module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Connected | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 16
} | [
{
"pp": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nC : Type u₂\ninst✝¹ : Category.{v₂, u₂} C\nX : C\ninst✝ : IsConnected J\ns : Cone ((Functor.const J).obj X)\nj : J\n⊢ s.π.app (Classical.arbitrary J) ≫ 𝟙 X = s.π.app j",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"Categ... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 342,
"column": 8
} | {
"line": 344,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : WithTerminal C ⥤ D\nx y : C\nf : x ⟶ y\n⊢ ((𝟭 (C ⥤ D)).obj (incl ⋙ F)).map f ≫ F.map (starTerminal.from (of y)) =\n F.map (starTerminal.from (of x)) ≫ ((Functor.const C).obj (F.obj star)).map f",
"usedConst... | dsimp
rw [Category.comp_id, ← F.map_comp]
congr 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 342,
"column": 8
} | {
"line": 344,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : WithTerminal C ⥤ D\nx y : C\nf : x ⟶ y\n⊢ ((𝟭 (C ⥤ D)).obj (incl ⋙ F)).map f ≫ F.map (starTerminal.from (of y)) =\n F.map (starTerminal.from (of x)) ≫ ((Functor.const C).obj (F.obj star)).map f",
"usedConst... | dsimp
rw [Category.comp_id, ← F.map_comp]
congr 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 495,
"column": 6
} | {
"line": 496,
"column": 58
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nZ✝ : C\ng₁ g₂ : Z✝ ⟶ X\nhg : g₁ ≫ f = g₂ ≫ f\n⊢ g₁ ≫ Cofork.π s = g₂ ≫ Cofork.π s",
"usedConstants": [
"Eq.mpr",
"Cate... | rw [← PullbackCone.IsLimit.lift_snd hc g₁ g₂ hg,
Category.assoc, ← Cofork.app_zero_eq_comp_π_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π = c.π",
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg",
"Quiver.Hom.op",
... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π = c.π",
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg",
"Quiver.Hom.op",
... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : C\nf g : X ⟶ Y\nc : Cofork f g\n⊢ c.op.unop.π = c.π",
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg",
"Quiver.Hom.op",
... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nc : Fork f g\n⊢ c.unop.op.ι = c.ι",
"usedConstants": [
"Opposite",
"Quiver.opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg"... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nc : Fork f g\n⊢ c.unop.op.ι = c.ι",
"usedConstants": [
"Opposite",
"Quiver.opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg"... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Cᵒᵖ\nf g : X ⟶ Y\nc : Fork f g\n⊢ c.unop.op.ι = c.ι",
"usedConstants": [
"Opposite",
"Quiver.opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Limits.WalkingParallelPair",
"congrArg"... | simp [Fork.unop_π, Cofork.op_ι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case zero\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case zero\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case zero\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case one\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.a... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case one\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.a... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {
"line": 499,
"column": 14
} | {
"line": 499,
"column": 22
} | [
{
"pp": "case one\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y B X : C\nf : X ⟶ B\ninst✝ : EffectiveEpi f\nc : PullbackCone f f\nhc : IsLimit c\ns : Cocone (parallelPair c.fst c.snd)\nthis : f ≫ EffectiveEpi.desc f (s.ι.app WalkingParallelPair.one) ⋯ = s.ι.app WalkingParallelPair.one\n⊢ (Cofork.ofπ f ⋯).ι.a... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 322,
"column": 35
} | {
"line": 325,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\ninst✝ : P.RespectsIso\nH : ∀ {X Y Z : C} (f : Z ⟶ X) (g : Z ⟶ Y) [HasPushout f g], P f → ∃ T inl inr, IsPushout f g inl inr ∧ P inr\n⊢ P.IsStableUnderCobaseChange",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsPushout.iso... | by
refine .mk' fun X Y S f g _ hg ↦ ?_
obtain ⟨T, inl, inr, h, hinl⟩ := H f g hg
rwa [← h.inr_isoPushout_hom, P.cancel_right_of_respectsIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.Small | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\ninst✝ : ObjectProperty.EssentiallySmall.{w, v, u} P\nQ : ObjectProperty C\nw✝ : ObjectProperty.Small.{w, v, u} Q\nhQ : P ≤ Q.isoClosure\nP' : ObjectProperty C := Q ⊓ P.isoClosure\nφ : Subtype P' → Subtype P\nhφ : ∀ (X' : Subtype P'), Nonempt... | refine ⟨fun X ↦ X ∈ Set.range (Subtype.val ∘ φ), ?_, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 888,
"column": 49
} | {
"line": 892,
"column": 90
