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.Sites.Descent.Precoverage | {
"line": 222,
"column": 49
} | {
"line": 222,
"column": 60
} | [
{
"pp": "case h\nC : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 243,
"column": 12
} | {
"line": 243,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 252,
"column": 10
} | {
"line": 252,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\nf g : E.Hom F\ni✝ j✝ : (cylinder f g).I₀\nk : (cylinder f g).I₁ i✝ j✝\n⊢ pullback.snd\n (pullback.map (cylinderf f g i✝.snd) (cylinderf f g j✝.snd) (E.f i✝.fst) (E.f j✝.fst)\n ... | have : E.p₁ k.down = pullback.lift _ _ (E.w k.down) ≫ pullback.fst _ _ := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 257,
"column": 12
} | {
"line": 257,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 258,
"column": 2
} | {
"line": 260,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nsq₃ : (i₁ i₂ i₃ : ι) → ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)\nD : F.DescentData' sq sq₃\ni₁ i₂ : ι\n⊢ IsIso (D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 268,
"column": 75
} | {
"line": 291,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | by
rw [← cancel_mono (D₂.hom q f₂ f₁), Category.assoc,
Category.assoc, DescentData.hom_comp, D₂.hom_self _ _ hf₁, Category.comp_id]
have H : (Sieve.overEquiv (Over.mk f₁)).symm
(Sieve.pullback q (Sieve.ofArrows X' f')) ∈ J.over _ _ := by
rw [J.mem_over_iff, Equiv.apply_symm_apply]
exact J.pullback... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 395,
"column": 4
} | {
"line": 396,
"column": 11
} | [
{
"pp": "case hF\nC : Type u\ninst✝⁴ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\ninst✝³ : HasPullbacks C\nJ : Precoverage C\ninst✝² : J.HasIsos\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : J.IsStableUnderComposition\nhF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R\nS : C\nM N : ↑(F.obj { as := o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 131,
"column": 37
} | {
"line": 131,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 134,
"column": 12
} | {
"line": 134,
"column": 23
} | [
{
"pp": "case left.right\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 136,
"column": 12
} | {
"line": 136,
"column": 23
} | [
{
"pp": "case right.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 137,
"column": 37
} | {
"line": 137,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Monoidal | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 17
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nA : Type u₃\ninst✝³ : Category.{v₃, u₃} A\ninst✝² : MonoidalCategory A\ninst✝¹ : MonoidalClosed A\nM : A\nF G : Cᵒᵖ ⥤ A\ninst✝ : HasFunctorEnrichedHom A F G\nX : C\ng : (presheafHom (F ⊗ (Functor.const Cᵒᵖ).obj M) G).obj (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1 | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ GrpCat\nI : Type w'\nU : I → C\nγ : OneCocycle G U\ni : I\nT : C\na : T ⟶ U i\n⊢ γ.ev i i a a = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 28
} | [
{
"pp": "case h.toFun.h.w.h\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nY : Over (unop (op X))\n⊢ ((ConcreteCategory.hom (↾(Over.map (𝟙 (op X)).u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 30
} | [
{
"pp": "case h.toFun.h.w.h\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ Y\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nW : Over (unop (op Z))\n⊢ ((ConcreteCategory.hom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 182,
"column": 59
} | {
"line": 182,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY₁ Y₂ : Over X\nφ : Y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 189,
"column": 43
} | {
