quotient.out ∘ g)) α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} this : #((i : ι) × ((Quotient.out ∘ f) i ⊕ (Quotient.out ∘ g) i)) = #((i : ι) × (Quotient.out ∘ f) i ⊕ (i : ι) × (Quotient.out ∘ g) i) ⊢ sum (f + g) = sum f + sum g simp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} this : (sum fun i => f i + g i) = (sum fun i => f i) + sum fun i => g i ⊢ sum (f + g) = sum f + sum g exact this ** Qed
Cardinal.lift_sum α β ι : Type u f : ι → Cardinal.{v} a : ι ⊢ Nonempty (Quotient.out (f a) ≃ Quotient.out ((fun i => lift.{w, v} (f i)) a)) rw [← lift_mk_eq.{_,_,v}, mk_out, mk_out, lift_lift] ** Qed
Cardinal.sum_le_sum α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g (↑(Embedding.refl ι) i))) have := H i α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι this : f i ≤ g i ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g (↑(Embedding.refl ι) i))) rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this ** Qed
Cardinal.mk_le_mk_mul_of_mk_preimage_le ** α β : Type u c : Cardinal.{u} f : α → β hf : ∀ (b : β), #↑(f ⁻¹' {b}) ≤ c ⊢ #α ≤ #β * c simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using sum_le_sum _ _ hf Qed
Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β ⊢ ↑(↑Equiv.ulift '' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) rw [Equiv.image_eq_preimage] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β ⊢ ↑(↑Equiv.ulift.symm ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) have : FunLike.coe (Equiv.symm (Equiv.ulift (α := α))) = ULift.up (α := α) := rfl α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ ↑(↑Equiv.ulift.symm ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) rw [this] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ ↑(ULift.up ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ { x // f x = b } ≃ { x // f x = b } exact Equiv.refl _ ** Qed
Cardinal.bddAbove_range α β ι : Type u f : ι → Cardinal.{max u v} ⊢ ?m.81700 f ∈ upperBounds (range f) rintro a ⟨i, rfl⟩ case intro α β ι : Type u f : ι → Cardinal.{max u v} i : ι ⊢ f i ≤ ?m.81700 f exact le_sum.{v,u} f i ** Qed
Cardinal.bddAbove_iff_small α β : Type u s : Set Cardinal.{u} ⊢ Small.{u, u + 1} ↑s → BddAbove s rintro ⟨ι, ⟨e⟩⟩ case mk.intro.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι ⊢ BddAbove s suffices (range fun x : ι => (e.symm x).1) = s by rw [← this] apply bddAbove_range.{u, u} case mk.intro.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι ⊢ (range fun x => ↑(↑e.symm x)) = s ext x case mk.intro.intro.h α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} ⊢ (x ∈ range fun x => ↑(↑e.symm x)) ↔ x ∈ s refine' ⟨_, fun hx => ⟨e ⟨x, hx⟩, _⟩⟩ α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι this : (range fun x => ↑(↑e.symm x)) = s ⊢ BddAbove s rw [← this] α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι this : (range fun x => ↑(↑e.symm x)) = s ⊢ BddAbove (range fun x => ↑(↑e.symm x)) apply bddAbove_range.{u, u} case mk.intro.intro.h.refine'_1 α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} ⊢ (x ∈ range fun x => ↑(↑e.symm x)) → x ∈ s rintro ⟨a, rfl⟩ case mk.intro.intro.h.refine'_1.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι a : ι ⊢ (fun x => ↑(↑e.symm x)) a ∈ s exact (e.symm a).2 case mk.intro.intro.h.refine'_2 α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} hx : x ∈ s ⊢ (fun x => ↑(↑e.symm x)) (↑e { val := x, property := hx }) = x simp_rw [Equiv.symm_apply_apply] ** Qed
Cardinal.bddAbove_image α β : Type u f : Cardinal.{u} → Cardinal.{max u v} s : Set Cardinal.{u} hs : BddAbove s ⊢ BddAbove (f '' s) rw [bddAbove_iff_small] at hs ⊢ α β : Type u f : Cardinal.{u} → Cardinal.{max u v} s : Set Cardinal.{u} hs : Small.{u, u + 1} ↑s ⊢ Small.{max u v, (max u v) + 1} ↑(f '' s) exact small_lift.{_,v,_} _ ** Qed
Cardinal.bddAbove_range_comp α β ι : Type u f : ι → Cardinal.{v} hf : BddAbove (range f) g : Cardinal.{v} → Cardinal.{max v w} ⊢ BddAbove (range (g ∘ f)) rw [range_comp] α β ι : Type u f : ι → Cardinal.{v} hf : BddAbove (range f) g : Cardinal.{v} → Cardinal.{max v w} ⊢ BddAbove (g '' range f) exact bddAbove_image.{v,w} g hf ** Qed
Cardinal.sum_le_iSup_lift ** α β ι : Type u f : ι → Cardinal.{max u v} ⊢ sum f ≤ lift.{v, u} #ι * iSup f rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const] α β ι : Type u f : ι → Cardinal.{max u v} ⊢ sum f ≤ sum fun x => iSup f exact sum_le_sum _ _ (le_ciSup <\| bddAbove_range.{u, v} f) Qed
Cardinal.sum_le_iSup ** α β ι : Type u f : ι → Cardinal.{u} ⊢ sum f ≤ #ι * iSup f rw [← lift_id #ι] α β ι : Type u f : ι → Cardinal.{u} ⊢ sum f ≤ lift.{u, u} #ι * iSup f exact sum_le_iSup_lift f Qed
Cardinal.sum_nat_eq_add_sum_succ α β : Type u f : ℕ → Cardinal.{u} ⊢ sum f = f 0 + sum fun i => f (i + 1) refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _ α β : Type u f : ℕ → Cardinal.{u} ⊢ #(Quotient.out (f 0) ⊕ (n : ℕ) × Quotient.out (f (n + 1))) = f 0 + sum fun i => f (i + 1) simp only [mk_sum, mk_out, lift_id, mk_sigma] ** Qed
Cardinal.lift_mk_shrink'' α✝ β : Type u α : Type (max u v) inst✝ : Small.{v, max u v} α ⊢ lift.{u, v} #(Shrink α) = #α rw [← lift_umax', lift_mk_shrink.{max u v, v, 0} α, ← lift_umax, lift_id] ** Qed
Cardinal.prod_le_prod α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g i)) have := H i α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι this : f i ≤ g i ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g i)) rwa [← mk_out (f i), ← mk_out (g i)] at this ** Qed
Cardinal.prod_eq_zero α β : Type u ι : Type u_1 f : ι → Cardinal.{u} ⊢ prod f = 0 ↔ ∃ i, f i = 0 lift f to ι → Type u using fun _ => trivial case intro α β : Type u ι : Type u_1 f : ι → Type u ⊢ (prod fun i => #(f i)) = 0 ↔ ∃ i, (fun i => #(f i)) i = 0 simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi] ** Qed
Cardinal.prod_ne_zero α β : Type u ι : Type u_1 f : ι → Cardinal.{u_2} ⊢ prod f ≠ 0 ↔ ∀ (i : ι), f i ≠ 0 simp [prod_eq_zero] ** Qed
Cardinal.lift_prod α β ι : Type u c : ι → Cardinal.{v} ⊢ lift.{w, max v u} (prod c) = prod fun i => lift.{w, v} (c i) lift c to ι → Type v using fun _ => trivial case intro α β ι : Type u c : ι → Type v ⊢ lift.{w, max v u} (prod fun i => #(c i)) = prod fun i => lift.{w, v} ((fun i => #(c i)) i) simp only [← mk_pi, ← mk_uLift] case intro α β ι : Type u c : ι → Type v ⊢ #(ULift.{w, max u v} ((i : ι) → c i)) = #((i : ι) → ULift.{w, v} (c i)) exact mk_congr (Equiv.ulift.trans <\| Equiv.piCongrRight fun i => Equiv.ulift.symm) ** Qed
Cardinal.prod_eq_of_fintype α✝ β α : Type u h : Fintype α f : α → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : α, f i) revert f α✝ β α : Type u h : Fintype α ⊢ ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) refine' Fintype.induction_empty_option _ _ _ α (h_fintype := h) case refine'_1 α✝ β α : Type u h : Fintype α ⊢ ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β), (∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i)) → ∀ (f : β → Cardinal.{v}), prod f = lift.{u, v} (∏ i : β, f i) intro α β hβ e h f case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : β, f i) letI := Fintype.ofEquiv β e.symm case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} this : Fintype α := Fintype.ofEquiv β e.symm ⊢ prod f = lift.{u, v} (∏ i : β, f i) rw [← e.prod_comp f, ← h] case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} this : Fintype α := Fintype.ofEquiv β e.symm ⊢ prod f = prod fun i => f (↑e i) exact mk_congr (e.piCongrLeft _).symm case refine'_2 α✝ β α : Type u h : Fintype α ⊢ ∀ (f : PEmpty.{u + 1} → Cardinal.{v}), prod f = lift.{u, v} (∏ i : PEmpty.{u + 1}, f i) intro f case refine'_2 α✝ β α : Type u h : Fintype α f : PEmpty.{u + 1} → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : PEmpty.{u + 1}, f i) rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one] case refine'_3 α✝ β α : Type u h : Fintype α ⊢ ∀ (α : Type u) [inst : Fintype α], (∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i)) → ∀ (f : Option α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : Option α, f i) intro α hα h f case refine'_3 α✝¹ β α✝ : Type u h✝ : Fintype α✝ α : Type u hα : Fintype α h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : Option α → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : Option α, f i) rw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, ← Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, ← h fun a => f (some a)] ** case refine'_3 α✝¹ β α✝ : Type u h✝ : Fintype α✝ α : Type u hα : Fintype α h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : Option α → Cardinal.{v} ⊢ (lift.{u, v} (f none) * prod fun i => lift.{v, v} (f (some i))) = lift.{u, v} (f none) * prod fun a => f (some a) simp only [lift_id] Qed
Cardinal.lift_sInf α β : Type u s : Set Cardinal.{v} ⊢ lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) rcases eq_empty_or_nonempty s with (rfl \| hs) case inl α β : Type u ⊢ lift.{u, v} (sInf ∅) = sInf (lift.{u, v} '' ∅) simp case inr α β : Type u s : Set Cardinal.{v} hs : Set.Nonempty s ⊢ lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) exact lift_monotone.map_csInf hs ** Qed
Cardinal.lift_iInf α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ lift.{u, v} (iInf f) = ⨅ i, lift.{u, v} (f i) unfold iInf α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ lift.{u, v} (sInf (range f)) = sInf (range fun i => lift.{u, v} (f i)) convert lift_sInf (range f) case h.e'_3.h.e'_3 α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ (range fun i => lift.{u, v} (f i)) = lift.{u, v} '' range f simp_rw [←comp_apply (f := lift), range_comp] ** Qed
Cardinal.lift_down α✝ β✝ : Type u a : Cardinal.{u} b : Cardinal.{max u v} α : Type u β : Type (max u v) ⊢ #β ≤ lift.{v, u} #α → ∃ a', lift.{v, u} a' = #β rw [← lift_id #β, ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}] α✝ β✝ : Type u a : Cardinal.{u} b : Cardinal.{max u v} α : Type u β : Type (max u v) ⊢ Nonempty (β ↪ α) → ∃ a', lift.{max u v, u} a' = lift.{max u v, max u v} #β exact fun ⟨f⟩ => ⟨#(Set.range f), Eq.symm <\| lift_mk_eq.{_, _, v}.2 ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self) fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩ ** Qed
Cardinal.lift_succ α β : Type u a : Cardinal.{u} h : lift.{v, u} (succ a) > succ (lift.{v, u} a) ⊢ False rcases lt_lift_iff.1 h with ⟨b, e, h⟩ case intro.intro α β : Type u a : Cardinal.{u} h✝ : lift.{v, u} (succ a) > succ (lift.{v, u} a) b : Cardinal.{u} e : lift.{v, u} b = succ (lift.{v, u} a) h : b < succ a ⊢ False rw [lt_succ_iff, ← lift_le, e] at h case intro.intro α β : Type u a : Cardinal.{u} h✝ : lift.{v, u} (succ a) > succ (lift.{v, u} a) b : Cardinal.{u} e : lift.{v, u} b = succ (lift.{v, u} a) h : succ (lift.{v, u} a) ≤ lift.{v, u} a ⊢ False exact h.not_lt (lt_succ _) ** Qed
Cardinal.lift_umax_eq α β : Type u a : Cardinal.{u} b : Cardinal.{v} ⊢ lift.{max v w, u} a = lift.{max u w, v} b ↔ lift.{v, u} a = lift.{u, v} b rw [← lift_lift.{v, w, u}, ← lift_lift.{u, w, v}, lift_inj] ** Qed
Cardinal.lift_sSup α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ lift.{u, u_1} (sSup s) = sSup (lift.{u, u_1} '' s) apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _) α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ ∀ (c : Cardinal.{max u_1 u}), c ∈ upperBounds (lift.{u, u_1} '' s) → lift.{u, u_1} (sSup s) ≤ c intro c hc α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s c : Cardinal.{max u_1 u} hc : c ∈ upperBounds (lift.{u, u_1} '' s) ⊢ lift.{u, u_1} (sSup s) ≤ c by_contra h α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s c : Cardinal.{max u_1 u} hc : c ∈ upperBounds (lift.{u, u_1} '' s) h : ¬lift.{u, u_1} (sSup s) ≤ c ⊢ False obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬lift.{u, u_1} (sSup s) ≤ lift.{u, u_1} d ⊢ False simp_rw [lift_le] at h hc case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬sSup s ≤ d ⊢ False rw [csSup_le_iff' hs] at h case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬∀ (x : Cardinal.{u_1}), x ∈ s → x ≤ d ⊢ False exact h fun a ha => lift_le.1 <\| hc (mem_image_of_mem _ ha) α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ lift.{u, u_1} (sSup s) ∈ upperBounds (lift.{u, u_1} '' s) rintro i ⟨j, hj, rfl⟩ case intro.intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s j : Cardinal.{u_1} hj : j ∈ s ⊢ lift.{u, u_1} j ≤ lift.{u, u_1} (sSup s) exact lift_le.2 (le_csSup hs hj) ** Qed
Cardinal.lift_iSup α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) ⊢ lift.{u, w} (iSup f) = ⨆ i, lift.{u, w} (f i) rw [iSup, iSup, lift_sSup hf, ← range_comp] α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) ⊢ sSup (range (lift.{u, w} ∘ f)) = sSup (range fun i => lift.{u, w} (f i)) simp [Function.comp] ** Qed
Cardinal.lift_iSup_le α β : Type u ι : Type v f : ι → Cardinal.{w} t : Cardinal.{max u w} hf : BddAbove (range f) w : ∀ (i : ι), lift.{u, w} (f i) ≤ t ⊢ lift.{u, w} (iSup f) ≤ t rw [lift_iSup hf] α β : Type u ι : Type v f : ι → Cardinal.{w} t : Cardinal.{max u w} hf : BddAbove (range f) w : ∀ (i : ι), lift.{u, w} (f i) ≤ t ⊢ ⨆ i, lift.{u, w} (f i) ≤ t exact ciSup_le' w ** Qed
Cardinal.lift_iSup_le_iff α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) t : Cardinal.{max u w} ⊢ lift.{u, w} (iSup f) ≤ t ↔ ∀ (i : ι), lift.{u, w} (f i) ≤ t rw [lift_iSup hf] α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) t : Cardinal.{max u w} ⊢ ⨆ i, lift.{u, w} (f i) ≤ t ↔ ∀ (i : ι), lift.{u, w} (f i) ≤ t exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _) ** Qed
Cardinal.lift_iSup_le_lift_iSup α β : Type u ι : Type v ι' : Type v' f : ι → Cardinal.{w} f' : ι' → Cardinal.{w'} hf : BddAbove (range f) hf' : BddAbove (range f') g : ι → ι' h : ∀ (i : ι), lift.{w', w} (f i) ≤ lift.{w, w'} (f' (g i)) ⊢ lift.{w', w} (iSup f) ≤ lift.{w, w'} (iSup f') rw [lift_iSup hf, lift_iSup hf'] α β : Type u ι : Type v ι' : Type v' f : ι → Cardinal.{w} f' : ι' → Cardinal.{w'} hf : BddAbove (range f) hf' : BddAbove (range f') g : ι → ι' h : ∀ (i : ι), lift.{w', w} (f i) ≤ lift.{w, w'} (f' (g i)) ⊢ ⨆ i, lift.{w', w} (f i) ≤ ⨆ i, lift.{w, w'} (f' i) exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩ ** Qed
Cardinal.aleph0_le_lift α β : Type u c : Cardinal.{u} ⊢ ℵ₀ ≤ lift.{v, u} c ↔ ℵ₀ ≤ c rw [← lift_aleph0.{u,v}, lift_le] ** Qed
Cardinal.lift_le_aleph0 α β : Type u c : Cardinal.{u} ⊢ lift.{v, u} c ≤ ℵ₀ ↔ c ≤ ℵ₀ rw [← lift_aleph0.{u,v}, lift_le] ** Qed
Cardinal.aleph0_lt_lift α β : Type u c : Cardinal.{u} ⊢ ℵ₀ < lift.{v, u} c ↔ ℵ₀ < c rw [← lift_aleph0.{u,v}, lift_lt] ** Qed
Cardinal.lift_lt_aleph0 α β : Type u c : Cardinal.{u} ⊢ lift.{v, u} c < ℵ₀ ↔ c < ℵ₀ rw [← lift_aleph0.{u,v}, lift_lt] ** Qed
Cardinal.mk_fin α β : Type u n : ℕ ⊢ #(Fin n) = ↑n simp ** Qed
Cardinal.lift_natCast α β : Type u n : ℕ ⊢ lift.{u, v} ↑n = ↑n induction n <;> simp [] * Qed
Cardinal.nat_eq_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n = lift.{v, u} a ↔ ↑n = a rw [← lift_natCast.{v,u} n, lift_inj] ** Qed
Cardinal.lift_le_nat_iff α β : Type u a : Cardinal.{u} n : ℕ ⊢ lift.{v, u} a ≤ ↑n ↔ a ≤ ↑n rw [← lift_natCast.{v,u}, lift_le] ** Qed
Cardinal.nat_le_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n ≤ lift.{v, u} a ↔ ↑n ≤ a rw [← lift_natCast.{v,u}, lift_le] ** Qed
Cardinal.lift_lt_nat_iff α β : Type u a : Cardinal.{u} n : ℕ ⊢ lift.{v, u} a < ↑n ↔ a < ↑n rw [← lift_natCast.{v,u}, lift_lt] ** Qed
Cardinal.nat_lt_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n < lift.{v, u} a ↔ ↑n < a rw [← lift_natCast.{v,u}, lift_lt] ** Qed
Cardinal.mk_coe_finset α✝ β α : Type u s : Finset α ⊢ #{ x // x ∈ s } = ↑(Finset.card s) simp ** Qed
Cardinal.mk_finset_of_fintype α β : Type u inst✝ : Fintype α ⊢ #(Finset α) = 2 ^ Fintype.card α simp [Pow.pow] ** Qed
Cardinal.mk_finsupp_lift_of_fintype α✝ β✝ α : Type u β : Type v inst✝¹ : Fintype α inst✝ : Zero β ⊢ #(α →₀ β) = lift.{u, v} #β ^ Fintype.card α simpa using (@Finsupp.equivFunOnFinite α β _ _).cardinal_eq ** Qed
Cardinal.mk_finsupp_of_fintype α✝ β✝ α β : Type u inst✝¹ : Fintype α inst✝ : Zero β ⊢ #(α →₀ β) = #β ^ Fintype.card α simp ** Qed
Cardinal.natCast_pow α β : Type u m n : ℕ ⊢ ↑(m ^ n) = ↑m ^ ↑n induction n <;> simp [pow_succ', power_add, , Pow.pow] * Qed
Cardinal.natCast_le α β : Type u m n : ℕ ⊢ ↑m ≤ ↑n ↔ m ≤ n rw [← lift_mk_fin, ← lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le, Fintype.card_fin, Fintype.card_fin] ** Qed
Cardinal.natCast_lt α β : Type u m n : ℕ ⊢ ↑m < ↑n ↔ m < n rw [lt_iff_le_not_le, ← not_le] α β : Type u m n : ℕ ⊢ ↑m ≤ ↑n ∧ ¬↑n ≤ ↑m ↔ ¬n ≤ m simp only [natCast_le, not_le, and_iff_right_iff_imp] α β : Type u m n : ℕ ⊢ m < n → m ≤ n exact fun h ↦ le_of_lt h ** Qed
Cardinal.nat_succ α β : Type u n : ℕ ⊢ ↑(Nat.succ n) = succ ↑n rw [Nat.cast_succ] α β : Type u n : ℕ ⊢ ↑n + 1 = succ ↑n refine (add_one_le_succ _).antisymm (succ_le_of_lt ?_) α β : Type u n : ℕ ⊢ ↑n < ↑n + 1 rw [← Nat.cast_succ] α β : Type u n : ℕ ⊢ ↑n < ↑(Nat.succ n) exact natCast_lt.2 (Nat.lt_succ_self _) ** Qed
Cardinal.succ_zero α β : Type u ⊢ succ 0 = 1 norm_cast ** Qed
Cardinal.card_le_of α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n ⊢ #α ≤ ↑n refine' le_of_lt_succ (lt_of_not_ge fun hn => _) α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n hn : #α ≥ succ ↑n ⊢ False rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n hn : #α ≥ lift.{u, 0} #(Fin (Nat.succ n)) ⊢ False cases' hn with f case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ False refine' (H <\| Finset.univ.map f).not_lt _ case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ n < Finset.card (Finset.map f Finset.univ) rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin] case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ n < Nat.succ n exact n.lt_succ_self ** Qed
Cardinal.cantor' α β : Type u a b : Cardinal.{u_1} hb : 1 < b ⊢ a < b ^ a rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb α β : Type u a b : Cardinal.{u_1} hb✝ : 1 < b hb : 2 ≤ b ⊢ a < b ^ a exact (cantor a).trans_le (power_le_power_right hb) α β : Type u a b : Cardinal.{u_1} hb✝ : 1 < b hb : succ 1 ≤ b ⊢ succ 1 = 2 norm_cast ** Qed
Cardinal.one_le_iff_pos α β : Type u c : Cardinal.{u_1} ⊢ 1 ≤ c ↔ 0 < c rw [← succ_zero, succ_le_iff] ** Qed
Cardinal.one_le_iff_ne_zero α β : Type u c : Cardinal.{u_1} ⊢ 1 ≤ c ↔ c ≠ 0 rw [one_le_iff_pos, pos_iff_ne_zero] ** Qed
Cardinal.nat_lt_aleph0 α β : Type u n : ℕ ⊢ succ ↑n ≤ ℵ₀ rw [← nat_succ, ← lift_mk_fin, aleph0, lift_mk_le.{u}] α β : Type u n : ℕ ⊢ Nonempty (Fin (Nat.succ n) ↪ ℕ) exact ⟨⟨(↑), fun a b => Fin.ext⟩⟩ ** Qed
Cardinal.one_lt_aleph0 α β : Type u ⊢ 1 < ℵ₀ simpa using nat_lt_aleph0 1 ** Qed
Cardinal.lt_aleph0 α β : Type u c : Cardinal.{u_1} h : c < ℵ₀ ⊢ ∃ n, c = ↑n rcases lt_lift_iff.1 h with ⟨c, rfl, h'⟩ case intro.intro α β : Type u c : Cardinal.{0} h' : c < #ℕ h : lift.{u_1, 0} c < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} c = ↑n rcases le_mk_iff_exists_set.1 h'.1 with ⟨S, rfl⟩ case intro.intro.intro α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} #↑S = ↑n suffices S.Finite by lift S to Finset ℕ using this simp case intro.intro.intro α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ ⊢ Set.Finite S contrapose! h' case intro.intro.intro α β : Type u S : Set ℕ h : lift.{u_1, 0} #↑S < ℵ₀ h' : ¬Set.Finite S ⊢ #ℕ ≤ #↑S haveI := Infinite.to_subtype h' case intro.intro.intro α β : Type u S : Set ℕ h : lift.{u_1, 0} #↑S < ℵ₀ h' : ¬Set.Finite S this : Infinite ↑S ⊢ #ℕ ≤ #↑S exact ⟨Infinite.natEmbedding S⟩ α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ this : Set.Finite S ⊢ ∃ n, lift.{u_1, 0} #↑S = ↑n lift S to Finset ℕ using this case intro α β : Type u S : Finset ℕ h' : #↑↑S < #ℕ h : lift.{u_1, 0} #↑↑S < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} #↑↑S = ↑n simp ** Qed
Cardinal.aleph0_le α β : Type u c : Cardinal.{u_1} h : ∀ (n : ℕ), ↑n ≤ c hn : c < ℵ₀ ⊢ False rcases lt_aleph0.1 hn with ⟨n, rfl⟩ case intro α β : Type u n : ℕ h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n hn : ↑n < ℵ₀ ⊢ False exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1))) ** Qed
Cardinal.isSuccLimit_aleph0 α β : Type u a : Cardinal.{u_1} ha : a < ℵ₀ ⊢ succ a < ℵ₀ rcases lt_aleph0.1 ha with ⟨n, rfl⟩ case intro α β : Type u n : ℕ ha : ↑n < ℵ₀ ⊢ succ ↑n < ℵ₀ rw [← nat_succ] case intro α β : Type u n : ℕ ha : ↑n < ℵ₀ ⊢ ↑(Nat.succ n) < ℵ₀ apply nat_lt_aleph0 ** Qed
Cardinal.IsLimit.aleph0_le α β : Type u c : Cardinal.{u_1} h : IsLimit c ⊢ ℵ₀ ≤ c by_contra' h' α β : Type u c : Cardinal.{u_1} h : IsLimit c h' : c < ℵ₀ ⊢ False rcases lt_aleph0.1 h' with ⟨_ \| n, rfl⟩ case intro.zero α β : Type u h : IsLimit ↑Nat.zero h' : ↑Nat.zero < ℵ₀ ⊢ False exact h.ne_zero.irrefl case intro.succ α β : Type u n : ℕ h : IsLimit ↑(Nat.succ n) h' : ↑(Nat.succ n) < ℵ₀ ⊢ False rw [nat_succ] at h case intro.succ α β : Type u n : ℕ h : IsLimit (succ ↑n) h' : ↑(Nat.succ n) < ℵ₀ ⊢ False exact not_isSuccLimit_succ _ h.isSuccLimit ** Qed
Cardinal.range_natCast α β : Type u x : Cardinal.{u_1} ⊢ x ∈ range Nat.cast ↔ x ∈ Iio ℵ₀ simp only [mem_Iio, mem_range, eq_comm, lt_aleph0] ** Qed
Cardinal.mk_eq_nat_iff α✝ β α : Type u n : ℕ ⊢ #α = ↑n ↔ Nonempty (α ≃ Fin n) rw [← lift_mk_fin, ← lift_uzero #α, lift_mk_eq'] ** Qed
Cardinal.lt_aleph0_iff_finite α✝ β α : Type u ⊢ #α < ℵ₀ ↔ Finite α simp only [lt_aleph0, mk_eq_nat_iff, finite_iff_exists_equiv_fin] ** Qed
Cardinal.mk_le_aleph0_iff α β : Type u ⊢ #α ≤ ℵ₀ ↔ Countable α rw [countable_iff_nonempty_embedding, aleph0, ← lift_uzero #α, lift_mk_le'] ** Qed

formal stringlengths 41 427k	informal stringclasses 1 value
Cardinal.ord_univ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop ⊢ ord univ = Ordinal.univ refine' le_antisymm (ord_card_le _) <\| le_of_forall_lt fun o h => lt_ord.2 ?_ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ ⊢ card o < univ have := lift.principalSeg.{u, v}.down.1 (by simpa only [lift.principalSeg_coe] using h) α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ this : ∃ a, ↑lift.principalSeg.toRelEmbedding a = o ⊢ card o < univ rcases this with ⟨o, h'⟩ case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o✝ : Ordinal.{max (u + 1) v} h : o✝ < Ordinal.univ o : Ordinal.{u} h' : ↑lift.principalSeg.toRelEmbedding o = o✝ ⊢ card o✝ < univ rw [←h', lift.principalSeg_coe, ← lift_card] case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o✝ : Ordinal.{max (u + 1) v} h : o✝ < Ordinal.univ o : Ordinal.{u} h' : ↑lift.principalSeg.toRelEmbedding o = o✝ ⊢ lift.{max (u + 1) v, u} (card o) < univ apply lift_lt_univ' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{max (u + 1) v} h : o < Ordinal.univ ⊢ ?m.226369 < lift.principalSeg.top simpa only [lift.principalSeg_coe] using h ** Qed
Cardinal.lt_univ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ ⊢ ∃ c', c = lift.{u + 1, u} c' have := ord_lt_ord.2 h α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < ord univ ⊢ ∃ c', c = lift.{u + 1, u} c' rw [ord_univ] at this α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ ⊢ ∃ c', c = lift.{u + 1, u} c' cases' lift.principalSeg.{u, u + 1}.down.1 (by simpa only [lift.principalSeg_top] ) with o e case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c ⊢ ∃ c', c = lift.{u + 1, u} c' have := card_ord c case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this✝ : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c this : card (ord c) = c ⊢ ∃ c', c = lift.{u + 1, u} c' rw [← e, lift.principalSeg_coe, ← lift_card] at this case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this✝ : ord c < Ordinal.univ o : Ordinal.{u} e : ↑lift.principalSeg.toRelEmbedding o = ord c this : lift.{u + 1, u} (card o) = c ⊢ ∃ c', c = lift.{u + 1, u} c' exact ⟨_, this.symm⟩ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{u + 1} h : c < univ this : ord c < Ordinal.univ ⊢ ?m.226837 < lift.principalSeg.top simpa only [lift.principalSeg_top] ** Qed
Cardinal.lt_univ' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' let ⟨a, e, h'⟩ := lt_lift_iff.1 h α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ a : Cardinal.{u + 1} e : lift.{v, u + 1} a = c h' : a < #Ordinal.{u} ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' rw [← univ_id] at h' α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ a : Cardinal.{u + 1} e : lift.{v, u + 1} a = c h' : a < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩ case intro α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ c' : Cardinal.{u} e : lift.{v, u + 1} (lift.{u + 1, u} c') = c h' : lift.{u + 1, u} c' < univ ⊢ ∃ c', c = lift.{max (u + 1) v, u} c' exact ⟨c', by simp only [e.symm, lift_lift]⟩ α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop c : Cardinal.{max (u + 1) v} h : c < univ c' : Cardinal.{u} e : lift.{v, u + 1} (lift.{u + 1, u} c') = c h' : lift.{u + 1, u} c' < univ ⊢ c = lift.{max (u + 1) v, u} c' simp only [e.symm, lift_lift] ** Qed
Cardinal.small_iff_lift_mk_lt_univ α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α ↔ lift.{v + 1, u} #α < univ rw [lt_univ'] α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α ↔ ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' constructor case mp α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ Small.{v, u} α → ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' rintro ⟨β, e⟩ case mp.mk.intro α✝ : Type u β✝ : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β✝ → β✝ → Prop t : γ → γ → Prop α : Type u β : Type v e : Nonempty (α ≃ β) ⊢ ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c' exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩ case mpr α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u ⊢ (∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c') → Small.{v, u} α rintro ⟨c, hc⟩ case mpr.intro α✝ : Type u β : Type u_1 γ : Type u_2 r : α✝ → α✝ → Prop s : β → β → Prop t : γ → γ → Prop α : Type u c : Cardinal.{v} hc : lift.{v + 1, u} #α = lift.{max u (v + 1), v} c ⊢ Small.{v, u} α exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩ ** Qed
Ordinal.nat_le_card α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑n ≤ card o ↔ ↑n ≤ o rw [← Cardinal.ord_le, Cardinal.ord_nat] ** Qed
Ordinal.nat_lt_card α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑n < card o ↔ ↑n < o rw [← succ_le_iff, ← succ_le_iff, ← nat_succ, nat_le_card] α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ ↑(Nat.succ n) ≤ o ↔ succ ↑n ≤ o rfl ** Qed
Ordinal.card_eq_nat α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop o : Ordinal.{u_3} n : ℕ ⊢ card o = ↑n ↔ o = ↑n simp only [le_antisymm_iff, card_le_nat, nat_le_card] ** Qed
Ordinal.type_fintype α : Type u β : Type u_1 γ : Type u_2 r✝ : α → α → Prop s : β → β → Prop t : γ → γ → Prop r : α → α → Prop inst✝¹ : IsWellOrder α r inst✝ : Fintype α ⊢ type r = ↑(Fintype.card α) rw [← card_eq_nat, card_type, mk_fintype] ** Qed
Ordinal.type_fin α : Type u β : Type u_1 γ : Type u_2 r : α → α → Prop s : β → β → Prop t : γ → γ → Prop n : ℕ ⊢ (type fun x x_1 => x < x_1) = ↑n simp ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheaf F ↔ IsSheafPairwiseIntersections F rw [isSheaf_iff_isSheafOpensLeCover, isSheafOpensLeCover_iff_isSheafPairwiseIntersections] ** Qed
TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheaf F ↔ IsSheafPreservesLimitPairwiseIntersections F rw [isSheaf_iff_isSheafPairwiseIntersections] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPairwiseIntersections F ↔ IsSheafPreservesLimitPairwiseIntersections F constructor case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPairwiseIntersections F → IsSheafPreservesLimitPairwiseIntersections F intro h ι U case mp C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPairwiseIntersections F ι : Type w U : ι → Opens ↑X ⊢ Nonempty (PreservesLimit (Pairwise.diagram U).op F) exact ⟨preservesLimitOfPreservesLimitCone (Pairwise.coconeIsColimit U).op (h U).some⟩ case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X ⊢ IsSheafPreservesLimitPairwiseIntersections F → IsSheafPairwiseIntersections F intro h ι U case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPreservesLimitPairwiseIntersections F ι : Type w U : ι → Opens ↑X ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U)))) haveI := (h U).some case mpr C : Type u inst✝ : Category.{v, u} C X : TopCat F : Presheaf C X h : IsSheafPreservesLimitPairwiseIntersections F ι : Type w U : ι → Opens ↑X this : PreservesLimit (Pairwise.diagram U).op F ⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U)))) exact ⟨PreservesLimit.preserves (Pairwise.coconeIsColimit U).op⟩ ** Qed
TopCat.Sheaf.interUnionPullbackConeLift_left C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ interUnionPullbackConeLift F U V s ≫ F.val.map (homOfLE (_ : U ≤ U ⊔ V)).op = PullbackCone.fst s erw [Category.assoc] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Opposite.casesOn X_1 fun unop => Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s)) fun a a_1 => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ V)).op) } ≫ F.val.map (eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ F.val.map (homOfLE (_ : U ≤ U ⊔ V)).op = PullbackCone.fst s simp_rw [← F.1.map_comp] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down) (fun a a_1 => WalkingPair.rec (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ WalkingPair.rec U V a_1.down ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ WalkingPair.rec U V a_1.down ≤ V)).op) a.down) X_1.unop } ≫ F.val.map ((eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ (homOfLE (_ : U ≤ U ⊔ V)).op) = PullbackCone.fst s exact (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <\| op <\| Pairwise.single <\| ULift.up WalkingPair.left ** Qed
TopCat.Sheaf.interUnionPullbackConeLift_right C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ interUnionPullbackConeLift F U V s ≫ F.val.map (homOfLE (_ : V ≤ U ⊔ V)).op = PullbackCone.snd s erw [Category.assoc] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Opposite.casesOn X_1 fun unop => Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s)) fun a a_1 => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ (fun j => WalkingPair.casesOn j.down U V) a_1 ≤ V)).op) } ≫ F.val.map (eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ F.val.map (homOfLE (_ : V ≤ U ⊔ V)).op = PullbackCone.snd s simp_rw [← F.1.map_comp] C : Type u inst✝ : Category.{v, u} C X : TopCat F : Sheaf C X U V : Opens ↑X s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op) ⊢ IsLimit.lift (Nonempty.some (_ : Nonempty (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V)))))) { pt := s.pt, π := NatTrans.mk fun X_1 => Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down) (fun a a_1 => WalkingPair.rec (PullbackCone.fst s ≫ F.val.map (homOfLE (_ : U ⊓ WalkingPair.rec U V a_1.down ≤ U)).op) (PullbackCone.snd s ≫ F.val.map (homOfLE (_ : V ⊓ WalkingPair.rec U V a_1.down ≤ V)).op) a.down) X_1.unop } ≫ F.val.map ((eqToHom (_ : U ⊔ V = ⨆ j, WalkingPair.casesOn j.down U V)).op ≫ (homOfLE (_ : V ≤ U ⊔ V)).op) = PullbackCone.snd s exact (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <\| op <\| Pairwise.single <\| ULift.up WalkingPair.right ** Qed
Cardinal.out_embedding α β : Type u c c' : Cardinal.{u_1} ⊢ c ≤ c' ↔ Nonempty (Quotient.out c ↪ Quotient.out c') trans α β : Type u c c' : Cardinal.{u_1} ⊢ c ≤ c' ↔ ?m.14684 rw [← Quotient.out_eq c, ← Quotient.out_eq c'] α β : Type u c c' : Cardinal.{u_1} ⊢ Quotient.mk isEquivalent (Quotient.out c) ≤ Quotient.mk isEquivalent (Quotient.out c') ↔ Nonempty (Quotient.out c ↪ Quotient.out c') rw [mk'_def, mk'_def, le_def] ** Qed
Cardinal.lift_le α✝ β✝ : Type u a b : Cardinal.{v} α β : Type v ⊢ lift.{u, v} #α ≤ lift.{u, v} #β ↔ #α ≤ #β rw [← lift_umax] α✝ β✝ : Type u a b : Cardinal.{v} α β : Type v ⊢ lift.{max v u, v} #α ≤ lift.{max v u, v} #β ↔ #α ≤ #β exact lift_mk_le.{u} ** Qed
Cardinal.mk_option α✝ β α : Type u ⊢ #(Option α) = #α + 1 rw [(Equiv.optionEquivSumPUnit.{u, u} α).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id] ** Qed
Cardinal.mk_fintype α✝ β α : Type u h : Fintype α ⊢ Fintype.card α = Fintype.card (ULift.{u, 0} (Fin (Fintype.card α))) simp ** Qed
Cardinal.cast_succ α β : Type u n : ℕ ⊢ ↑(n + 1) = ↑n + 1 change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1 α β : Type u n : ℕ ⊢ #(ULift.{u, 0} (Fin (n + 1))) = #(ULift.{u, 0} (Fin n)) + 1 rw [← mk_option, mk_fintype, mk_fintype] α β : Type u n : ℕ ⊢ ↑(Fintype.card (ULift.{u, 0} (Fin (n + 1)))) = ↑(Fintype.card (Option (ULift.{u, 0} (Fin n)))) simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option] ** Qed
Cardinal.power_bit1 ** α β : Type u a b : Cardinal.{u_1} ⊢ a ^ bit1 b = a ^ b * a ^ b * a rw [bit1, ← power_bit0, power_add, power_one] Qed
Cardinal.mk_bool α β : Type u ⊢ #Bool = 2 simp ** Qed
Cardinal.mk_Prop α β : Type u ⊢ #Prop = 2 simp ** Qed
Cardinal.power_mul ** α β : Type u a b c : Cardinal.{u_1} ⊢ a ^ (b * c) = (a ^ b) ^ c rw [mul_comm b c] α β : Type u a b c : Cardinal.{u_1} ⊢ a ^ (c * b) = (a ^ b) ^ c exact inductionOn₃ a b c fun α β γ => mk_congr <\| Equiv.curry γ β α Qed
Cardinal.lift_bit1 α β : Type u a : Cardinal.{u_1} ⊢ lift.{v, u_1} (bit1 a) = bit1 (lift.{v, u_1} a) simp [bit1] ** Qed
Cardinal.lift_two α β : Type u ⊢ lift.{u, v} 2 = 2 simp [←one_add_one_eq_two] ** Qed
Cardinal.mk_set α✝ β α : Type u ⊢ #(Set α) = 2 ^ #α simp [←one_add_one_eq_two, Set, mk_arrow] ** Qed
Cardinal.lift_two_power α β : Type u a : Cardinal.{u_1} ⊢ lift.{v, u_1} (2 ^ a) = 2 ^ lift.{v, u_1} a simp [←one_add_one_eq_two] ** Qed
Cardinal.zero_le α β : Type u ⊢ ∀ (a : Cardinal.{u_1}), 0 ≤ a rintro ⟨α⟩ case mk α✝ β : Type u a✝ : Cardinal.{u_1} α : Type u_1 ⊢ 0 ≤ Quot.mk Setoid.r α exact ⟨Embedding.ofIsEmpty⟩ ** Qed
Cardinal.add_le_add' α β : Type u ⊢ ∀ {a b c d : Cardinal.{u_1}}, a ≤ b → c ≤ d → a + c ≤ b + d rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩ case mk.mk.mk.mk.intro.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 d✝ : Cardinal.{u_1} δ : Type u_1 e₁ : α ↪ β e₂ : γ ↪ δ ⊢ Quot.mk Setoid.r α + Quot.mk Setoid.r γ ≤ Quot.mk Setoid.r β + Quot.mk Setoid.r δ exact ⟨e₁.sumMap e₂⟩ ** Qed
Cardinal.zero_power_le α β : Type u c : Cardinal.{u} ⊢ 0 ^ c ≤ 1 by_cases h : c = 0 case pos α β : Type u c : Cardinal.{u} h : c = 0 ⊢ 0 ^ c ≤ 1 rw [h, power_zero] case neg α β : Type u c : Cardinal.{u} h : ¬c = 0 ⊢ 0 ^ c ≤ 1 rw [zero_power h] case neg α β : Type u c : Cardinal.{u} h : ¬c = 0 ⊢ 0 ≤ 1 apply zero_le ** Qed
Cardinal.power_le_power_left α β : Type u ⊢ ∀ {a b c : Cardinal.{u_1}}, a ≠ 0 → b ≤ c → a ^ b ≤ a ^ c rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ hα ⟨e⟩ case mk.mk.mk.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 hα : Quot.mk Setoid.r α ≠ 0 e : β ↪ γ ⊢ Quot.mk Setoid.r α ^ Quot.mk Setoid.r β ≤ Quot.mk Setoid.r α ^ Quot.mk Setoid.r γ let ⟨a⟩ := mk_ne_zero_iff.1 hα case mk.mk.mk.intro α✝ β✝ : Type u a✝ : Cardinal.{u_1} α : Type u_1 b✝ : Cardinal.{u_1} β : Type u_1 c✝ : Cardinal.{u_1} γ : Type u_1 hα : Quot.mk Setoid.r α ≠ 0 e : β ↪ γ a : α ⊢ Quot.mk Setoid.r α ^ Quot.mk Setoid.r β ≤ Quot.mk Setoid.r α ^ Quot.mk Setoid.r γ exact ⟨@Function.Embedding.arrowCongrLeft _ _ _ ⟨a⟩ e⟩ ** Qed
Cardinal.self_le_power α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ⊢ a ≤ a ^ b rcases eq_or_ne a 0 with (rfl \| ha) case inl α β : Type u b : Cardinal.{u_1} hb : 1 ≤ b ⊢ 0 ≤ 0 ^ b exact zero_le _ case inr α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ha : a ≠ 0 ⊢ a ≤ a ^ b convert power_le_power_left ha hb case h.e'_3 α β : Type u a b : Cardinal.{u_1} hb : 1 ≤ b ha : a ≠ 0 ⊢ a = a ^ 1 exact power_one.symm ** Qed
Cardinal.cantor α β : Type u a : Cardinal.{u} ⊢ a < 2 ^ a induction' a using Cardinal.inductionOn with α case h α✝ β α : Type u ⊢ #α < 2 ^ #α rw [← mk_set] case h α✝ β α : Type u ⊢ #α < #(Set α) refine' ⟨⟨⟨singleton, fun a b => singleton_eq_singleton_iff.1⟩⟩, _⟩ case h α✝ β α : Type u ⊢ ¬Quotient.liftOn₂ (#(Set α)) (#α) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 rintro ⟨⟨f, hf⟩⟩ case h.intro.mk α✝ β α : Type u f : Set α → α hf : Injective f ⊢ False exact cantor_injective f hf ** Qed
Cardinal.one_lt_iff_nontrivial α✝ β α : Type u ⊢ 1 < #α ↔ Nontrivial α rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not] ** Qed
Cardinal.power_le_max_power_one α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ⊢ a ^ b ≤ max (a ^ c) 1 by_cases ha : a = 0 case pos α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ha : a = 0 ⊢ a ^ b ≤ max (a ^ c) 1 simp [ha, zero_power_le] case neg α β : Type u a b c : Cardinal.{u_1} h : b ≤ c ha : ¬a = 0 ⊢ a ^ b ≤ max (a ^ c) 1 exact (power_le_power_left ha h).trans (le_max_left _ _) ** Qed
Cardinal.lt_wf α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (?u.61137 + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } ⊢ False let f : ι → Cardinal := Subtype.val α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (?u.61137 + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{?u.61137} := Subtype.val ⊢ False haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩ case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) ⊢ False refine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_) case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) j : Cardinal.{u} h' : j < c hj : ¬Acc (fun x x_1 => x < x_1) j ⊢ Quotient.liftOn₂ c j (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 have : #_ ≤ #_ := ⟨h_1 ⟨j, hj⟩⟩ case intro.mk.intro α β : Type u a : Cardinal.{u} h : ¬Acc (fun x x_1 => x < x_1) a ι : Type (u + 1) := { c // ¬Acc (fun x x_1 => x < x_1) c } f : ι → Cardinal.{u} := Subtype.val hι : Nonempty ι c : Cardinal.{u} hc : ¬Acc (fun x x_1 => x < x_1) c h_1 : (j : ι) → Quotient.out (f { val := c, property := hc }) ↪ Quotient.out (f j) j : Cardinal.{u} h' : j < c hj : ¬Acc (fun x x_1 => x < x_1) j this : #(Quotient.out (f { val := c, property := hc })) ≤ #(Quotient.out (f { val := j, property := hj })) ⊢ Quotient.liftOn₂ c j (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 simpa only [mk_out] using this ** Qed
Cardinal.add_one_le_succ α β : Type u c : Cardinal.{u} ⊢ c + 1 ≤ succ c have : Set.Nonempty { c' \| c < c' } := exists_gt c α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} ⊢ c + 1 ≤ succ c simp_rw [succ_def, le_csInf_iff'' this, mem_setOf] α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} ⊢ ∀ (b : Cardinal.{u}), c < b → c + 1 ≤ b intro b hlt α β : Type u c : Cardinal.{u} this : Set.Nonempty {c' \| c < c'} b : Cardinal.{u} hlt : c < b ⊢ c + 1 ≤ b rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩ case mk.mk α β✝ : Type u c b : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β cases' le_of_lt hlt with f case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β have : ¬Surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn) case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this✝ : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β this : ¬Surjective ↑f ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β simp only [Surjective, not_forall] at this case mk.mk.intro α β✝ : Type u c b : Cardinal.{u} β γ : Type u this✝ : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β this : ∃ x, ¬∃ a, ↑f a = x ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β rcases this with ⟨b, hb⟩ case mk.mk.intro.intro α β✝ : Type u c b✝ : Cardinal.{u} β γ : Type u this : Set.Nonempty {c' \| Quot.mk Setoid.r γ < c'} hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β f : γ ↪ β b : β hb : ¬∃ a, ↑f a = b ⊢ Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β calc #γ + 1 = #(Option γ) := mk_option.symm _ ≤ #β := (f.optionElim b hb).cardinal_le ** Qed
Cardinal.le_sum α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι ⊢ f i ≤ sum f rw [← Quotient.out_eq (f i)] α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι ⊢ Quotient.mk isEquivalent (Quotient.out (f i)) ≤ sum f exact ⟨⟨fun a => ⟨i, a⟩, fun a b h => by injection h⟩⟩ α β : Type u ι : Type u_1 f : ι → Cardinal.{max u_1 u_2} i : ι a b : Quotient.out (f i) h : (fun a => { fst := i, snd := a }) a = (fun a => { fst := i, snd := a }) b ⊢ a = b injection h ** Qed
Cardinal.sum_const' ** α β ι : Type u a : Cardinal.{u} ⊢ (sum fun x => a) = #ι * a simp Qed
Cardinal.sum_add_distrib α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} ⊢ sum (f + g) = sum f + sum g have := mk_congr (Equiv.sigmaSumDistrib (Quotient.out ∘ f) (Quotient.out ∘ g)) α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} this : #((i : ι) × ((Quotient.out ∘ f) i ⊕ (Quotient.out ∘ g) i)) = #((i : ι) × (Quotient.out ∘ f) i ⊕ (i : ι) × (Quotient.out ∘ g) i) ⊢ sum (f + g) = sum f + sum g simp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} this : (sum fun i => f i + g i) = (sum fun i => f i) + sum fun i => g i ⊢ sum (f + g) = sum f + sum g exact this ** Qed
Cardinal.lift_sum α β ι : Type u f : ι → Cardinal.{v} a : ι ⊢ Nonempty (Quotient.out (f a) ≃ Quotient.out ((fun i => lift.{w, v} (f i)) a)) rw [← lift_mk_eq.{_,_,v}, mk_out, mk_out, lift_lift] ** Qed
Cardinal.sum_le_sum α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g (↑(Embedding.refl ι) i))) have := H i α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι this : f i ≤ g i ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g (↑(Embedding.refl ι) i))) rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this ** Qed
Cardinal.mk_le_mk_mul_of_mk_preimage_le ** α β : Type u c : Cardinal.{u} f : α → β hf : ∀ (b : β), #↑(f ⁻¹' {b}) ≤ c ⊢ #α ≤ #β * c simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using sum_le_sum _ _ hf Qed
Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β ⊢ ↑(↑Equiv.ulift '' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) rw [Equiv.image_eq_preimage] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β ⊢ ↑(↑Equiv.ulift.symm ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) have : FunLike.coe (Equiv.symm (Equiv.ulift (α := α))) = ULift.up (α := α) := rfl α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ ↑(↑Equiv.ulift.symm ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) rw [this] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ ↑(ULift.up ⁻¹' ((fun x => { down := f x.down }) ⁻¹' {{ down := b }})) ≃ ↑(f ⁻¹' {b}) simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf] α✝ β✝ α : Type u β : Type v c : Cardinal.{max u v} f : α → β hf : ∀ (b : β), lift.{v, u} #↑(f ⁻¹' {b}) ≤ c b : β this : ↑Equiv.ulift.symm = ULift.up ⊢ { x // f x = b } ≃ { x // f x = b } exact Equiv.refl _ ** Qed
Cardinal.bddAbove_range α β ι : Type u f : ι → Cardinal.{max u v} ⊢ ?m.81700 f ∈ upperBounds (range f) rintro a ⟨i, rfl⟩ case intro α β ι : Type u f : ι → Cardinal.{max u v} i : ι ⊢ f i ≤ ?m.81700 f exact le_sum.{v,u} f i ** Qed
Cardinal.bddAbove_iff_small α β : Type u s : Set Cardinal.{u} ⊢ Small.{u, u + 1} ↑s → BddAbove s rintro ⟨ι, ⟨e⟩⟩ case mk.intro.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι ⊢ BddAbove s suffices (range fun x : ι => (e.symm x).1) = s by rw [← this] apply bddAbove_range.{u, u} case mk.intro.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι ⊢ (range fun x => ↑(↑e.symm x)) = s ext x case mk.intro.intro.h α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} ⊢ (x ∈ range fun x => ↑(↑e.symm x)) ↔ x ∈ s refine' ⟨_, fun hx => ⟨e ⟨x, hx⟩, _⟩⟩ α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι this : (range fun x => ↑(↑e.symm x)) = s ⊢ BddAbove s rw [← this] α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι this : (range fun x => ↑(↑e.symm x)) = s ⊢ BddAbove (range fun x => ↑(↑e.symm x)) apply bddAbove_range.{u, u} case mk.intro.intro.h.refine'_1 α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} ⊢ (x ∈ range fun x => ↑(↑e.symm x)) → x ∈ s rintro ⟨a, rfl⟩ case mk.intro.intro.h.refine'_1.intro α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι a : ι ⊢ (fun x => ↑(↑e.symm x)) a ∈ s exact (e.symm a).2 case mk.intro.intro.h.refine'_2 α β : Type u s : Set Cardinal.{u} ι : Type u e : ↑s ≃ ι x : Cardinal.{u} hx : x ∈ s ⊢ (fun x => ↑(↑e.symm x)) (↑e { val := x, property := hx }) = x simp_rw [Equiv.symm_apply_apply] ** Qed
Cardinal.bddAbove_image α β : Type u f : Cardinal.{u} → Cardinal.{max u v} s : Set Cardinal.{u} hs : BddAbove s ⊢ BddAbove (f '' s) rw [bddAbove_iff_small] at hs ⊢ α β : Type u f : Cardinal.{u} → Cardinal.{max u v} s : Set Cardinal.{u} hs : Small.{u, u + 1} ↑s ⊢ Small.{max u v, (max u v) + 1} ↑(f '' s) exact small_lift.{_,v,_} _ ** Qed
Cardinal.bddAbove_range_comp α β ι : Type u f : ι → Cardinal.{v} hf : BddAbove (range f) g : Cardinal.{v} → Cardinal.{max v w} ⊢ BddAbove (range (g ∘ f)) rw [range_comp] α β ι : Type u f : ι → Cardinal.{v} hf : BddAbove (range f) g : Cardinal.{v} → Cardinal.{max v w} ⊢ BddAbove (g '' range f) exact bddAbove_image.{v,w} g hf ** Qed
Cardinal.sum_le_iSup_lift ** α β ι : Type u f : ι → Cardinal.{max u v} ⊢ sum f ≤ lift.{v, u} #ι * iSup f rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const] α β ι : Type u f : ι → Cardinal.{max u v} ⊢ sum f ≤ sum fun x => iSup f exact sum_le_sum _ _ (le_ciSup <\| bddAbove_range.{u, v} f) Qed
Cardinal.sum_le_iSup ** α β ι : Type u f : ι → Cardinal.{u} ⊢ sum f ≤ #ι * iSup f rw [← lift_id #ι] α β ι : Type u f : ι → Cardinal.{u} ⊢ sum f ≤ lift.{u, u} #ι * iSup f exact sum_le_iSup_lift f Qed
Cardinal.sum_nat_eq_add_sum_succ α β : Type u f : ℕ → Cardinal.{u} ⊢ sum f = f 0 + sum fun i => f (i + 1) refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _ α β : Type u f : ℕ → Cardinal.{u} ⊢ #(Quotient.out (f 0) ⊕ (n : ℕ) × Quotient.out (f (n + 1))) = f 0 + sum fun i => f (i + 1) simp only [mk_sum, mk_out, lift_id, mk_sigma] ** Qed
Cardinal.lift_mk_shrink'' α✝ β : Type u α : Type (max u v) inst✝ : Small.{v, max u v} α ⊢ lift.{u, v} #(Shrink α) = #α rw [← lift_umax', lift_mk_shrink.{max u v, v, 0} α, ← lift_umax, lift_id] ** Qed
Cardinal.prod_le_prod α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g i)) have := H i α β : Type u ι : Type u_1 f g : ι → Cardinal.{u_2} H : ∀ (i : ι), f i ≤ g i i : ι this : f i ≤ g i ⊢ Nonempty (Quotient.out (f i) ↪ Quotient.out (g i)) rwa [← mk_out (f i), ← mk_out (g i)] at this ** Qed
Cardinal.prod_eq_zero α β : Type u ι : Type u_1 f : ι → Cardinal.{u} ⊢ prod f = 0 ↔ ∃ i, f i = 0 lift f to ι → Type u using fun _ => trivial case intro α β : Type u ι : Type u_1 f : ι → Type u ⊢ (prod fun i => #(f i)) = 0 ↔ ∃ i, (fun i => #(f i)) i = 0 simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi] ** Qed
Cardinal.prod_ne_zero α β : Type u ι : Type u_1 f : ι → Cardinal.{u_2} ⊢ prod f ≠ 0 ↔ ∀ (i : ι), f i ≠ 0 simp [prod_eq_zero] ** Qed
Cardinal.lift_prod α β ι : Type u c : ι → Cardinal.{v} ⊢ lift.{w, max v u} (prod c) = prod fun i => lift.{w, v} (c i) lift c to ι → Type v using fun _ => trivial case intro α β ι : Type u c : ι → Type v ⊢ lift.{w, max v u} (prod fun i => #(c i)) = prod fun i => lift.{w, v} ((fun i => #(c i)) i) simp only [← mk_pi, ← mk_uLift] case intro α β ι : Type u c : ι → Type v ⊢ #(ULift.{w, max u v} ((i : ι) → c i)) = #((i : ι) → ULift.{w, v} (c i)) exact mk_congr (Equiv.ulift.trans <\| Equiv.piCongrRight fun i => Equiv.ulift.symm) ** Qed
Cardinal.prod_eq_of_fintype α✝ β α : Type u h : Fintype α f : α → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : α, f i) revert f α✝ β α : Type u h : Fintype α ⊢ ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) refine' Fintype.induction_empty_option _ _ _ α (h_fintype := h) case refine'_1 α✝ β α : Type u h : Fintype α ⊢ ∀ (α β : Type u) [inst : Fintype β] (e : α ≃ β), (∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i)) → ∀ (f : β → Cardinal.{v}), prod f = lift.{u, v} (∏ i : β, f i) intro α β hβ e h f case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : β, f i) letI := Fintype.ofEquiv β e.symm case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} this : Fintype α := Fintype.ofEquiv β e.symm ⊢ prod f = lift.{u, v} (∏ i : β, f i) rw [← e.prod_comp f, ← h] case refine'_1 α✝¹ β✝ α✝ : Type u h✝ : Fintype α✝ α β : Type u hβ : Fintype β e : α ≃ β h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : β → Cardinal.{v} this : Fintype α := Fintype.ofEquiv β e.symm ⊢ prod f = prod fun i => f (↑e i) exact mk_congr (e.piCongrLeft _).symm case refine'_2 α✝ β α : Type u h : Fintype α ⊢ ∀ (f : PEmpty.{u + 1} → Cardinal.{v}), prod f = lift.{u, v} (∏ i : PEmpty.{u + 1}, f i) intro f case refine'_2 α✝ β α : Type u h : Fintype α f : PEmpty.{u + 1} → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : PEmpty.{u + 1}, f i) rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one] case refine'_3 α✝ β α : Type u h : Fintype α ⊢ ∀ (α : Type u) [inst : Fintype α], (∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i)) → ∀ (f : Option α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : Option α, f i) intro α hα h f case refine'_3 α✝¹ β α✝ : Type u h✝ : Fintype α✝ α : Type u hα : Fintype α h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : Option α → Cardinal.{v} ⊢ prod f = lift.{u, v} (∏ i : Option α, f i) rw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, ← Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, ← h fun a => f (some a)] ** case refine'_3 α✝¹ β α✝ : Type u h✝ : Fintype α✝ α : Type u hα : Fintype α h : ∀ (f : α → Cardinal.{v}), prod f = lift.{u, v} (∏ i : α, f i) f : Option α → Cardinal.{v} ⊢ (lift.{u, v} (f none) * prod fun i => lift.{v, v} (f (some i))) = lift.{u, v} (f none) * prod fun a => f (some a) simp only [lift_id] Qed
Cardinal.lift_sInf α β : Type u s : Set Cardinal.{v} ⊢ lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) rcases eq_empty_or_nonempty s with (rfl \| hs) case inl α β : Type u ⊢ lift.{u, v} (sInf ∅) = sInf (lift.{u, v} '' ∅) simp case inr α β : Type u s : Set Cardinal.{v} hs : Set.Nonempty s ⊢ lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) exact lift_monotone.map_csInf hs ** Qed
Cardinal.lift_iInf α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ lift.{u, v} (iInf f) = ⨅ i, lift.{u, v} (f i) unfold iInf α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ lift.{u, v} (sInf (range f)) = sInf (range fun i => lift.{u, v} (f i)) convert lift_sInf (range f) case h.e'_3.h.e'_3 α β : Type u ι : Sort u_1 f : ι → Cardinal.{v} ⊢ (range fun i => lift.{u, v} (f i)) = lift.{u, v} '' range f simp_rw [←comp_apply (f := lift), range_comp] ** Qed
Cardinal.lift_down α✝ β✝ : Type u a : Cardinal.{u} b : Cardinal.{max u v} α : Type u β : Type (max u v) ⊢ #β ≤ lift.{v, u} #α → ∃ a', lift.{v, u} a' = #β rw [← lift_id #β, ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}] α✝ β✝ : Type u a : Cardinal.{u} b : Cardinal.{max u v} α : Type u β : Type (max u v) ⊢ Nonempty (β ↪ α) → ∃ a', lift.{max u v, u} a' = lift.{max u v, max u v} #β exact fun ⟨f⟩ => ⟨#(Set.range f), Eq.symm <\| lift_mk_eq.{_, _, v}.2 ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self) fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩ ** Qed
Cardinal.lift_succ α β : Type u a : Cardinal.{u} h : lift.{v, u} (succ a) > succ (lift.{v, u} a) ⊢ False rcases lt_lift_iff.1 h with ⟨b, e, h⟩ case intro.intro α β : Type u a : Cardinal.{u} h✝ : lift.{v, u} (succ a) > succ (lift.{v, u} a) b : Cardinal.{u} e : lift.{v, u} b = succ (lift.{v, u} a) h : b < succ a ⊢ False rw [lt_succ_iff, ← lift_le, e] at h case intro.intro α β : Type u a : Cardinal.{u} h✝ : lift.{v, u} (succ a) > succ (lift.{v, u} a) b : Cardinal.{u} e : lift.{v, u} b = succ (lift.{v, u} a) h : succ (lift.{v, u} a) ≤ lift.{v, u} a ⊢ False exact h.not_lt (lt_succ _) ** Qed
Cardinal.lift_umax_eq α β : Type u a : Cardinal.{u} b : Cardinal.{v} ⊢ lift.{max v w, u} a = lift.{max u w, v} b ↔ lift.{v, u} a = lift.{u, v} b rw [← lift_lift.{v, w, u}, ← lift_lift.{u, w, v}, lift_inj] ** Qed
Cardinal.lift_sSup α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ lift.{u, u_1} (sSup s) = sSup (lift.{u, u_1} '' s) apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _) α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ ∀ (c : Cardinal.{max u_1 u}), c ∈ upperBounds (lift.{u, u_1} '' s) → lift.{u, u_1} (sSup s) ≤ c intro c hc α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s c : Cardinal.{max u_1 u} hc : c ∈ upperBounds (lift.{u, u_1} '' s) ⊢ lift.{u, u_1} (sSup s) ≤ c by_contra h α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s c : Cardinal.{max u_1 u} hc : c ∈ upperBounds (lift.{u, u_1} '' s) h : ¬lift.{u, u_1} (sSup s) ≤ c ⊢ False obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬lift.{u, u_1} (sSup s) ≤ lift.{u, u_1} d ⊢ False simp_rw [lift_le] at h hc case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬sSup s ≤ d ⊢ False rw [csSup_le_iff' hs] at h case intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s d : Cardinal.{u_1} hc : lift.{u, u_1} d ∈ upperBounds (lift.{u, u_1} '' s) h : ¬∀ (x : Cardinal.{u_1}), x ∈ s → x ≤ d ⊢ False exact h fun a ha => lift_le.1 <\| hc (mem_image_of_mem _ ha) α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s ⊢ lift.{u, u_1} (sSup s) ∈ upperBounds (lift.{u, u_1} '' s) rintro i ⟨j, hj, rfl⟩ case intro.intro α β : Type u s : Set Cardinal.{u_1} hs : BddAbove s j : Cardinal.{u_1} hj : j ∈ s ⊢ lift.{u, u_1} j ≤ lift.{u, u_1} (sSup s) exact lift_le.2 (le_csSup hs hj) ** Qed
Cardinal.lift_iSup α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) ⊢ lift.{u, w} (iSup f) = ⨆ i, lift.{u, w} (f i) rw [iSup, iSup, lift_sSup hf, ← range_comp] α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) ⊢ sSup (range (lift.{u, w} ∘ f)) = sSup (range fun i => lift.{u, w} (f i)) simp [Function.comp] ** Qed
Cardinal.lift_iSup_le α β : Type u ι : Type v f : ι → Cardinal.{w} t : Cardinal.{max u w} hf : BddAbove (range f) w : ∀ (i : ι), lift.{u, w} (f i) ≤ t ⊢ lift.{u, w} (iSup f) ≤ t rw [lift_iSup hf] α β : Type u ι : Type v f : ι → Cardinal.{w} t : Cardinal.{max u w} hf : BddAbove (range f) w : ∀ (i : ι), lift.{u, w} (f i) ≤ t ⊢ ⨆ i, lift.{u, w} (f i) ≤ t exact ciSup_le' w ** Qed
Cardinal.lift_iSup_le_iff α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) t : Cardinal.{max u w} ⊢ lift.{u, w} (iSup f) ≤ t ↔ ∀ (i : ι), lift.{u, w} (f i) ≤ t rw [lift_iSup hf] α β : Type u ι : Type v f : ι → Cardinal.{w} hf : BddAbove (range f) t : Cardinal.{max u w} ⊢ ⨆ i, lift.{u, w} (f i) ≤ t ↔ ∀ (i : ι), lift.{u, w} (f i) ≤ t exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _) ** Qed
Cardinal.lift_iSup_le_lift_iSup α β : Type u ι : Type v ι' : Type v' f : ι → Cardinal.{w} f' : ι' → Cardinal.{w'} hf : BddAbove (range f) hf' : BddAbove (range f') g : ι → ι' h : ∀ (i : ι), lift.{w', w} (f i) ≤ lift.{w, w'} (f' (g i)) ⊢ lift.{w', w} (iSup f) ≤ lift.{w, w'} (iSup f') rw [lift_iSup hf, lift_iSup hf'] α β : Type u ι : Type v ι' : Type v' f : ι → Cardinal.{w} f' : ι' → Cardinal.{w'} hf : BddAbove (range f) hf' : BddAbove (range f') g : ι → ι' h : ∀ (i : ι), lift.{w', w} (f i) ≤ lift.{w, w'} (f' (g i)) ⊢ ⨆ i, lift.{w', w} (f i) ≤ ⨆ i, lift.{w, w'} (f' i) exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩ ** Qed
Cardinal.aleph0_le_lift α β : Type u c : Cardinal.{u} ⊢ ℵ₀ ≤ lift.{v, u} c ↔ ℵ₀ ≤ c rw [← lift_aleph0.{u,v}, lift_le] ** Qed
Cardinal.lift_le_aleph0 α β : Type u c : Cardinal.{u} ⊢ lift.{v, u} c ≤ ℵ₀ ↔ c ≤ ℵ₀ rw [← lift_aleph0.{u,v}, lift_le] ** Qed
Cardinal.aleph0_lt_lift α β : Type u c : Cardinal.{u} ⊢ ℵ₀ < lift.{v, u} c ↔ ℵ₀ < c rw [← lift_aleph0.{u,v}, lift_lt] ** Qed
Cardinal.lift_lt_aleph0 α β : Type u c : Cardinal.{u} ⊢ lift.{v, u} c < ℵ₀ ↔ c < ℵ₀ rw [← lift_aleph0.{u,v}, lift_lt] ** Qed
Cardinal.mk_fin α β : Type u n : ℕ ⊢ #(Fin n) = ↑n simp ** Qed
Cardinal.lift_natCast α β : Type u n : ℕ ⊢ lift.{u, v} ↑n = ↑n induction n <;> simp [] * Qed
Cardinal.nat_eq_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n = lift.{v, u} a ↔ ↑n = a rw [← lift_natCast.{v,u} n, lift_inj] ** Qed
Cardinal.lift_le_nat_iff α β : Type u a : Cardinal.{u} n : ℕ ⊢ lift.{v, u} a ≤ ↑n ↔ a ≤ ↑n rw [← lift_natCast.{v,u}, lift_le] ** Qed
Cardinal.nat_le_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n ≤ lift.{v, u} a ↔ ↑n ≤ a rw [← lift_natCast.{v,u}, lift_le] ** Qed
Cardinal.lift_lt_nat_iff α β : Type u a : Cardinal.{u} n : ℕ ⊢ lift.{v, u} a < ↑n ↔ a < ↑n rw [← lift_natCast.{v,u}, lift_lt] ** Qed
Cardinal.nat_lt_lift_iff α β : Type u n : ℕ a : Cardinal.{u} ⊢ ↑n < lift.{v, u} a ↔ ↑n < a rw [← lift_natCast.{v,u}, lift_lt] ** Qed
Cardinal.mk_coe_finset α✝ β α : Type u s : Finset α ⊢ #{ x // x ∈ s } = ↑(Finset.card s) simp ** Qed
Cardinal.mk_finset_of_fintype α β : Type u inst✝ : Fintype α ⊢ #(Finset α) = 2 ^ Fintype.card α simp [Pow.pow] ** Qed
Cardinal.mk_finsupp_lift_of_fintype α✝ β✝ α : Type u β : Type v inst✝¹ : Fintype α inst✝ : Zero β ⊢ #(α →₀ β) = lift.{u, v} #β ^ Fintype.card α simpa using (@Finsupp.equivFunOnFinite α β _ _).cardinal_eq ** Qed
Cardinal.mk_finsupp_of_fintype α✝ β✝ α β : Type u inst✝¹ : Fintype α inst✝ : Zero β ⊢ #(α →₀ β) = #β ^ Fintype.card α simp ** Qed
Cardinal.natCast_pow α β : Type u m n : ℕ ⊢ ↑(m ^ n) = ↑m ^ ↑n induction n <;> simp [pow_succ', power_add, , Pow.pow] * Qed
Cardinal.natCast_le α β : Type u m n : ℕ ⊢ ↑m ≤ ↑n ↔ m ≤ n rw [← lift_mk_fin, ← lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le, Fintype.card_fin, Fintype.card_fin] ** Qed
Cardinal.natCast_lt α β : Type u m n : ℕ ⊢ ↑m < ↑n ↔ m < n rw [lt_iff_le_not_le, ← not_le] α β : Type u m n : ℕ ⊢ ↑m ≤ ↑n ∧ ¬↑n ≤ ↑m ↔ ¬n ≤ m simp only [natCast_le, not_le, and_iff_right_iff_imp] α β : Type u m n : ℕ ⊢ m < n → m ≤ n exact fun h ↦ le_of_lt h ** Qed
Cardinal.nat_succ α β : Type u n : ℕ ⊢ ↑(Nat.succ n) = succ ↑n rw [Nat.cast_succ] α β : Type u n : ℕ ⊢ ↑n + 1 = succ ↑n refine (add_one_le_succ _).antisymm (succ_le_of_lt ?_) α β : Type u n : ℕ ⊢ ↑n < ↑n + 1 rw [← Nat.cast_succ] α β : Type u n : ℕ ⊢ ↑n < ↑(Nat.succ n) exact natCast_lt.2 (Nat.lt_succ_self _) ** Qed
Cardinal.succ_zero α β : Type u ⊢ succ 0 = 1 norm_cast ** Qed
Cardinal.card_le_of α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n ⊢ #α ≤ ↑n refine' le_of_lt_succ (lt_of_not_ge fun hn => _) α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n hn : #α ≥ succ ↑n ⊢ False rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n hn : #α ≥ lift.{u, 0} #(Fin (Nat.succ n)) ⊢ False cases' hn with f case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ False refine' (H <\| Finset.univ.map f).not_lt _ case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ n < Finset.card (Finset.map f Finset.univ) rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin] case intro α✝ β α : Type u n : ℕ H : ∀ (s : Finset α), Finset.card s ≤ n f : ULift.{u, 0} (Fin (Nat.succ n)) ↪ α ⊢ n < Nat.succ n exact n.lt_succ_self ** Qed
Cardinal.cantor' α β : Type u a b : Cardinal.{u_1} hb : 1 < b ⊢ a < b ^ a rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb α β : Type u a b : Cardinal.{u_1} hb✝ : 1 < b hb : 2 ≤ b ⊢ a < b ^ a exact (cantor a).trans_le (power_le_power_right hb) α β : Type u a b : Cardinal.{u_1} hb✝ : 1 < b hb : succ 1 ≤ b ⊢ succ 1 = 2 norm_cast ** Qed
Cardinal.one_le_iff_pos α β : Type u c : Cardinal.{u_1} ⊢ 1 ≤ c ↔ 0 < c rw [← succ_zero, succ_le_iff] ** Qed
Cardinal.one_le_iff_ne_zero α β : Type u c : Cardinal.{u_1} ⊢ 1 ≤ c ↔ c ≠ 0 rw [one_le_iff_pos, pos_iff_ne_zero] ** Qed
Cardinal.nat_lt_aleph0 α β : Type u n : ℕ ⊢ succ ↑n ≤ ℵ₀ rw [← nat_succ, ← lift_mk_fin, aleph0, lift_mk_le.{u}] α β : Type u n : ℕ ⊢ Nonempty (Fin (Nat.succ n) ↪ ℕ) exact ⟨⟨(↑), fun a b => Fin.ext⟩⟩ ** Qed
Cardinal.one_lt_aleph0 α β : Type u ⊢ 1 < ℵ₀ simpa using nat_lt_aleph0 1 ** Qed
Cardinal.lt_aleph0 α β : Type u c : Cardinal.{u_1} h : c < ℵ₀ ⊢ ∃ n, c = ↑n rcases lt_lift_iff.1 h with ⟨c, rfl, h'⟩ case intro.intro α β : Type u c : Cardinal.{0} h' : c < #ℕ h : lift.{u_1, 0} c < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} c = ↑n rcases le_mk_iff_exists_set.1 h'.1 with ⟨S, rfl⟩ case intro.intro.intro α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} #↑S = ↑n suffices S.Finite by lift S to Finset ℕ using this simp case intro.intro.intro α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ ⊢ Set.Finite S contrapose! h' case intro.intro.intro α β : Type u S : Set ℕ h : lift.{u_1, 0} #↑S < ℵ₀ h' : ¬Set.Finite S ⊢ #ℕ ≤ #↑S haveI := Infinite.to_subtype h' case intro.intro.intro α β : Type u S : Set ℕ h : lift.{u_1, 0} #↑S < ℵ₀ h' : ¬Set.Finite S this : Infinite ↑S ⊢ #ℕ ≤ #↑S exact ⟨Infinite.natEmbedding S⟩ α β : Type u S : Set ℕ h' : #↑S < #ℕ h : lift.{u_1, 0} #↑S < ℵ₀ this : Set.Finite S ⊢ ∃ n, lift.{u_1, 0} #↑S = ↑n lift S to Finset ℕ using this case intro α β : Type u S : Finset ℕ h' : #↑↑S < #ℕ h : lift.{u_1, 0} #↑↑S < ℵ₀ ⊢ ∃ n, lift.{u_1, 0} #↑↑S = ↑n simp ** Qed
Cardinal.aleph0_le α β : Type u c : Cardinal.{u_1} h : ∀ (n : ℕ), ↑n ≤ c hn : c < ℵ₀ ⊢ False rcases lt_aleph0.1 hn with ⟨n, rfl⟩ case intro α β : Type u n : ℕ h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n hn : ↑n < ℵ₀ ⊢ False exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1))) ** Qed
Cardinal.isSuccLimit_aleph0 α β : Type u a : Cardinal.{u_1} ha : a < ℵ₀ ⊢ succ a < ℵ₀ rcases lt_aleph0.1 ha with ⟨n, rfl⟩ case intro α β : Type u n : ℕ ha : ↑n < ℵ₀ ⊢ succ ↑n < ℵ₀ rw [← nat_succ] case intro α β : Type u n : ℕ ha : ↑n < ℵ₀ ⊢ ↑(Nat.succ n) < ℵ₀ apply nat_lt_aleph0 ** Qed
Cardinal.IsLimit.aleph0_le α β : Type u c : Cardinal.{u_1} h : IsLimit c ⊢ ℵ₀ ≤ c by_contra' h' α β : Type u c : Cardinal.{u_1} h : IsLimit c h' : c < ℵ₀ ⊢ False rcases lt_aleph0.1 h' with ⟨_ \| n, rfl⟩ case intro.zero α β : Type u h : IsLimit ↑Nat.zero h' : ↑Nat.zero < ℵ₀ ⊢ False exact h.ne_zero.irrefl case intro.succ α β : Type u n : ℕ h : IsLimit ↑(Nat.succ n) h' : ↑(Nat.succ n) < ℵ₀ ⊢ False rw [nat_succ] at h case intro.succ α β : Type u n : ℕ h : IsLimit (succ ↑n) h' : ↑(Nat.succ n) < ℵ₀ ⊢ False exact not_isSuccLimit_succ _ h.isSuccLimit ** Qed
Cardinal.range_natCast α β : Type u x : Cardinal.{u_1} ⊢ x ∈ range Nat.cast ↔ x ∈ Iio ℵ₀ simp only [mem_Iio, mem_range, eq_comm, lt_aleph0] ** Qed
Cardinal.mk_eq_nat_iff α✝ β α : Type u n : ℕ ⊢ #α = ↑n ↔ Nonempty (α ≃ Fin n) rw [← lift_mk_fin, ← lift_uzero #α, lift_mk_eq'] ** Qed
Cardinal.lt_aleph0_iff_finite α✝ β α : Type u ⊢ #α < ℵ₀ ↔ Finite α simp only [lt_aleph0, mk_eq_nat_iff, finite_iff_exists_equiv_fin] ** Qed
Cardinal.mk_le_aleph0_iff α β : Type u ⊢ #α ≤ ℵ₀ ↔ Countable α rw [countable_iff_nonempty_embedding, aleph0, ← lift_uzero #α, lift_mk_le'] ** Qed