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : P.IsStableUnderComposition\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : P.universally f\nhg : P.universally g\nX' Z' : C\ni₁ : X' ⟶ X\ni₂ : Z' ⟶ Z\nf' : X' ⟶ Z'\nH : IsPullback f' i₁ i₂ (f ≫ g)\n⊢ P f'",
"usedCon... | by
have := pullback.lift_fst _ _ (H.w.trans (Category.assoc _ _ _).symm)
rw [← this] at H ⊢
apply P.comp_mem _ _ _ (hg _ _ _ <| IsPullback.of_hasPullback _ _)
exact hf _ _ _ (H.of_right (pullback.lift_snd _ _ _) (IsPullback.of_hasPullback i₂ g)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape | {
"line": 147,
"column": 12
} | {
"line": 147,
"column": 50
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nJ' : Type u''\ninst✝ : Category.{v'', u''} J'\n⊢ ∀ {X Y : C} (x : X ≅ Y), P.limitsOfShape J X → P.limitsOfShape J Y",
"usedConstants": [
... | by rintro _ _ e ⟨h⟩; exact ⟨h.ofIso e⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | {
"line": 155,
"column": 12
} | {
"line": 155,
"column": 50
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nP : ObjectProperty C\nJ : Type u'\ninst✝¹ : Category.{v', u'} J\nJ' : Type u''\ninst✝ : Category.{v'', u''} J'\n⊢ ∀ {X Y : C} (x : X ≅ Y), P.colimitsOfShape J X → P.colimitsOfShape J Y",
"usedConstants": [
... | by rintro _ _ e ⟨h⟩; exact ⟨h.ofIso e⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ConcreteCategory.EpiMono | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 48
} | [
{
"pp": "case mpr\nC : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\na✝ : (forget C).PreservesMonomorphisms\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nhf : monomorphisms C f\nthis : Mono f\n⊢ MorphismProperty.injecti... | change Function.Injective ((forget C).map f) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 108,
"column": 10
} | {
"line": 108,
"column": 79
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nG : C ⥤ D\ninst✝³ : HasBinaryProducts C\ninst✝² : HasPullbacks C\ninst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst✝ : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair ⥤ C\nc : ... | apply (mapIsLimitOfPreservesOfIsLimit G _ _ (prodIsProd _ _)).hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 73
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n⊢ PreservesLimit (parallelPair f g) G",
"usedConstants": [
"Catego... | apply preservesLimit_of_preserves_limit_cone (equalizerIsEqualizer f g) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 62,
"column": 55
} | {
"line": 62,
"column": 88
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C... | rw [PullbackCone.condition_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 62,
"column": 55
} | {
"line": 62,
"column": 88
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C... | rw [PullbackCone.condition_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 62,
"column": 55
} | {
"line": 62,
"column": 88
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C... | rw [PullbackCone.condition_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.NonPreadditive | {
"line": 251,
"column": 2
} | {
"line": 252,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0\nhp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) (fun s ↦ s.ι ≫ prod.fst) ⋯ ⋯\n⊢ Epi (r A)",
"usedConstants... | let hp2 : IsColimit (CokernelCofork.ofπ (Limits.prod.snd : A ⨯ A ⟶ A) hlp) :=
epiIsCokernelOfKernel _ hp1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Category.CommAlgCat.Monoidal | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 70
} | [
{
"pp": "case hf\nR : Type u\ninst✝ : CommRing R\nA B C D T✝ : CommAlgCat R\nf : A ⟶ T✝\ng : B ⟶ T✝\nm : (A.binaryCofan B).pt ⟶ T✝\nhm₁ : (A.binaryCofan B).inl ≫ m = f\nhm₂ : (A.binaryCofan B).inr ≫ m = g\n⊢ Algebra.TensorProduct.liftEquiv.symm (Hom.hom m) = ⟨(ConcreteCategory.hom f, ConcreteCategory.hom g), ⋯⟩... | exact Subtype.ext (Prod.ext congr(($hm₁).hom) congr(($hm₂).hom)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Closure | {
"line": 265,
"column": 68
} | {
"line": 266,
"column": 58
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ (closure s).Nonempty ↔ s.Nonempty",
"usedConstants": [
"congrArg",
"Ne",
"closure_empty_iff._simp_1",
"iff_self",
"Iff",
"Set.Nonempty",
"closure",
"_private.Mathlib.Topology.Closure.0.closure_nonem... | by
simp only [nonempty_iff_ne_empty, Ne, closure_empty_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Closure | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 85
} | [
{
"pp": "case h\nX : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nh : Codisjoint (interior s) (interior t)\n⊢ Disjoint (closure sᶜ) (closure tᶜ)",
"usedConstants": [
"Eq.mpr",
"Codisjoint",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Compl.compl",
... | simpa only [closure_compl, disjoint_compl_left_iff, ← codisjoint_iff_compl_le_left] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.NhdsSet | {
"line": 106,
"column": 41
} | {
"line": 106,
"column": 58
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ 𝓝ˢ s ≤ 𝓟 s ↔ IsOpen[inst✝] s",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.le_principal_iff",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"id",
... | le_principal_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order | {
"line": 486,
"column": 21
} | {
"line": 486,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set (TopologicalSpace α)\n⊢ TopologicalSpace.coinduced f (⨆ a, ↑a) = sSup (TopologicalSpace.coinduced f '' s)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"Membership.mem",
"CompleteLattice.toConditionallyCompleteLattice... | sSup_image', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : TopologicalSpace β\nf : α → β\ns : Set α\n⊢ IsClosed[induced f t] s ↔ ∀ (a : α), f a ∈ closure[t] (f '' s) → a ∈ s",
"usedConstants": [
"congrArg",
"_private.Mathlib.Topology.Order.0.isClosed_induced_iff'._simp_1_1",
"Membership.mem",
"HasSubs... | simp only [← closure_subset_iff_isClosed, subset_def, closure_induced] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : TopologicalSpace β\nf : α → β\ns : Set α\n⊢ IsClosed[induced f t] s ↔ ∀ (a : α), f a ∈ closure[t] (f '' s) → a ∈ s",
"usedConstants": [
"congrArg",
"_private.Mathlib.Topology.Order.0.isClosed_induced_iff'._simp_1_1",
"Membership.mem",
"HasSubs... | simp only [← closure_subset_iff_isClosed, subset_def, closure_induced] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : TopologicalSpace β\nf : α → β\ns : Set α\n⊢ IsClosed[induced f t] s ↔ ∀ (a : α), f a ∈ closure[t] (f '' s) → a ∈ s",
"usedConstants": [
"congrArg",
"_private.Mathlib.Topology.Order.0.isClosed_induced_iff'._simp_1_1",
"Membership.mem",
"HasSubs... | simp only [← closure_subset_iff_isClosed, subset_def, closure_induced] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Basic | {
"line": 110,
"column": 47
} | {
"line": 110,
"column": 60
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace X\nhf : IsInducing f\nx : X\nl : Filter X\n⊢ (map f (𝓝 x ⊓ l)).NeBot ↔ (𝓝 x ⊓ l).NeBot",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"Filter.NeBot",
"nhds",
... | map_neBot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Maps.Basic | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace X\nhf : IsInducing f\ns : Set X\nhs : IsOpen[inst✝] s\n⊢ ∃ c, IsOpen[inst✝¹] c ∧ f '' s = c ∩ range f",
"usedConstants": [
"Topology.IsInducing.isOpen_iff",
"Exists",
"And",
"Set.pre... | obtain ⟨c, hc, rfl⟩ := hf.isOpen_iff.1 hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Separation.SeparatedNhds | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 24
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhst : SeparatedNhds s t\nhst' : IsOpen[inst✝] (s ∪ t)\n⊢ IsOpen[inst✝] s",
"usedConstants": [
"Set.inter_eq_left",
"Iff.mpr",
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Set.union_empty",
"c... | obtain ⟨u, v, hu, hv, hsu, htv, huv⟩ := hst
suffices s = (s ∪ t) ∩ u from this ▸ hst'.inter hu
rw [union_inter_distrib_right, (huv.symm.mono_left htv).inter_eq, union_empty,
inter_eq_left.2 hsu] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.SeparatedNhds | {
"line": 164,
"column": 2
} | {
"line": 167,
"column": 24
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhst : SeparatedNhds s t\nhst' : IsOpen[inst✝] (s ∪ t)\n⊢ IsOpen[inst✝] s",
"usedConstants": [
"Set.inter_eq_left",
"Iff.mpr",
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Set.union_empty",
"c... | obtain ⟨u, v, hu, hv, hsu, htv, huv⟩ := hst
suffices s = (s ∪ t) ∩ u from this ▸ hst'.inter hu
rw [union_inter_distrib_right, (huv.symm.mono_left htv).inter_eq, union_empty,
inter_eq_left.2 hsu] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions.SumProd | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 68
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous[instTopologicalSpaceProd, tc1] fun p ↦ f p.1 p.2\nha : Continuous[ta1 ⊓ ta2, ta1] id\nhb : Continuous[tb1 ⊓ tb2, tb1] id\nh_continuous_id : Conti... | exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructions.SumProd | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 68
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\nta1 ta2 : TopologicalSpace X\ntb1 tb2 : TopologicalSpace Y\ntc1 : TopologicalSpace Z\nh : Continuous[instTopologicalSpaceProd, tc1] fun p ↦ f p.1 p.2\nha : Continuous[ta1 ⊓ ta2, ta2] id\nhb : Continuous[tb1 ⊓ tb2, tb2] id\nh_continuous_id : Conti... | exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 66
} | [
{
"pp": "case refine_6\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ Codisjoint a (￢(a ⊓ b))",
"usedConstants": [
"CoheytingAlgebra.toHNot",
"codisjoint_hnot_right",
"SemilatticeInf.toPartialOrder",
"CoheytingAlgebra.toOrderTop",
"SemilatticeInf.toMin",
"HNot.hnot... | exact codisjoint_hnot_right.mono_right (hnot_anti inf_le_left) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Cofinite | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 60
} | [
{
"pp": "N : ℕ\nx✝ : True\n⊢ Ici N ∈ cofinite",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Set.Ici",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"Set.Finite",
"Membership.mem",
"DistribLattice.toLat... | simpa only [mem_cofinite, compl_Ici] using finite_lt_nat N | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.Cofinite | {
"line": 322,
"column": 27
} | {
"line": 322,
"column": 44
} | [
{
"pp": "case refine_4.refine_2\nα : Type u_2\nf : Filter α\nq : Set α × Filter α\nhq : (fun p ↦ p.2 ≤ cofinite ∧ Disjoint (𝓟 p.1) p.2 ∧ f = 𝓟 p.1 ⊔ p.2) q\nhqk : f.ker = q.1\n⊢ q.2 ≤ 𝓟 f.kerᶜ",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.le_principal_iff",
"cong... | le_principal_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.NhdsWithin | {
"line": 284,
"column": 83
} | {
"line": 286,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\ns : Set α\n⊢ insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Compl.compl",
"nhdsWithin",
"_private.Mathlib.Topology.NhdsWithin.0.insert_mem_nhds_iff._simp_1_4",
"setOf",
... | by
simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
insert_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.NhdsWithin | {
"line": 596,
"column": 35
} | {
"line": 596,
"column": 48
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\nt : TopologicalSpace β\nf : α → β\ns u : Set α\nthis : TopologicalSpace α := TopologicalSpace.induced f t\nx✝ : ∃ U, (∃ t_1, IsOpen[t] t_1 ∧ f ⁻¹' t_1 = U) ∧ s ⊆ U ∧ U ⊆ u\nv : Set α\nv' : Set β\nhv' : IsOpen[t] v' ∧ f ⁻¹' v' = v\nhv : s ⊆ v ∧ v ⊆ u\n⊢ f '' v ⊆ v'",
"use... | by simp [hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousOn | {
"line": 812,
"column": 87
} | {
"line": 813,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\ng : β → γ\nhg : IsInducing g\ns : Set α\nx : α\n⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x",
"usedConstants": [
"Eq.mpr",
"Continuo... | by
simp_rw [ContinuousWithinAt, hg.tendsto_nhds_iff]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Units | {
"line": 321,
"column": 86
} | {
"line": 321,
"column": 94
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\nH : Subgroup G\nx : Gˣ\n⊢ ↑x ∈ H ∧ ↑x ∈ H ↔ ↑x ∈ H",
"usedConstants": [
"Units.val",
"congrArg",
"and_self",
"Group.toDivisionMonoid",
"Membership.mem",
"DivInvMonoid.toMonoid",
"Subgroup",
"DivisionMonoid.toDivInvMonoid... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 33
} | [
{
"pp": "case inl\nG₀ : Type u_1\ninst✝¹ : Inv G₀\ninst✝ : LinearOrder G₀\nx y : G₀\nh✝ : x ≤ y\n⊢ min x⁻¹ y⁻¹ ≤ (max x y)⁻¹",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"Semilatt... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 33
} | [
{
"pp": "case inr\nG₀ : Type u_1\ninst✝¹ : Inv G₀\ninst✝ : LinearOrder G₀\nx y : G₀\nh✝ : y ≤ x\n⊢ min x⁻¹ y⁻¹ ≤ (max x y)⁻¹",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"Semilatt... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Constructions | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 30
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsOpenEmbedding f\ns : Set X\nt : Set Y\nH : MapsTo f s t\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[instTopologicalSpaceSubtype] (range (MapsTo.restrict f s t H))",
"usedConstants": [
"Eq.mpr",
"cong... | rw [MapsTo.range_restrict] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Constructions | {
"line": 572,
"column": 31
} | {
"line": 574,
"column": 56
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsOpenEmbedding f\ns : Set X\nt : Set Y\nH : MapsTo f s t\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[instTopologicalSpaceSubtype] (range (MapsTo.restrict f s t H))",
"usedConstants": [
"Eq.mpr",
"Topo... | by
rw [MapsTo.range_restrict]
exact continuous_subtype_val.1 _ (hf.isOpenMap _ hs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.NAry | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 59
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nγ' : Type u_6\nδ : Type u_7\nε : Type u_9\nf : Filter α\ng : Filter β\nh : Filter γ\nm : α → δ → ε\nn : β → γ → δ\nm₁ : α → β → β'\nm₂ : α → γ → γ'\nn' : β' → γ' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), m a (n b c) = n' (m₁ a b)... | · exact (image2_subset_right inter_subset_left).trans ht₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Constructions | {
"line": 1050,
"column": 11
} | {
"line": 1050,
"column": 28
} | [
{
"pp": "ι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ns : Set ((a : ι) → A a)\n⊢ (∀ x ∈ s, 𝓝 x ≤ 𝓟 s) ↔ ∀ f ∈ s, ∃ I u, (∀ a ∈ I, IsOpen[T a] (u a) ∧ f a ∈ u a) ∧ (↑I).pi u ⊆ s",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Pi.topologicalSpace",
"... | le_principal_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Constructions | {
"line": 1077,
"column": 11
} | {
"line": 1077,
"column": 28
} | [
{
"pp": "case intro\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : Finite ι\ns : Set ((a : ι) → A a)\nval✝ : Fintype ι\n⊢ (∀ x ∈ s, 𝓝 x ≤ 𝓟 s) ↔ ∀ f ∈ s, ∃ u, (∀ (a : ι), IsOpen[T a] (u a) ∧ f a ∈ u a) ∧ univ.pi u ⊆ s",
"usedConstants": [
"Filter.instMembership",
... | le_principal_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 385,
"column": 4
} | {
"line": 389,
"column": 53
} | [] | Filter.map m (of f) ≤ Filter.map m f := map_mono (of_le _)
_ ≤ (Filter.map m <| Filter.comap m g) ⊓ Filter.map m (𝓟 s) := map_inf_le
_ = (Filter.map m <| Filter.comap m g) ⊓ (𝓟 <| m '' s) := by rw [map_principal]
_ ≤ ↑g ⊓ (𝓟 <| m '' s) := inf_le_inf_right _ map_comap_le
_ = ↑g := inf_of_le_left (le_p... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Order.Filter.AtTopBot.CountablyGenerated | {
"line": 154,
"column": 27
} | {
"line": 154,
"column": 40
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ng : Filter ι\ninst✝¹ : g.IsCountablyGenerated\nu : ι → α\nf : Filter α\ninst✝ : f.IsCountablyGenerated\nhx : (map u (comap u f ⊓ g)).NeBot\n⊢ ∃ θ, Tendsto θ atTop g ∧ Tendsto (u ∘ θ) atTop f",
"usedConstants": [
"congrArg",
"Filter.map",
"Filter.NeBot",... | map_neBot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b c d : α\nthis✝ : MulLeftMono α\nthis : MulRightMono α\ny : α\nhya : a ≤ y\nhyb : y < b\nz : α\nhzc : c ≤ z\nhzd : z ≤ d\n⊢ (fun x1 x2 ↦ x1 * x2) y z ∈ Ico (a * c) (b * d)",
"usedCo... | exact ⟨mul_le_mul' hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 72
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na b c d : α\nthis✝ : MulLeftMono α\nthis : MulRightMono α\ny : α\nhya : a ≤ y\nhyb : y < b\nz : α\nhzc : c < z\nhzd : z ≤ d\n⊢ (fun x1 x2 ↦ x1 * x2) y z ∈ Ioo (a * c) (b * d)",
"usedCo... | exact ⟨mul_lt_mul_of_le_of_lt hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 574,
"column": 2
} | {
"line": 577,
"column": 40
} | [
{
"pp": "G₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : MulPosReflectLT G₀\na b c : G₀\nhab : a ≤ b\nhc : 0 ≤ c\n⊢ (fun x ↦ x * c) '' Icc a b = Icc (a * c) (b * c)",
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"Set.Icc_self",
"Preorder... | cases eq_or_lt_of_le hc
· subst c
simp [(nonempty_Icc.2 hab).image_const]
exact image_mul_right_Icc' a b ‹0 < c› | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 574,
"column": 2
} | {
"line": 577,
"column": 40
} | [
{
"pp": "G₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : PartialOrder G₀\ninst✝ : MulPosReflectLT G₀\na b c : G₀\nhab : a ≤ b\nhc : 0 ≤ c\n⊢ (fun x ↦ x * c) '' Icc a b = Icc (a * c) (b * c)",
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"Set.Icc_self",
"Preorder... | cases eq_or_lt_of_le hc
· subst c
simp [(nonempty_Icc.2 hab).image_const]
exact image_mul_right_Icc' a b ‹0 < c› | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Ultrafilter | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 78
} | [
{
"pp": "X : Type u\nx : X\ns : Set X\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x",
"usedConstants": [
"Filter.instMembership",
"closure_eq_cluster_pts",
"congrArg",
"Filter.NeBot",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreor... | simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Ultrafilter | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 78
} | [
{
"pp": "X : Type u\nx : X\ns : Set X\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x",
"usedConstants": [
"Filter.instMembership",
"closure_eq_cluster_pts",
"congrArg",
"Filter.NeBot",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreor... | simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Ultrafilter | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 78
} | [
{
"pp": "X : Type u\nx : X\ns : Set X\ninst✝ : TopologicalSpace X\n⊢ x ∈ closure[inst✝] s ↔ ∃ u, s ∈ u ∧ ↑u ≤ 𝓝 x",
"usedConstants": [
"Filter.instMembership",
"closure_eq_cluster_pts",
"congrArg",
"Filter.NeBot",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreor... | simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 65
} | [
{
"pp": "case inr.refine_2\nX✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝⁵ : TopologicalSpace X✝\ninst✝⁴ : TopologicalSpace Y\ns t : Set X✝\ninst✝³ : SigmaCompactSpace X✝\ninst✝² : Countable ι\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), SigmaCompactSpace (X i)\nh✝ : Nonem... | refine ⟨max k n, k, le_max_left _ _, mem_image_of_mem _ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Bases | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 19
} | [
{
"pp": "β : Type u_1\nι : Type u_2\nt : ι → TopologicalSpace β\nT : ι → Set (Set β)\nh_basis : ∀ (i : ι), IsTopologicalBasis (T i)\n⊢ IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}",
"usedConstants": [
"iInf",
"CompleteLattice.toConditionallyCompleteLattice",
"Top... | let _ := ⨅ i, t i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Bases | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 51
} | [
{
"pp": "β : Type u_1\nι : Type u_2\nt : ι → TopologicalSpace β\nT : ι → Set (Set β)\nh_basis : ∀ (i : ι), IsTopologicalBasis (T i)\nx✝ : TopologicalSpace β := ⨅ i, t i\n⊢ IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}",
"usedConstants": [
"iInf",
"Set.iInter",
"Fi... | refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Bases | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 34
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : t.Countable ∧ ∅ ∉ t ∧ IsTopologicalBasis t\nhn : t = ∅\n⊢ IsTopologicalBasis (range fun n ↦ ∅)",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"instInha... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Bases | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 34
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : t.Countable ∧ ∅ ∉ t ∧ IsTopologicalBasis t\nhn : t = ∅\n⊢ IsTopologicalBasis (range fun n ↦ ∅)",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"instInha... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 34
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : t.Countable ∧ ∅ ∉ t ∧ IsTopologicalBasis t\nhn : t = ∅\n⊢ IsTopologicalBasis (range fun n ↦ ∅)",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"instInha... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.UpperLower.Fibration | {
"line": 24,
"column": 2
} | {
"line": 24,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : LE α\ninst✝ : LE β\nhf : Fibration (fun x1 x2 ↦ x1 ≤ x2) (fun x1 x2 ↦ x1 ≤ x2) f\ns : Set α\nhs : IsLowerSet s\n⊢ IsLowerSet (f '' s)",
"usedConstants": [
"Membership.mem",
"Exists",
"LE.le",
"And.casesOn",
"And",
"... | rintro _ y' e ⟨x, hx, rfl⟩; obtain ⟨y, e', rfl⟩ := hf e; exact ⟨_, hs e' hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.UpperLower.Fibration | {
"line": 24,
"column": 2
} | {
"line": 24,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : LE α\ninst✝ : LE β\nhf : Fibration (fun x1 x2 ↦ x1 ≤ x2) (fun x1 x2 ↦ x1 ≤ x2) f\ns : Set α\nhs : IsLowerSet s\n⊢ IsLowerSet (f '' s)",
"usedConstants": [
"Membership.mem",
"Exists",
"LE.le",
"And.casesOn",
"And",
"... | rintro _ y' e ⟨x, hx, rfl⟩; obtain ⟨y, e', rfl⟩ := hf e; exact ⟨_, hs e' hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 29
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (cocompact X) (𝓝 y)\nhfc : Continuous[inst✝¹, inst✝] f\nl : Filter Y\nhne : l.NeBot\nhle : l ≤ 𝓟 (insert y (range f))\n⊢ ∃ x ∈ insert y (range f), ClusterPt x l",
"usedConstants": [
... | by_cases hy : ClusterPt y l | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.GDelta.Basic | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 51
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\nT : Set (Set X)\nhT : ∀ t ∈ T, IsOpen[inst✝] t\nT_count : T.Countable\n⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n",
"usedConstants": [
"Set.eq_empty_or_nonempty",
"Set"
]
}
] | rcases Set.eq_empty_or_nonempty T with rfl | hT | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.GDelta.Basic | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 27
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nT : Set (Set X)\nhT✝ : ∀ t ∈ T, IsOpen[inst✝] t\nT_count : T.Countable\nhT : T.Nonempty\nf : ℕ → Set X\nhf : T = range f\n⊢ ∀ (n : ℕ), IsOpen[inst✝] (f n)",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
"i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.GDelta.Basic | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 27
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nT : Set (Set X)\nhT✝ : ∀ t ∈ T, IsOpen[inst✝] t\nT_count : T.Countable\nhT : T.Nonempty\nf : ℕ → Set X\nhf : T = range f\n⊢ ∀ (n : ℕ), IsOpen[inst✝] (f n)",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
"i... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.GDelta.Basic | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 27
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nT : Set (Set X)\nhT✝ : ∀ t ∈ T, IsOpen[inst✝] t\nT_count : T.Countable\nhT : T.Nonempty\nf : ℕ → Set X\nhf : T = range f\n⊢ ∀ (n : ℕ), IsOpen[inst✝] (f n)",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
"i... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 1147,
"column": 2
} | {