"line": 189,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y ⟶ X\nhg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\ninst✝ : IsCofiltered N\nU : N\nX : C\nf₁ f₂ : p.obj U ⟶ X\nhf : fiberMk f₁ = fiberMk f₂\nV : Nᵒᵖ\ng : op U ⟶ V\nhg :\n (hom ((p.op ⋙ shrinkYoneda.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU V : N\ng : V ⟶ U\n⊢ fiberMk (p.map g) = fiberMk (𝟙 (p.obj U))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 115,
"column": 47
} | {
"line": 115,
"column": 58
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\nJ : GrothendieckTopology C\ninst✝ : IsCofiltered N\nX : C\nV U : N\nf : p.obj U ⟶ X\nφ₁ : ((functor p).obj V).fst ⟶ ⟨X, fiberMk f⟩.fst\nhφ₁ : (hom ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 170,
"column": 2
} | {
"line": 179,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (... | refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩
· let ι' : Type _ := Σ (Φ : P.FullSubcategory), Φ.obj.fiber.obj X
choose i y hy using fun (j : ι') ↦ hf j.1 j.2
refine J.superset_covering (S := S... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 170,
"column": 2
} | {
"line": 179,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\ninst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (... | refine ⟨fun hf Φ x ↦ ?_, fun hf ↦ ?_⟩
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
exact ⟨i, Φ.obj.fiber.map p z, by simp⟩
· let ι' : Type _ := Σ (Φ : P.FullSubcategory), Φ.obj.fiber.obj X
choose i y hy using fun (j : ι') ↦ hf j.1 j.2
refine J.superset_covering (S := S... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Point.Skyscraper | {
"line": 154,
"column": 35
} | {
"line": 154,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasProducts A\nM : A\nX : C\nR : Sieve X\nhR : R ∈ J X\ns : Cone (R.arrows.diagram.op ⋙ Φ.skyscraperPresheaf M)\nl : Φ.fiber.obj X → (s.pt ⟶ M)\nhl :\n ∀ (x : Φ.fiber.obj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Map | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nJ : GrothendieckTopology C\nΦ : J.Point\nF : C ⥤ D\nK : GrothendieckTopology D\ninst✝³ : F.IsCocontinuous J K\ninst✝² : LocallySmall.{w, v', u'} D\nA : Type u''\ninst✝¹ : Category.{v'', u''} A\ninst✝ : HasColimitsOfSize.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 15
} | [
{
"pp": "case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 15
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Skyscraper | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasProducts A\nM : A\nX : C\nR : Sieve X\nhR : R ∈ J X\ns : Cone (R.arrows.diagram.op ⋙ Φ.skyscraperPresheaf M)\nx : Φ.fiber.obj X\nY₁ : C\nf₁ : Y₁ ⟶ X\nhf₁ : R.arrows f₁\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Skyscraper | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 17
} | [
{
"pp": "case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasProducts A\nM : A\nX : C\nR : Sieve X\nhR : R ∈ J X\ns : Cone (R.arrows.diagram.op ⋙ Φ.skyscraperPresheaf M)\nj : R.arrows.categoryᵒᵖ\ny : Φ.fiber.obj (unop j).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Skyscraper | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 58
} | [
{
"pp": "case h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasProducts A\nM : A\nX : C\nR : Sieve X\nhR : R ∈ J X\ns : Cone (R.arrows.diagram.op ⋙ Φ.skyscraperPresheaf M)\nm : s.pt ⟶ ((Φ.skyscraperPresheaf M).mapCone R.arr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Presheaf | {
"line": 41,
"column": 29
} | {
"line": 41,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX U : C\nR : Sieve U\nhR : R ∈ ⊥ U\nx : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U\n⊢ R = ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Presheaf | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 11
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nS : Sieve X\nhS :\n ∀ (Φ : (pointsBot C).FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x\nY : C\na : Y ⟶ X\nha : S.arrows a\nb : X ⟶ Y\nhb :\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver | {
"line": 43,
"column": 26
} | {
"line": 43,
"column": 71
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nb₀✝ b₁✝ b₂✝ b₃✝ : Cᵒᵖ\nf : b₀✝ ⟶ b₁✝\ng : b₁✝ ⟶ b₂✝\nh : b₂✝ ⟶ b₃✝\n⊢ (((fun {b₀ b₁ b₂} f g ↦ Cat.Hom.isoMk (J.overMapPullbackComp A g.unop f.unop).symm) (f ≫ g) h).hom ≫\n Bicat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver | {
"line": 45,
"column": 22
} | {
"line": 45,
"column": 67
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nb₀✝ b₁✝ : Cᵒᵖ\nf : b₀✝ ⟶ b₁✝\n⊢ (((fun {b₀ b₁ b₂} f g ↦ Cat.Hom.isoMk (J.overMapPullbackComp A g.unop f.unop).symm) (𝟙 b₀✝) f).hom ≫\n Bicategory.whiskerRight ((fun X ↦ Cat.Hom.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.RegularEpi | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 38
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{u_4, u_2} D\nJ : GrothendieckTopology C\ninst✝⁴ : HasPullbacks D\ninst✝³ : HasPushouts D\ninst✝² : IsRegularEpiCategory D\nh : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom\ninst✝¹ : HasShe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver | {
"line": 47,
"column": 22
} | {
"line": 47,
"column": 67
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nb₀✝ b₁✝ : Cᵒᵖ\nf : b₀✝ ⟶ b₁✝\n⊢ (((fun {b₀ b₁ b₂} f g ↦ Cat.Hom.isoMk (J.overMapPullbackComp A g.unop f.unop).symm) f (𝟙 b₁✝)).hom ≫\n Bicategory.whiskerLeft ((fun {b b'} f ↦ (J... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Types | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 15
} | [
{
"pp": "case h.a\nX : Type u\n⊢ typesGlue (yoneda.obj X) ⋯ X ⇑(ConcreteCategory.hom (𝟙 X ≫ ↾fun x ↦ ↾fun x_1 ↦ x)) = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ObjectProperty.FullSubcategory.mk",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.CategoryStruct.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Finite | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 43
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\n⊢ (Subfunctor.range f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Finite | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\ninst✝ : Epi f\n⊢ PresheafIsGeneratedBy F' fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)",
"usedConstants": [
"Oppos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Subobject | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 17
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nA B : Subfunctor F\nh :\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ } A ≤\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ }... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nX : C\nf : X ⟶ Ω\n⊢ h.homEquiv f = (Subobject.pullback f).obj h.Ω₀",
"usedConstants": [
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 13
} | [
{
"pp": "case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nU X : C\nm : U ⟶ X\ninst✝ : Mono m\nχ' : X ⟶ Ω\nπ : U ⟶ underlying.obj h.Ω₀\nsq : IsPullback m π χ' h.Ω₀.arrow\n⊢ (Subobject.pullback χ').obj h.Ω₀ = Subobject.mk m",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 530,
"column": 29
} | {
"line": 530,
"column": 40
} | [
{
"pp": "C✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nX : C\nπ' : X ⟶ underlying.obj h.Ω₀\ns : PullbackCone (π' ≫ h.Ω₀.arrow) h.Ω₀.arrow\nm : s.pt ⟶ X\nhm : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ π' = s.snd\n⊢ m = (fun s ↦ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 452,
"column": 8
} | {
"line": 454,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nM N : ↑(F.obj { as := op S })\ng : (F.toDescentData f).obj M ⟶ (F.toDescentData f).obj N\ni₁ i₂ : ι\nZ : Over S\nf₁ : Z ⟶ (fun i ↦ Over.mk (f i)) i₁\nf₂ : Z ⟶ (fun i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 678,
"column": 4
} | {
"line": 678,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n𝒞 : Classifier C\nΩ₀ Ω : C\neΩ : 𝒞.Ω ≅ Ω\neΩ₀ : 𝒞.Ω₀ ≅ Ω₀\nfrom' : (C_1 : C) → C_1 ⟶ Ω₀\nt : Ω₀ ⟶ Ω\nht : t = eΩ₀.inv ≫ 𝒞.truth ≫ eΩ.hom\nF G : C\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ Ω₀\nχ' : G ⟶ Ω\nhχ' : IsPullback m χ₀' χ' t\nthis : χ' ≫ eΩ.inv = 𝒞.χ m\n⊢ χ' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\n𝒞₁ : Classifier C\ne : C ≌ D\nF G : D\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ e.functor.obj 𝒞₁.Ω₀\nχ' : G ⟶ e.functor.obj 𝒞₁.Ω\nhχ' : IsPullback m χ₀' χ' (e.functor.map 𝒞₁.truth)\nthis : e.inverse.map χ' ≫ e.unitInv.app... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 567,
"column": 45
} | {
"line": 567,
"column": 56
} | [
{
"pp": "case h\nC : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nM N : ↑(F.obj { as := op S })\nX : C\ng : X ⟶ S\nf : Over.mk g ⟶ Over.mk (𝟙 S)\nhf : R.arrows (Over.Hom.left f)\n⊢ Over.Hom.left f = Over.Hom.left (Over.homMk g ⋯)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 584,
"column": 67
} | {
"line": 584,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 591,
"column": 4
} | {
"line": 591,
"column": 21
} | [
{
"pp": "case h.e'_5.h.e'_4.refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh✝ :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\nS : C\na : S ⟶ S₀\nR : Sieve (Over.mk a)\nhF :\n ∀ ⦃S₀_1 : C⦄ (M N : ↑(F.obj { as := op S₀_1 })) (a_1 : (Over.mk a).left ⟶ S₀_1),\n Presieve.IsSheafFor (F.presheafHom M N)\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nX : Cᵒᵖ\n⊢ ∀ (x₁ y₁ : (fun X ↦ X) (F.obj X)),\n (ConcreteCategory.hom (m.app X)) x₁ = (ConcreteCategory.hom (m.app X)) y₁ ∧\n (ConcreteCategory.hom (((Functor.isTerminalConst Cᵒᵖ Type... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 25
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nX : Cᵒᵖ\np : G.obj X\nhp :\n { arrows := fun Y f ↦ ∃ a, (ConcreteCategory.hom (G.map f.op)) p = (ConcreteCategory.hom (m.app (Opposite.op Y))) a,\n downward_closed := ⋯ } =\n ⊤\n⊢ ∃ x₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | [
{
"pp": "case w.h.h.toFun.h.mpr\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nχ' : G ⟶ Functor.sieves C\nX : Cᵒᵖ\nx : G.obj X\nh₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ ↾fun x_1 ↦ ⊤\nh₂ : ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x), (ConcreteCategory.hom (m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 140,
"column": 2
} | {
"line": 150,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nhF : Presieve.IsSheaf J F\nhG : Presieve.IsSeparated J G\nX : Cᵒᵖ\nx : G.obj X\n⊢ J.IsClosed ((ConcreteCategory.hom ((χ m).app X)) x)",
"usedConstants": [
"CategoryTheory... | intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturality_app... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 140,
"column": 2
} | {
"line": 150,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nhF : Presieve.IsSheaf J F\nhG : Presieve.IsSeparated J G\nX : Cᵒᵖ\nx : G.obj X\n⊢ J.IsClosed ((ConcreteCategory.hom ((χ m).app X)) x)",
"usedConstants": [
"CategoryTheory... | intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturality_app... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 196,
"column": 49
} | {
"line": 196,
"column": 60
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\n⊢ (m.hom ≫ Subfunctor.lift (Presheaf.χ m.hom) ⋯) ≫ (closedSieves J).ι =\n (((isTerminalTerminal J Types.isTerminalPUnit).from F).hom ≫ Subfunctor.lift (Presheaf.truth C) ⋯) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSieves J).ι) (closedSiev... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 13
} | [
{
"pp": "case h.hχ'.refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type (max u v))\nm : F ⟶ G\ninst✝ : Mono m\nχ' : G ⟶ Ω J\nhχ' : IsPullback m ((isTerminalTerminal J Types.isTerminalPUnit).from F) χ' (truth J)\npb : IsPullback (𝟙 G.obj) χ'.hom (χ'.hom ