module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 450,
"column": 61
} | {
"line": 450,
"column": 69
} | {
"line": 450,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y✝ : C\nS R : Sieve X\nJ : GrothendieckTopology C\nx✝¹ x✝ : J.Cover X\nh1 : x✝¹ ≤ x✝\nh2 : x✝ ≤ x✝¹\nY : C\nf : Y ⟶ X\n⊢ (↑x✝¹).arrows f → (↑x✝).arrows f",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h1",
... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 450,
"column": 61
} | {
"line": 450,
"column": 69
} | {
"line": 450,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y✝ : C\nS R : Sieve X\nJ : GrothendieckTopology C\nx✝¹ x✝ : J.Cover X\nh1 : x✝¹ ≤ x✝\nh2 : x✝ ≤ x✝¹\nY : C\nf : Y ⟶ X\n⊢ (↑x✝¹).arrows f → (↑x✝).arrows f",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h1",
... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 21
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : Y ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\ninst✝ : ∀ (i : ι), HasPullback (g i) f\nT : C\nh : T ⟶ Y\n⊢ pullbackArrows f (ofArrows Z g) h → ofArrows (fun i ↦ pullback (g i) f) (fun x ↦ pullback.snd (g x) f) h",
"ppTerm": "... | [] | rintro ⟨W, k, ⟨_⟩⟩
apply ofArrows.mk | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 21
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : Y ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\ninst✝ : ∀ (i : ι), HasPullback (g i) f\nT : C\nh : T ⟶ Y\n⊢ pullbackArrows f (ofArrows Z g) h → ofArrows (fun i ↦ pullback (g i) f) (fun x ↦ pullback.snd (g x) f) h",
"ppTerm": "... | [] | rintro ⟨W, k, ⟨_⟩⟩
apply ofArrows.mk | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 314,
"column": 4
} | {
"line": 314,
"column": 16
} | {
"line": 315,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nι : Type u_1\nU : ι → C\nX Y : C\ng : (i : ι) → U i ⟶ X\nf : X ⟶ Y\ni : ι\n⊢ pushforward f (ofArrows U g) (g i ≫ f)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Q... | [
"case right\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nι : Type u_1\nU : ι → C\nX Y : C\ng : (i : ι) → U i ⟶ X\nf : X ⟶ Y\ni : ι\n⊢ ofArrows U g (g i)"
] | use g i, rfl | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Sites.Precoverage | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 94
} | {
"line": 192,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : Precoverage C\ninst✝¹ : J.IsStableUnderBaseChange\nX Y : C\nf : X ⟶ Y\nι : Type (max u v)\nZ : ι → C\ng : (i : ι) → Z i ⟶ Y\nhR : Presieve.ofArrows Z g ∈ J.coverings Y\ninst✝ : (Presieve.ofArrows Z g).HasPullbacks f\n⊢ Presieve.pullbackArrows f (Presieve.ofAr... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : Precoverage C\ninst✝¹ : J.IsStableUnderBaseChange\nX Y : C\nf : X ⟶ Y\nι : Type (max u v)\nZ : ι → C\ng : (i : ι) → Z i ⟶ Y\nhR : Presieve.ofArrows Z g ∈ J.coverings Y\ninst✝ : (Presieve.ofArrows Z g).HasPullbacks f\nthis : ∀ (i : ι), Limits.HasPullback (g i) f\n⊢ Presie... | have (i : ι) : Limits.HasPullback (g i) f := Presieve.hasPullback f (Presieve.ofArrows.mk i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 868,
"column": 4
} | {
"line": 868,
"column": 29
} | {
"line": 869,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf : Y✝ ⟶ X✝\nS R : Sieve X✝\nI : Type u_1\nY : I → C\nX : C\n⊢ ∀ {Y_1 Z : C} {f : Y_1 ⟶ X}, (∃ i, Nonempty (Y_1 ⟶ Y i)) → ∀ (g : Z ⟶ Y_1), ∃ i, Nonempty (Z ⟶ Y i)",
"ppTerm": "?m.20",
"a... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : Y✝ ⟶ X✝\nS R : Sieve X✝\nI : Type u_1\nY : I → C\nX Z₁ Z₂ : C\np : Z₁ ⟶ X\ni : I\nf : Z₁ ⟶ Y i\ng : Z₂ ⟶ Z₁\n⊢ ∃ i, Nonempty (Z₂ ⟶ Y i)"
] | rintro Z₁ Z₂ p ⟨i, ⟨f⟩⟩ g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Limits.Types.Equalizers | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 16
} | {
"line": 43,
"column": 6
} | [
{
"pp": "case refine_3\nX Y Z : Type u\nf : X ⟶ Y\ng h : Y ⟶ Z\nw : f ≫ g = f ≫ h\nt : ∀ (y : Y), (hom g) y = (hom h) y → ∃! x, (hom f) x = y\ns : Fork g h\n⊢ ∀ {m : s.pt ⟶ (Fork.ofι f w).pt}, m ≫ (Fork.ofι f w).ι = s.ι → m = ↾fun i ↦ Classical.choose ⋯",
"ppTerm": "?refine_3",
"assigned": true,
"us... | [
"case refine_3\nX Y Z : Type u\nf : X ⟶ Y\ng h : Y ⟶ Z\nw : f ≫ g = f ≫ h\nt : ∀ (y : Y), (hom g) y = (hom h) y → ∃! x, (hom f) x = y\ns : Fork g h\nm : s.pt ⟶ (Fork.ofι f w).pt\nhm : m ≫ (Fork.ofι f w).ι = s.ι\n⊢ m = ↾fun i ↦ Classical.choose ⋯"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1109,
"column": 4
} | {
"line": 1109,
"column": 19
} | {
"line": 1110,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nR : Sieve X\n⊢ ∀ {Y Z : D} {f : Y ⟶ F.obj X},\n Presieve.functorPushforward F R.arrows f → ∀ (g : Z ⟶ Y), Presie... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nR : Sieve X\nY✝ Z✝ : D\nf : Y✝ ⟶ F.obj X\nh : Presieve.functorPushforward F R.arrows f\ng : Z✝ ⟶ Y✝\n⊢ Presieve.functorPushfor... | intro _ _ f h g | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 282,
"column": 11
} | {
"line": 284,
"column": 7
} | {
"line": 286,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx₁ x₂ : FamilyOfElements P (generate R).arrows\nt₁ : x₁.Compatible\nt₂ : x₂.Compatible\nh : FamilyOfElements.restrict ⋯ x₁ = FamilyOfElements.restrict ⋯ x₂\n⊢ x₁ = x₂",
"ppTerm": "?m.60",
"assigned": true,
"u... | [] | by
rw [← extend_restrict t₁, ← extend_restrict t₂]
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1240,
"column": 4
} | {
"line": 1240,
"column": 76
} | {
"line": 1241,
"column": 2
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : C\nS : Sieve X\ne : C ≌ D\nY : D\nZ : C\niZX : Z ⟶ X\niYZ : Y ⟶ e.functor.obj Z\nhiZX : S.arrows iZX\n⊢ (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows (iYZ ≫ e.functor.map iZX)",
"pp... | [] | simpa using S.downward_closed hiZX (e.inverse.map iYZ ≫ e.unitInv.app Z) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1239,
"column": 2
} | {
"line": 1240,
"column": 76
} | {
"line": 1241,
"column": 2
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : C\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPushforward e.functor S).arrows iYX → (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX",
"ppTerm": "?mp",
... | [
"case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX : C\nS : Sieve X\ne : C ≌ D\nY : D\niYX : Y ⟶ e.functor.obj X\n⊢ (functorPullback e.inverse (pullback (e.unitInv.app X) S)).arrows iYX → (functorPushforward e.functor S).arrows iYX"
] | · rintro ⟨Z, iZX, iYZ, hiZX, rfl⟩
simpa using S.downward_closed hiZX (e.inverse.map iYZ ≫ e.unitInv.app Z) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 25
} | {
"line": 147,
"column": 4
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝ : Category.{v₂, u₂} A\nP : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhu : ∀ (i : Cone (S.arrows.diagram.op ⋙ P)), Nonempty (Unique (i ⟶ P.mapCone S.arrows.cocone.op))\nE : Aᵒᵖ\nx : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows\nhx : x.SieveComp... | [
"case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝ : Category.{v₂, u₂} A\nP : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nE : Aᵒᵖ\nx : FamilyOfElements (P ⋙ coyoneda.obj E) S.arrows\nhx : x.SieveCompatible\nhu : Nonempty (Unique (hx.cone ⟶ P.mapCone S.arrows.cocone.op))\n⊢ ∃! t, x.IsAmalgamation t"
] | specialize hu hx.cone | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 37
} | {
"line": 350,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type w\nB : C\nI : Type t\ninst✝¹ : Small.{w, t} I\nX : I → C\nπ : (i : I) → X i ⟶ B\ninst✝ : (Presieve.ofArrows X π).HasPairwisePullbacks\nx : FirstObj P X\n⊢ Arrows.PullbackCompatible P π\n ((equivShrink ((i : I) → P.obj (op (X i)))).symm\n ... | [
"case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type w\nB : C\nI : Type t\ninst✝¹ : Small.{w, t} I\nX : I → C\nπ : (i : I) → X i ⟶ B\ninst✝ : (Presieve.ofArrows X π).HasPairwisePullbacks\nx : FirstObj P X\nt :\n Arrows.PullbackCompatible P π\n ((equivShrink ((i : I) → P.obj (op (X i)))).symm\n... | refine ⟨fun t ↦ ?_, fun t i j ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 8
} | {
"line": 542,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nA' : Type u₂\ninst✝¹ : Category.{max v₁ u₁, u₂} A'\nB : Type u₃\ninst✝ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : Cᵒᵖ ⥤ A\nP' : Cᵒᵖ ⥤ A'\nX : C\nS : J.Cover X\nhP : IsSheaf J P\nE : ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nA' : Type u₂\ninst✝¹ : Category.{max v₁ u₁, u₂} A'\nB : Type u₃\ninst✝ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : Cᵒᵖ ⥤ A\nP' : Cᵒᵖ ⥤ A'\nX : C\nS : J.Cover X\nhP : IsSheaf J P\nE : Multifork (S... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 28
} | {
"line": 577,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ninst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\n⊢ IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P)",
"ppTerm": "?m.36",
"assigned... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ninst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\n⊢ (∀ (X : C) (S : J.Cover X), Nonempty (IsLimit (S.multifork P))) ↔\n ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequali... | rw [isSheaf_iff_multifork] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 10
} | {
"line": 587,
"column": 6
} | [
{
"pp": "case refine_2.refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ninst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nX : C\nS : J.Cover X\nh : IsIso (S.toMultiequalizer P)\na : WalkingMulticospan S.shape... | [
"case refine_2.refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : Type u₂\ninst✝¹ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ninst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nX : C\nS : J.Cover X\nh : IsIso (S.toMultiequalizer P)\na : WalkingMulticospan S.shape\n⊢ (asIso (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 887,
"column": 8
} | {
"line": 888,
"column": 36
} | {
"line": 888,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : Y ⟶ X\ninst✝ : IsIso f\nι : Type (max u₁ v₁)\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nthis : Sieve.pullback f (Sieve.ofArrows Z g) = Sieve.ofArrows Z fun i ↦ g i ≫ inv f\ns : Subtype (Arrows.Compatible P g)\ni₁ i₂ : ι\nW : C\ng₁ : W ⟶ ... | [] | simp only [← cancel_mono f, assoc, IsIso.inv_hom_id, comp_id] at h
exact s.property _ _ _ _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 887,
"column": 8
} | {
"line": 888,
"column": 36
} | {
"line": 888,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : Y ⟶ X\ninst✝ : IsIso f\nι : Type (max u₁ v₁)\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nthis : Sieve.pullback f (Sieve.ofArrows Z g) = Sieve.ofArrows Z fun i ↦ g i ≫ inv f\ns : Subtype (Arrows.Compatible P g)\ni₁ i₂ : ι\nW : C\ng₁ : W ⟶ ... | [] | simp only [← cancel_mono f, assoc, IsIso.inv_hom_id, comp_id] at h
exact s.property _ _ _ _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 33
} | {
"line": 699,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : IsSheaf J (P ⋙ s)\n⊢ IsSheaf J P",
"ppTerm": "?m.2... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : ∀ ⦃X : C⦄, ∀ S ∈ J X, Nonempty (IsLimit ((P ⋙ s).mapCone S.arrows.... | rw [isSheaf_iff_isLimit] at h ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 703,
"column": 2
} | {
"line": 703,
"column": 33
} | {
"line": 704,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : PreservesLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : IsSheaf J P\n⊢ IsSheaf J (P ⋙ s)",
"ppTerm": "?m.... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : PreservesLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : ∀ ⦃X : C⦄, ∀ S ∈ J X, Nonempty (IsLimit (P.mapCone S.arrows.cocon... | rw [isSheaf_iff_isLimit] at h ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.ColimitLimit | {
"line": 78,
"column": 24
} | {
"line": 85,
"column": 62
} | {
"line": 85,
"column": 63
} | [
{
"pp": "J : Type u₁\nK : Type u₂\ninst✝⁴ : Category.{v₁, u₁} J\ninst✝³ : Category.{v₂, u₂} K\nC : Type u\ninst✝² : Category.{v, u} C\nF : J × K ⥤ C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\n⊢ ∀ ⦃X Y : J⦄ (f : X ⟶ Y),\n ((const J).obj (colimit.cocone (curry.obj (swap K J ⋙ F) ⋙ lim)).1)... | [] | by
intro j j' f
dsimp
ext k
simp only [Functor.comp_obj, lim_obj, Category.id_comp, colimit.ι_desc,
colimit.ι_desc_assoc, Category.assoc, ι_colimMap,
curry_obj_obj_obj, curry_obj_map_app]
rw [map_id_right_eq_curry_swap_map, limit.w_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 52
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case right\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShap... | [
"case right\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMu... | rintro (x : ToType (multiequalizer (S.index _))) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 592,
"column": 38
} | {
"line": 592,
"column": 42
} | {
"line": 593,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMu... | [
"C : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 208,
"column": 69
} | {
"line": 216,
"column": 45
} | {
"line": 218,
"column": 0
} | [
{
"pp": "X Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\n⊢ Set.range ⇑(ConcreteCategory.hom (pullback.fst f g)) =\n {x | ∃ y, (ConcreteCategory.hom f) x = (ConcreteCategory.hom g) y}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
... | [] | by
ext x
constructor
· rintro ⟨y, rfl⟩
use pullback.snd f g y
exact CategoryTheory.congr_fun pullback.condition y
· rintro ⟨y, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_fst_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 68,
"column": 48
} | {
"line": 72,
"column": 42
} | {
"line": 74,
"column": 0
} | [
{
"pp": "J : Type v'\ninst✝³ : Category.{u', v'} J\nC : Type u\ninst✝² : Category.{v, u} C\nK : Type u_1\ninst✝¹ : Category.{v_1, u_1} K\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nF : J ⥤ C\nc : Cocone F\nh : IsUniversalColimit c\n⊢ IsColimit c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstant... | [] | by
refine ((h c (𝟙 F) (𝟙 c.pt :) (by rw [Functor.map_id, Category.comp_id, Category.id_comp])
(.of_isIso _)) fun j => ?_).some
haveI : IsIso (𝟙 c.pt) := inferInstance
exact IsPullback.of_vert_isIso ⟨by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 29
} | {
"line": 167,
"column": 8
} | [
{
"pp": "case mpr.mpr.h₀\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' :\n ∀ ⦃X' Y' Z' : C⦄ (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z),\n CommSq h' αX αZ h →\n CommSq i' αY αZ i →\n ∀ [HasPu... | [
"case mpr.mpr.h₁\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' :\n ∀ ⦃X' Y' Z' : C⦄ (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z),\n CommSq h' αX αZ h →\n CommSq i' αY αZ i →\n ∀ [HasPullback αX f]... | · simp [← cs.w, hP.w] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 209,
"column": 8
} | {
"line": 209,
"column": 12
} | {
"line": 209,
"column": 12
} | [
{
"pp": "case mpr.uniq.h₀\nC : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f c.inl iY fE\nH : IsPushout f c.inl iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (c.inr ≫ fE) ... | [
"case mpr.uniq.h₀\nC : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f c.inl iY fE\nH : IsPushout f c.inl iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (c.inr ≫ fE) ≫ m = s.inl\... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 216,
"column": 66
} | {
"line": 216,
"column": 70
} | {
"line": 216,
"column": 70
} | [
{
"pp": "case mpr.uniq.h₁.h₂\nC : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f c.inl iY fE\nH : IsPushout f c.inl iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (c.inr ≫ f... | [
"case mpr.uniq.h₁.h₂\nC : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f c.inl iY fE\nH : IsPushout f c.inl iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (c.inr ≫ fE) ≫ m = s.i... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 181,
"column": 87
} | {
"line": 216,
"column": 88
} | {
"line": 218,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f c.inl iY fE\n⊢ Nonempty (IsColimit (BinaryCofan.mk (c.inr ≫ fE) iY)) ↔ IsPushout f c.inl iY fE",
"ppTerm": "?m.66",
"assigned": true,
"usedConstan... | [] | by
constructor
· rintro ⟨h⟩
refine ⟨H, ⟨Limits.PushoutCocone.isColimitAux' _ ?_⟩⟩
intro s
dsimp
refine ⟨BinaryCofan.IsColimit.desc h (c.inr ≫ s.inr) s.inl,
BinaryCofan.IsColimit.inr_desc h _ _, ?_, ?_⟩
· apply BinaryCofan.IsColimit.hom_ext hc
· rw [← H.w_assoc]; erw [h.fac _ ⟨Walki... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Extensive | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 55
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case refine_3\nJ : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u''\ninst✝ : Category.{v'', u''} D\nX Y : C\nZ : TopCat\nf : Z ⟶ TopCat.of (PUnit.{u + 1} ⊕ PUnit.{u + 1})\nh₁ :\n Set.range\n ⇑(ConcreteCategory.hom\n (TopCat.pullbackFst f ((TopCa... | [] | · convert! Set.isCompl_range_inl_range_inr.preimage f | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 250,
"column": 4
} | {
"line": 253,
"column": 77
} | {
"line": 254,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh... | [
"C : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\... | have : cmp = (hc₂.coconePointUniqueUpToIso hc₄).hom := by
apply BinaryCofan.IsColimit.hom_ext hc₂
exacts [(hc₂.comp_coconePointUniqueUpToIso_hom hc₄ ⟨WalkingPair.left⟩).symm,
(hc₂.comp_coconePointUniqueUpToIso_hom hc₄ ⟨WalkingPair.right⟩).symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Extensive | {
"line": 410,
"column": 2
} | {
"line": 417,
"column": 65
} | {
"line": 419,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\nD : Type u''\ninst✝⁷ : Category.{v'', u''} D\nF : C ⥤ D\ninst✝⁶ : FinitaryExtensive D\ninst✝⁵ : HasFiniteCoproducts C\ninst✝⁴ : HasPullbacksOfInclusions C\ninst✝³ : PreservesPullbacksOfInclusions F\ninst✝² : ReflectsLimitsOfShape WalkingCospan F\ninst✝¹ : Preserv... | [] | constructor
intro X Y c hc
refine IsVanKampenColimit.of_iso ?_ (hc.uniqueUpToIso (coprodIsCoprod X Y)).symm
have (i : Discrete WalkingPair) (Z : C) (f : Z ⟶ X ⨿ Y) :
PreservesLimit (cospan f ((BinaryCofan.mk coprod.inl coprod.inr).ι.app i)) F := by
rcases i with ⟨_ | _⟩ <;> dsimp <;> infer_instance
refi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Extensive | {
"line": 410,
"column": 2
} | {
"line": 417,
"column": 65
} | {
"line": 419,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\nD : Type u''\ninst✝⁷ : Category.{v'', u''} D\nF : C ⥤ D\ninst✝⁶ : FinitaryExtensive D\ninst✝⁵ : HasFiniteCoproducts C\ninst✝⁴ : HasPullbacksOfInclusions C\ninst✝³ : PreservesPullbacksOfInclusions F\ninst✝² : ReflectsLimitsOfShape WalkingCospan F\ninst✝¹ : Preserv... | [] | constructor
intro X Y c hc
refine IsVanKampenColimit.of_iso ?_ (hc.uniqueUpToIso (coprodIsCoprod X Y)).symm
have (i : Discrete WalkingPair) (Z : C) (f : Z ⟶ X ⨿ Y) :
PreservesLimit (cospan f ((BinaryCofan.mk coprod.inl coprod.inr).ι.app i)) F := by
rcases i with ⟨_ | _⟩ <;> dsimp <;> infer_instance
refi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 8
} | {
"line": 391,
"column": 4
} | [
{
"pp": "J : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ninst... | [
"J : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ninst✝² : ∀ (X : ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Category.ModuleCat.Sheaf | {
"line": 240,
"column": 32
} | {
"line": 240,
"column": 35
} | {
"line": 240,
"column": 35
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR✝ : Sheaf J RingCat\nR : Cᵒᵖ ⥤ RingCat\nM₁ M₂ : PresheafOfModules R\nf : M₁ ⟶ M₂\nN : PresheafOfModules R\nhN : Presheaf.IsSheaf J N.presheaf\ninst✝² : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : IsLocallySurjective J f\ninst... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR✝ : Sheaf J RingCat\nR : Cᵒᵖ ⥤ RingCat\nM₁ M₂ : PresheafOfModules R\nf : M₁ ⟶ M₂\nN : PresheafOfModules R\nhN : Presheaf.IsSheaf J N.presheaf\ninst✝² : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : IsLocallySurjective J f\ninst✝ : IsLocall... | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 763,
"column": 39
} | {
"line": 763,
"column": 68
} | {
"line": 763,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Type u_3\nS B : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nf : (i : ι) → X i ⟶ S\nu : a.pt ⟶ S\nv : B ⟶ S\ns : (i : ι) → PullbackCone v (f i)\nhs : (i : ι) → IsLimit (s i)\nt : PullbackCone v u\nht : IsLimit t\nd : Cofan fun i ↦ (s i).pt\ne : d.pt ... | [] | by simp [hu, (s i).condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 51
} | {
"line": 70,
"column": 4
} | [
{
"pp": "case x.rj\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\n... | [] | exact Presheaf.equalizerSieve_mem J α _ _ hr₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 51
} | {
"line": 70,
"column": 4
} | [
{
"pp": "case x.rj\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\n... | [] | exact Presheaf.equalizerSieve_mem J α _ _ hr₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 51
} | {
"line": 70,
"column": 4
} | [
{
"pp": "case x.rj\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\n... | [] | exact Presheaf.equalizerSieve_mem J α _ _ hr₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 71,
"column": 2
} | {
"line": 74,
"column": 7
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nY :... | [] | · intro Z g hg
rw [← NatTrans.naturality_apply (D := Ab), ← NatTrans.naturality_apply (D := Ab)]
erw [M₀.map_smul, M₀.map_smul, hg.1, hg.2]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 827,
"column": 41
} | {
"line": 827,
"column": 70
} | {
"line": 827,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Type u_3\nS B : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nf : (i : ι) → X i ⟶ S\nu : a.pt ⟶ S\nv : B ⟶ S\ns : (i : ι) → PullbackCone (f i) v\nhs : (i : ι) → IsLimit (s i)\nt : PullbackCone u v\nht : IsLimit t\nd : Cofan fun i ↦ (s i).pt\ne : d.pt ... | [] | by simp [hu, (s i).condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 60,
"column": 4
} | {
"line": 63,
"column": 7
} | {
"line": 64,
"column": 2
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | [] | ext1
apply (toPresheaf _).map_injective
simp
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 60,
"column": 4
} | {
"line": 63,
"column": 7
} | {
"line": 64,
"column": 2
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | [] | ext1
apply (toPresheaf _).map_injective
simp
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 65,
"column": 4
} | {
"line": 68,
"column": 7
} | {
"line": 70,
"column": 0
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | [] | ext1
apply (toPresheaf _).map_injective
simp
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 65,
"column": 4
} | {
"line": 68,
"column": 7
} | {
"line": 70,
"column": 0
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | [] | ext1
apply (toPresheaf _).map_injective
simp
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 75
} | {
"line": 70,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nhX : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0\ni : ℕ\nhi : n ≤ i\nY : C\n⊢ Subsingleton (Ext X Y i)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"Categ... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nhX : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0\ni : ℕ\nhi : n ≤ i\nY : C\nthis : Subsingleton (Ext X Y i)\n⊢ Subsingleton (Ext X Y i)"
] | have : Subsingleton (Ext X Y i) := ⟨fun e₁ e₂ ↦ by simp only [hX i hi]⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 291,
"column": 77
} | {
"line": 291,
"column": 83
} | {
"line": 291,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasProjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_dec... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 291,
"column": 77
} | {
"line": 291,
"column": 83
} | {
"line": 291,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasProjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_dec... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 291,
"column": 77
} | {
"line": 291,
"column": 83
} | {
"line": 291,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nh : HasProjectiveDimensionLT X 0\n⊢ ⊥ < ↑0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instCompleteLinearOrderENat",
"of_dec... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 62
} | {
"line": 314,
"column": 4
} | [
{
"pp": "case bot\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : projectiveDimension X = ⊥\n⊢ ⊥ ≠ ⊤ ↔ ∃ n, HasProjectiveDimensionLE X n",
"ppTerm": "?bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot.instBoundedOrder",
"False",
"_private.M... | [
"case bot\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : projectiveDimension X = ⊥\n⊢ ∃ n, HasProjectiveDimensionLE X n"
] | simp only [ne_eq, bot_ne_top, not_false_eq_true, true_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 311,
"column": 2
} | {
"line": 325,
"column": 74
} | {
"line": 327,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nd : WithBot ℕ∞\nhd : projectiveDimension X = d\n⊢ d ≠ ⊤ ↔ ∃ n, HasProjectiveDimensionLE X n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Mathlib.Tactic.Push.not_exists._simp_1",... | [] | induction d with
| bot =>
simp only [ne_eq, bot_ne_top, not_false_eq_true, true_iff]
exact ⟨0, by simp [← projectiveDimension_le_iff, hd]⟩
| coe d =>
induction d with
| top =>
by_contra!
simp only [WithBot.coe_top, ne_eq, not_true_eq_false, false_and, true_and, false_or] at this
ob... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Limits.Preserves.SigmaConst | {
"line": 112,
"column": 6
} | {
"line": 114,
"column": 65
} | {
"line": 114,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nR : C\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝² : HasCoproduct fun x ↦ R\ninst✝¹ : HasCoproduct fun x ↦ R\ninst✝ : HasCoproduct fun x ↦ R\ns : Cofork (Sigma.map' f fun x ↦ 𝟙 R) 0\nm : (sigmaConstCokernelCofork R f).pt ⟶ s.pt\nhm... | [] | dsimp
ext ⟨b, hb⟩
rw [Sigma.ι_desc, ← hm, ι_sigmaConstCokernelCofork_π_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.SigmaConst | {
"line": 112,
"column": 6
} | {
"line": 114,
"column": 65
} | {
"line": 114,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nR : C\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝² : HasCoproduct fun x ↦ R\ninst✝¹ : HasCoproduct fun x ↦ R\ninst✝ : HasCoproduct fun x ↦ R\ns : Cofork (Sigma.map' f fun x ↦ 𝟙 R) 0\nm : (sigmaConstCokernelCofork R f).pt ⟶ s.pt\nhm... | [] | dsimp
ext ⟨b, hb⟩
rw [Sigma.ι_desc, ← hm, ι_sigmaConstCokernelCofork_π_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Opposites | {
"line": 1004,
"column": 16
} | {
"line": 1006,
"column": 69
} | {
"line": 1008,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ReflectsFiniteProducts F\nx✝ : ℕ\n⊢ ReflectsColimitsOfShape (Discrete (Fin x✝)) F.op",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.Lim... | [] | by
apply +allowSynthFailures reflectsColimitsOfShape_op
exact reflectsLimitsOfShape_of_equiv (Discrete.opposite _).symm _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 91,
"column": 20
} | {
"line": 91,
"column": 51
} | {
"line": 91,
"column": 52
} | [
{
"pp": "J : Type u₁\ninst✝ : Category.{v₁, u₁} J\nW✝ X✝ Y✝ Z✝ : Bicone J\nf : W✝ ⟶ X✝\ng : X✝ ⟶ Y✝\nh : Y✝ ⟶ Z✝\n⊢ (f ≫ g) ≫ h = f ≫ g ≫ h",
"ppTerm": "?m.392",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.ctorIdx",
"CategoryTheory.Bic... | [
"case left_id.left_id.left_id\nJ : Type u₁\ninst✝ : Category.{v₁, u₁} J\n⊢ (BiconeHom.left_id ≫ BiconeHom.left_id) ≫ BiconeHom.left_id =\n BiconeHom.left_id ≫ BiconeHom.left_id ≫ BiconeHom.left_id",
"case left_id.left_id.left\nJ : Type u₁\ninst✝ : Category.{v₁, u₁} J\nj✝ : J\n⊢ (BiconeHom.left_id ≫ BiconeHom.l... | cases f <;> cases g <;> cases h | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 144,
"column": 2
} | {
"line": 147,
"column": 33
} | {
"line": 149,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX : C\nS : Sieve X\n⊢ S ∈ J₁ X → J₁.close S = ⊤",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"CategoryTheory.CategoryStruct.t... | [] | · intro hS
rw [_root_.eq_top_iff]
intro Y f _
apply J₁.pullback_stable _ hS | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 280,
"column": 8
} | {
"line": 280,
"column": 25
} | {
"line": 280,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX Y : C\nS : Sieve X\nf : Y ⟶ X\nhS : (c X) S = ⊤\n⊢ Sieve.pullback f S ∈ {S | (c Y) S = ⊤... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX Y : C\nS : Sieve X\nf : Y ⟶ X\nhS : (c X) S = ⊤\n⊢ (c Y) (Sieve.pullback f S) = ⊤"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 25
} | {
"line": 283,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pull... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 283,
"column": 46
} | {
"line": 283,
"column": 57
} | {
"line": 283,
"column": 58
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pull... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 10
} | {
"line": 244,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCon... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type v₁\ninst✝² : SmallCategory J\ninst✝¹ : FinCategory J\nF : C ⥤ D\ninst✝ : RepresentablyFlat F\nK : J ⥤ C\nc : Cone K\nhc : IsLimit c\ns : Cone (K ⋙ F)\nf₁ f₂ : s.pt ⟶ F.obj c.pt\nh₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 317,
"column": 26
} | {
"line": 317,
"column": 52
} | {
"line": 317,
"column": 53
} | [
{
"pp": "C D : Type u₁\ninst✝³ : SmallCategory C\ninst✝² : SmallCategory D\nE : Type u₂\ninst✝¹ : Category.{u₁, u₂} E\nF : C ⥤ D\nX : D\ninst✝ : ∀ (X : D), HasColimitsOfShape (CostructuredArrow F X) E\nG₁ G₂ : C ⥤ E\nφ : G₁ ⟶ G₂\nT : CostructuredArrow F X\nh₂ :\n φ.app T.left ≫ (F.leftKanExtensionUnit G₂).app ... | [
"C D : Type u₁\ninst✝³ : SmallCategory C\ninst✝² : SmallCategory D\nE : Type u₂\ninst✝¹ : Category.{u₁, u₂} E\nF : C ⥤ D\nX : D\ninst✝ : ∀ (X : D), HasColimitsOfShape (CostructuredArrow F X) E\nG₁ G₂ : C ⥤ E\nφ : G₁ ⟶ G₂\nT : CostructuredArrow F X\nh₂ :\n φ.app T.left ≫ (F.leftKanExtensionUnit G₂).app T.left =\n ... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 654,
"column": 32
} | {
"line": 654,
"column": 36
} | {
"line": 654,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\ninst✝ : J.RespectsIso\nS : C\nR T : Presieve S\nhR : R ∈ J.coverings S\nYR : ⦃Z : C⦄ → (g : Z ⟶ S) → R g → C\neR : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : R g) → YR g a ≅ Z\nhTeg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : R g), T ((eR g a).hom ≫ g)\nYT : ⦃Z : C⦄ → (g :... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\ninst✝ : J.RespectsIso\nS : C\nR T : Presieve S\nhR : R ∈ J.coverings S\nYR : ⦃Z : C⦄ → (g : Z ⟶ S) → R g → C\neR : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : R g) → YR g a ≅ Z\nhTeg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : R g), T ((eR g a).hom ≫ g)\nYT : ⦃Z : C⦄ → (g : Z ⟶ S) → T ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 293,
"column": 38
} | {
"line": 293,
"column": 64
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case mpr.transitive\nC : Type ?u.2\nD : Type ?u.4\ninst✝¹ : Category.{v_1, ?u.2} C\ninst✝ : Category.{v_2, ?u.4} D\nK : Coverage C\nJ : GrothendieckTopology C\nH : K ≤ J.toCoverage\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 293,
"column": 38
} | {
"line": 293,
"column": 64
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case mpr.transitive\nC : Type ?u.2\nD : Type ?u.4\ninst✝¹ : Category.{v_1, ?u.2} C\ninst✝ : Category.{v_2, ?u.4} D\nK : Coverage C\nJ : GrothendieckTopology C\nH : K ≤ J.toCoverage\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 293,
"column": 38
} | {
"line": 293,
"column": 64
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case mpr.transitive\nC : Type ?u.2\nD : Type ?u.4\ninst✝¹ : Category.{v_1, ?u.2} C\ninst✝ : Category.{v_2, ?u.4} D\nK : Coverage C\nJ : GrothendieckTopology C\nH : K ≤ J.toCoverage\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 62
} | {
"line": 308,
"column": 2
} | [
{
"pp": "case a.transitive\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nK : Coverage C\nJ : GrothendieckTopology C\nhJ : J ∈ {J | K ≤ J.toCoverage}\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)\nH1 : R ∈ J X\nH2 : ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 62
} | {
"line": 308,
"column": 2
} | [
{
"pp": "case a.transitive\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nK : Coverage C\nJ : GrothendieckTopology C\nhJ : J ∈ {J | K ≤ J.toCoverage}\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)\nH1 : R ∈ J X\nH2 : ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 62
} | {
"line": 308,
"column": 2
} | [
{
"pp": "case a.transitive\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nK : Coverage C\nJ : GrothendieckTopology C\nhJ : J ∈ {J | K ≤ J.toCoverage}\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝¹ : K.Saturate X R\na✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)\nH1 : R ∈ J X\nH2 : ... | [] | exact J.transitive H1 _ H2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 356,
"column": 19
} | {
"line": 356,
"column": 49
} | {
"line": 357,
"column": 4
} | [
{
"pp": "case refine_1.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nJ : Pretopology C\nT : C\nS✝ : Sieve T\nX : C\nS : Presieve X\nhS : S ∈ J.toCoverage.coverings X\n⊢ ∃ R ∈ J.coverings X, R ≤ (Sieve.generate S).arrows",
"ppTerm": "?refine_1.of",
"assigned": true,
"usedC... | [] | use S, hS, Sieve.le_generate S | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 356,
"column": 19
} | {
"line": 356,
"column": 49
} | {
"line": 357,
"column": 4
} | [
{
"pp": "case refine_1.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nJ : Pretopology C\nT : C\nS✝ : Sieve T\nX : C\nS : Presieve X\nhS : S ∈ J.toCoverage.coverings X\n⊢ ∃ R ∈ J.coverings X, R ≤ (Sieve.generate S).arrows",
"ppTerm": "?refine_1.of",
"assigned": true,
"usedC... | [] | use S, hS, Sieve.le_generate S | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 356,
"column": 19
} | {
"line": 356,
"column": 49
} | {
"line": 357,
"column": 4
} | [
{
"pp": "case refine_1.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nJ : Pretopology C\nT : C\nS✝ : Sieve T\nX : C\nS : Presieve X\nhS : S ∈ J.toCoverage.coverings X\n⊢ ∃ R ∈ J.coverings X, R ≤ (Sieve.generate S).arrows",
"ppTerm": "?refine_1.of",
"assigned": true,
"usedC... | [] | use S, hS, Sieve.le_generate S | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 66
} | {
"line": 789,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : Precoverage C\nS T : C\ninst✝¹ : J.IsStableUnderComposition\ninst✝ : J.HasIsos\nX Y : C\nf : X ⟶ Y\nhf : Presieve.singleton f ∈ J.coverings Y\nE : J.ZeroHypercover X\n⊢ (PreZeroHypercover.pushforward f E.toPreZeroHypercover).presieve₀ ∈ J.coverings Y",
"p... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : Precoverage C\nS T : C\ninst✝¹ : J.IsStableUnderComposition\ninst✝ : J.HasIsos\nX Y : C\nf : X ⟶ Y\nhf : Presieve.singleton f ∈ J.coverings Y\nE : J.ZeroHypercover X\n⊢ ((PreZeroHypercover.singleton f).bind fun x ↦ E.toPreZeroHypercover).presieve₀ ∈ J.coverings Y"
] | rw [PreZeroHypercover.mem_iff_of_iso (E.pushforwardIsoBind _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 76,
"column": 70
} | {
"line": 79,
"column": 13
} | {
"line": 81,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\ni₁ i₂ : E.I₀\nW : C\np₁ : W ⟶ E.X i₁\np₂ : W ⟶ E.X i₂\nT : C\nf : T ⟶ W\n⊢ Sieve.pullback f (E.sieve₁ p₁ p₂) = E.sieve₁ (f ≫ p₁) (f ≫ p₂)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Sie... | [] | by
refine le_antisymm ?_ ?_ <;>
· intro Z g ⟨k, u, hu₁, hu₂⟩
cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 35
} | {
"line": 110,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\nh✝ : ∀ (B : D), ∃ X f, Sieve.generate (Presieve.singleton f) ∈ K B\nB : D\nX : C\nf : G.obj X ⟶ B\nh : Sieve.generate (Presieve.singleton f) ∈ K B\n⊢ Sieve.coverByImage G B ... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\nh✝ : ∀ (B : D), ∃ X f, Sieve.generate (Presieve.singleton f) ∈ K B\nB : D\nX : C\nf : G.obj X ⟶ B\nh : Sieve.generate (Presieve.singleton f) ∈ K B\n⊢ Sieve.generate (Presieve.singleton ... | refine K.superset_covering ?_ h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 8
} | {
"line": 323,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝³ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense ... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝³ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 6
} | {
"line": 398,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\nX : Cᵒᵖ\nU : ... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\nX : Cᵒᵖ\nU : Aᵒᵖ\nx✝ : (y... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 6
} | {
"line": 428,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.obj\nX : Dᵒᵖ\nU : Aᵒᵖ\nx✝ : (yon... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.obj\nX : Dᵒᵖ\nU : Aᵒᵖ\nx✝ : (yoneda.obj (ℱ.o... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 6
} | {
"line": 345,
"column": 2
} | [
{
"pp": "J✝ : Type u₁\nK : Type u₂\ninst✝⁴ : SmallCategory J✝\ninst✝³ : Category.{v₂, u₂} K\ninst✝² : Small.{v, u₂} K\ninst✝¹ : FinCategory J✝\nF✝ : J✝ × K ⥤ Type v\ninst✝ : IsFiltered K\nJ : Type v₂\n𝒥✝ : SmallCategory J\nx✝ : FinCategory J\nF : J ⥤ K ⥤ Type v\nc : Cone F\nhc : IsLimit c\n⊢ limit.cone (F ⋙ co... | [
"J✝ : Type u₁\nK : Type u₂\ninst✝⁴ : SmallCategory J✝\ninst✝³ : Category.{v₂, u₂} K\ninst✝² : Small.{v, u₂} K\ninst✝¹ : FinCategory J✝\nF✝ : J✝ × K ⥤ Type v\ninst✝ : IsFiltered K\nJ : Type v₂\n𝒥✝ : SmallCategory J\nx✝ : FinCategory J\nF : J ⥤ K ⥤ Type v\nc : Cone F\nhc : IsLimit c\n⊢ colim.mapCone c ≅ limit.cone (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 39
} | {
"line": 151,
"column": 4
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝³ : Category.{u, v} A\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nX : ... | [] | exact T.val.downward_closed hg g' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 29
} | {
"line": 180,
"column": 4
} | [
{
"pp": "C : Type u_3\nD : Type u_4\ninst✝⁵ : Category.{v_3, u_3} C\ninst✝⁴ : Category.{v_4, u_4} D\nF : C ⥤ D\nK : Precoverage D\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nH : ∀ {S : C} {R : Presieve (F.obj S)}, R ∈ K.coverings (F.obj S... | [
"C : Type u_3\nD : Type u_4\ninst✝⁵ : Category.{v_3, u_3} C\ninst✝⁴ : Category.{v_4, u_4} D\nF : C ⥤ D\nK : Precoverage D\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nH : ∀ {S : C} {R : Presieve (F.obj S)}, R ∈ K.coverings (F.obj S) → map F (f... | obtain ⟨R, hR, hle⟩ := hT | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 29
} | {
"line": 197,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u_3\nD : Type u_4\ninst✝⁷ : Category.{v_3, u_3} C\ninst✝⁶ : Category.{v_4, u_4} D\nF : C ⥤ D\nK : Precoverage D\ninst✝⁵ : K.HasIsos\ninst✝⁴ : K.IsStableUnderBaseChange\ninst✝³ : K.IsStableUnderComposition\ninst✝² : K.HasPullbacks\ninst✝¹ : F.Faithful\ninst✝ : F.Full\nH : ∀ {S : ... | [
"case refine_2\nC : Type u_3\nD : Type u_4\ninst✝⁷ : Category.{v_3, u_3} C\ninst✝⁶ : Category.{v_4, u_4} D\nF : C ⥤ D\nK : Precoverage D\ninst✝⁵ : K.HasIsos\ninst✝⁴ : K.IsStableUnderBaseChange\ninst✝³ : K.IsStableUnderComposition\ninst✝² : K.HasPullbacks\ninst✝¹ : F.Faithful\ninst✝ : F.Full\nH : ∀ {S : C} {R : Pres... | obtain ⟨R, hR, hle⟩ := hT | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Over | {
"line": 464,
"column": 48
} | {
"line": 471,
"column": 11
} | {
"line": 473,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y : C\nf : X ⟶ Y\n⊢ (J.overMapPullbackComp A (𝟙 X) f).inv ≫\n (J.overMapPullback A f).whiskerLeft (J.overMapPullbackId A X).hom ≫ (J.overMapPullback A f).rightUnitor.hom =\n (J.overM... | [] | by
ext
dsimp
simp only [overMapPullbackComp_inv_app_hom_app, overMapPullbackId_hom_app_hom_app,
Functor.sheafPushforwardContinuous_obj_obj_map, Quiver.Hom.unop_op,
comp_id, ← Functor.map_comp, ← op_comp]
congr
cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Simple | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 95
} | {
"line": 116,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\nthis : IsIso (image.ι f)\n⊢ Epi (factorThruImage f ≫ image.ι f)"
] | haveI : IsIso (image.ι f) := isIso_of_mono_of_nonzero fun h => w (eq_zero_of_image_eq_zero h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 457,
"column": 46
} | {
"line": 463,
"column": 23
} | {
"line": 465,
"column": 0
} | [
{
"pp": "X : TopCat\nU : Opens ↑X\n⊢ Set.range ⇑(ConcreteCategory.hom U.inclusion') = ↑U",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.ext",
"TopologicalSpace.Opens.instPartialOrder",
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCategory",
"Continu... | [] | by
ext x
constructor
· rintro ⟨x, rfl⟩
exact x.2
· intro h
exact ⟨⟨x, h⟩, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Lattice | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 46
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nthis : ∀ (x y : α), HasColimit (pair x y)\n⊢ HasBinaryCoproducts α",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasBinaryCoprodu... | [] | apply hasBinaryCoproducts_of_hasColimit_pair | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous | {
"line": 425,
"column": 6
} | {
"line": 425,
"column": 67
} | {
"line": 426,
"column": 6
} | [
{
"pp": "case refine_1.refine_1\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nD' : Type u₃\ninst✝² : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝¹ : Category.{v₄, u₄} D''\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nS : Sheaf J RingCat\nR : Sheaf K R... | [
"case refine_1.refine_1\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nD' : Type u₃\ninst✝² : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝¹ : Category.{v₄, u₄} D''\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nS : Sheaf J RingCat\nR : Sheaf K RingCat\ninst... | rw [← RingCat.hom_comp, ← RingCat.hom_comp, φ.hom.naturality] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent | {
"line": 281,
"column": 32
} | {
"line": 283,
"column": 48
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝² : ∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat\ninst✝¹ : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat\nM : SheafOfModules R\ninst✝ : M.IsFinitePresentation\n⊢ ∃ σ, σ.IsFiniteType",
... | [] | by
obtain ⟨σ, _⟩ := IsFinitePresentation.exists_quasicoherentData M
exact ⟨σ.localGeneratorsData, inferInstance⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 8
} | {
"line": 217,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝³ : ConcreteCategory C FC\ninst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C\ninst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms\ninst✝ : PreservesLimitsOfSiz... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝³ : ConcreteCategory C FC\ninst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C\ninst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms\ninst✝ : PreservesLimitsOfSize.{x, x, v_1... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 340,
"column": 2
} | {
"line": 370,
"column": 24
} | {
"line": 372,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\n⊢ s.pt ⟶ F.obj.obj (op (U ⊔ V))",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Set.ext",
"CategoryTheory.Fu... | [] | let ι : ULift.{w} WalkingPair → Opens X := fun j => WalkingPair.casesOn j.down U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 340,
"column": 2
} | {
"line": 370,
"column": 24
} | {
"line": 372,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\n⊢ s.pt ⟶ F.obj.obj (op (U ⊔ V))",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Set.ext",
"CategoryTheory.Fu... | [] | let ι : ULift.{w} WalkingPair → Opens X := fun j => WalkingPair.casesOn j.down U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Stalks | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 6
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case e'_5\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nhf : IsInducing ⇑(ConcreteCategory.hom f)\nF : Presheaf C X\nx : ↑X\nthis : (OpenNhds.map f x).Initial\ne_3✝ :\n ((pushforward C f).obj F).stalk ((ConcreteCategory.hom f) x) =\n colimit ((OpenNhds.map... | [
"case e'_5\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nhf : IsInducing ⇑(ConcreteCategory.hom f)\nF : Presheaf C X\nx : ↑X\nthis : (OpenNhds.map f x).Initial\ne_3✝ :\n ((pushforward C f).obj F).stalk ((ConcreteCategory.hom f) x) =\n colimit ((OpenNhds.map f x).op ⋙ (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Sheaves.Stalks | {
"line": 496,
"column": 2
} | {
"line": 502,
"column": 58
} | {
"line": 504,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF G : Presheaf... | [] | intro s t hst
obtain ⟨U₁, hxU₁, hU₁, s, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x s
obtain ⟨U₂, hxU₂, hU₂, t, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x t
rw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply] at hst
obtain ⟨W, hxW, hW, iWU₁, iWU₂, heq⟩ := germ_eq_of_isBasis hB _ _ hxU₁ hxU₂ hst
s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Stalks | {
"line": 496,
"column": 2
} | {
"line": 502,
"column": 58
} | {
"line": 504,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF G : Presheaf... | [] | intro s t hst
obtain ⟨U₁, hxU₁, hU₁, s, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x s
obtain ⟨U₂, hxU₂, hU₂, t, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x t
rw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply] at hst
obtain ⟨W, hxW, hW, iWU₁, iWU₂, heq⟩ := germ_eq_of_isBasis hB _ _ hxU₁ hxU₂ hst
s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Stalk | {
"line": 48,
"column": 2
} | {
"line": 53,
"column": 71
} | {
"line": 54,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r... | [
"case refine_2\nC : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (Concrete... | · rintro ⟨U, a⟩ ⟨V₁, b₁⟩ ⟨V₂, b₂⟩ ⟨f : V₁ ⟶ V₂, rfl : b₂ = M.map _ b₁⟩
obtain ⟨s, α, β, h₁, h₂⟩ :=
bowtie (leftToMax U V₁) (leftToMax U V₂)
(rightToMax U V₁) (f ≫ rightToMax U V₂)
refine Functor.ιColimitType_eq_of_map_eq_map _ _ _ α β ?_
simp [*, ← R.map_comp_apply, ← M.map_comp_apply, -Functo... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 41
} | {
"line": 639,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\ninst✝³ : Semiring R₁\nM₁ : Type u_4\ninst✝² : TopologicalSpace M₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nT : (M₁ →L[R₁] M₁)ˣ\n⊢ IsHomeomorph ⇑↑T",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"ContinuousLinearMap.homeomorphOfUnit",
"Homeomorp... | [] | exact (homeomorphOfUnit T).isHomeomorph | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Preadditive.Injective.Preserves | {
"line": 63,
"column": 25
} | {
"line": 66,
"column": 85
} | {
"line": 68,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : EnoughInjectives D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : G.PreservesInjectiveObjects\nX Y : C\nf : X ⟶ Y\nx✝ : Mono f\n⊢ Mono (F.map f)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants"... | [] | by
suffices ∃ h, F.map f ≫ h = Injective.ι (F.obj X) from mono_of_mono_fac this.choose_spec
exact ⟨F.map (Injective.factorThru (adj.unit.app X ≫ G.map (Injective.ι _)) f) ≫
adj.counit.app (Injective.under (F.obj X)), by simp [← Functor.map_comp_assoc]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 649,
"column": 56
} | {
"line": 649,
"column": 82
} | {
"line": 649,
"column": 82
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomIn... | [] | simpa using congr($(h₁) x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 649,
"column": 56
} | {
"line": 649,
"column": 82
} | {
"line": 649,
"column": 82
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomIn... | [] | simpa using congr($(h₁) x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 649,
"column": 56
} | {
"line": 649,
"column": 82
} | {
"line": 649,
"column": 82
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomIn... | [] | simpa using congr($(h₁) x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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