"line": 1148,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : SecondCountableTopology β\nhf : IsInducing f\n⊢ SecondCountableTopology α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SecondCountableTopology",
"id",
"TopologicalSp... | rw [hf.1]
exact secondCountableTopology_induced α β f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 1147,
"column": 2
} | {
"line": 1148,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : SecondCountableTopology β\nhf : IsInducing f\n⊢ SecondCountableTopology α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SecondCountableTopology",
"id",
"TopologicalSp... | rw [hf.1]
exact secondCountableTopology_induced α β f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.GDelta.Basic | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 23
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsMeagre s\nht : IsMeagre t\n⊢ sᶜ ∩ tᶜ ∈ residual X",
"usedConstants": [
"Compl.compl",
"Filter.inter_mem",
"residual",
"Set.instCompl",
"Set"
]
}
] | exact inter_mem hs ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Baire.Lemmas | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 69
} | [
{
"pp": "case insert\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (Set X)\na✝ : Set X\ns✝ : Set (Set X)\nha : a✝ ∉ s✝\nhsf : s✝.Finite\nih : (∀ t ∈ s✝, IsOpen[inst✝] t) → (∀ t ∈ s✝, Dense t) → Dense (⋂₀ s✝)\nho : ∀ t ∈ insert a✝ s✝, IsOpen[inst✝] t\nhd : Dense a✝ ∧ ∀ x ∈ s✝, Dense x\n⊢ Dense (⋂₀ s✝)",
... | exact ih ((fun y hy => ho y (Or.inr hy))) (fun y hy => hd.2 y hy) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Baire.Lemmas | {
"line": 77,
"column": 57
} | {
"line": 77,
"column": 65
} | [
{
"pp": "case h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : BaireSpace X\ns : Set X\nhG : IsGδ s\nhd : Dense s\nf : ℕ → Set ↑s\nhof : ∀ (n : ℕ), IsOpen[instTopologicalSpaceSubtype] (f n)\nhdf : ∀ (n : ℕ), Dense (f n)\nV : ℕ → Set X\nhV : (∀ (n : ℕ), IsOpen[inst✝¹] (V n)) ∧ s = ⋂ n, V n\ng : ℕ → Set X\nh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Compactness.Compact | {
"line": 1114,
"column": 4
} | {
"line": 1114,
"column": 12
} | [
{
"pp": "case right.h\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nu : Set ((i : ι) × X i)\nhu : IsCompact u\ns : Finset ι\nhs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ\nx : (i : ι) × X i\nhx : ⟨x.fst, x.snd⟩ ∈ u\ni : ι\nhi : ⟨x.fst, x.snd⟩ ∈ ⋃ (_ : i ∈ s), Sigma.mk i '' Sig... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Baire.Lemmas | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 26
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : BaireSpace X\ns t : Set X\nhs : IsGδ s\nht : IsGδ t\nhsc : Dense s\nhtc : Dense t\n⊢ Dense (⋂ b, bif b then s else t)",
"usedConstants": [
"cond",
"dense_iInter_of_Gδ",
"Bool.countable",
"Bool",
"Set"
]
}
] | apply dense_iInter_of_Gδ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Compactness.Compact | {
"line": 1212,
"column": 8
} | {
"line": 1212,
"column": 40
} | [
{
"pp": "case pos\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}\nc : Set (Set X)\nhc : c ⊆ opens\nhz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c\nU₀ : Set X\nhU₀ : U₀ ∈ c\nthis : ... | obtain ⟨U₀compl, -, -⟩ := hc hU₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Piecewise | {
"line": 57,
"column": 8
} | {
"line": 58,
"column": 81
} | [
{
"pp": "case neg.inr.hs\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns : Set α\np : α → Prop\nf g : α → β\ninst✝ : (a : α) → Decidable (p a)\nhpf : ∀ a ∈ s ∩ frontier {a | p a}, Tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 (if p a then f a else g a))\nhpg : ∀ a ∈ s ∩ frontier {... | have : x ∉ closure { a | p a } := fun h =>
hx' ⟨h, fun h' : x ∈ interior { a | p a } => hx.2 (interior_subset h')⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Inseparable | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 86
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nhf : SpecializingMap f\nhg : SpecializingMap g\n⊢ SpecializingMap (g ∘ f)",
"usedConstants": [
"Eq.mpr",
"StableUnderSpecialization",
... | simp only [specializingMap_iff_stableUnderSpecialization_image, Set.image_comp] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Piecewise | {
"line": 105,
"column": 52
} | {
"line": 107,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf g : α → β\np : α → Prop\ninst✝ : (a : α) → Decidable (p a)\nhp : ∀ a ∈ frontier {x | p x}, f a = g a\nhf : ContinuousOn f (closure[inst✝²] {x | p x})\nhg : ContinuousOn g (closure[inst✝²] {x | ¬p x})\n⊢ Continuous[i... | by
rw [← continuousOn_univ]
apply ContinuousOn.if <;> simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Irreducible | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nH : ∀ ⦃U V : Set X⦄, IsOpen[inst✝] U → IsOpen[inst✝] V → U.Nonempty → V.Nonempty → (U ∩ V).Nonempty\nx✝ : Set X\n⊢ ∀ (v : Set X),\n IsOpen[inst✝] x✝ → IsOpen[inst✝] v → (univ ∩ x✝).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (x✝ ∩ v)).Nonempty",
"usedConsta... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Irreducible | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nH : ∀ ⦃U V : Set X⦄, IsOpen[inst✝] U → IsOpen[inst✝] V → U.Nonempty → V.Nonempty → (U ∩ V).Nonempty\nx✝ : Set X\n⊢ ∀ (v : Set X),\n IsOpen[inst✝] x✝ → IsOpen[inst✝] v → (univ ∩ x✝).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (x✝ ∩ v)).Nonempty",
"usedConsta... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Irreducible | {
"line": 191,
"column": 32
} | {
"line": 191,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nH : ∀ ⦃U V : Set X⦄, IsOpen[inst✝] U → IsOpen[inst✝] V → U.Nonempty → V.Nonempty → (U ∩ V).Nonempty\nx✝ : Set X\n⊢ ∀ (v : Set X),\n IsOpen[inst✝] x✝ → IsOpen[inst✝] v → (univ ∩ x✝).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (x✝ ∩ v)).Nonempty",
"usedConsta... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Irreducible | {
"line": 263,
"column": 23
} | {
"line": 263,
"column": 31
} | [
{
"pp": "case inl\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : IndiscreteTopology X\nv : Set X\n⊢ v = ∅ ∨ v = univ → ∅.Nonempty → v.Nonempty → (∅ ∩ v).Nonempty",
"usedConstants": [
"False",
"congrArg",
"Set.univ",
"Set.em... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Irreducible | {
"line": 263,
"column": 23
} | {
"line": 263,
"column": 31
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : IndiscreteTopology X\nu v : Set X\nh✝ : u = univ\n⊢ v = ∅ ∨ v = univ → u.Nonempty → v.Nonempty → (u ∩ v).Nonempty",
"usedConstants": [
"congrArg",
"Set.univ",
"Set.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.DiscreteSubset | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 12
} | [
{
"pp": "case insert\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns t✝ : Set X\nh : t✝.Finite\nτ : X\nt : Set X\nhτ : τ ∉ t\nh₁t : t.Finite\nh₂t : tᶜ ∈ codiscreteWithin s\nthis : (insert τ t)ᶜ = {τ}ᶜ ∩ tᶜ\n⊢ (insert τ t)ᶜ ∈ codiscreteWithin s",
"usedConstants": [
"Filter.instMembersh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.DiscreteSubset | {
"line": 365,
"column": 56
} | {
"line": 366,
"column": 75
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set X\nU : Set ↑S\n⊢ U ∈ codiscrete ↑S ↔ Subtype.val '' U ∈ codiscreteWithin S",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"Set.Elem",
"Subtype",
"Subtype.range_coe_subtype",
"iff... | by
simp [← Topology.IsEmbedding.subtypeVal.image_mem_codiscreteWithin_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.Hausdorff | {
"line": 664,
"column": 6
} | {
"line": 664,
"column": 14
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nx : X\nf g : X → Y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ninst✝ : (𝓝[≠] x).NeBot\nhfg : f =ᶠ[𝓝[≠] x] g\nhCon : ¬f x = g x\nh₁ : ∀ᶠ (x : X) in 𝓝[≠] x, f x ≠ g x ∧ f x = g x\nh₂ : ∅ ∉ 𝓝[≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 53,
"column": 4
} | {
"line": 55,
"column": 86
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : TopologicalSpace α\nx✝ : α\n⊢ (∀ U ∈ 𝓝 x✝, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x✝ ∈ V ∧ IsConnected V) →\n (𝓝 x✝).HasBasis (fun s ↦ IsOpen[inst✝] s ∧ x✝ ∈ s ∧ IsConnected s) id",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"_private.Mathlib.Top... | exact fun h => ⟨fun U => ⟨fun hU =>
let ⟨V, hVU, hV⟩ := h U hU
⟨V, hV, hVU⟩, fun ⟨V, ⟨hV, hxV, _⟩, hVU⟩ => mem_nhds_iff.mpr ⟨V, hVU, hV, hxV⟩⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 53,
"column": 4
} | {
"line": 55,
"column": 86
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : TopologicalSpace α\nx✝ : α\n⊢ (∀ U ∈ 𝓝 x✝, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x✝ ∈ V ∧ IsConnected V) →\n (𝓝 x✝).HasBasis (fun s ↦ IsOpen[inst✝] s ∧ x✝ ∈ s ∧ IsConnected s) id",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"_private.Mathlib.Top... | exact fun h => ⟨fun U => ⟨fun hU =>
let ⟨V, hVU, hV⟩ := h U hU
⟨V, hV, hVU⟩, fun ⟨V, ⟨hV, hxV, _⟩, hVU⟩ => mem_nhds_iff.mpr ⟨V, hVU, hV, hxV⟩⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 53,
"column": 4
} | {
"line": 55,
"column": 86
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : TopologicalSpace α\nx✝ : α\n⊢ (∀ U ∈ 𝓝 x✝, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x✝ ∈ V ∧ IsConnected V) →\n (𝓝 x✝).HasBasis (fun s ↦ IsOpen[inst✝] s ∧ x✝ ∈ s ∧ IsConnected s) id",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"_private.Mathlib.Top... | exact fun h => ⟨fun U => ⟨fun hU =>
let ⟨V, hVU, hV⟩ := h U hU
⟨V, hV, hVU⟩, fun ⟨V, ⟨hV, hxV, _⟩, hVU⟩ => mem_nhds_iff.mpr ⟨V, hVU, hV, hxV⟩⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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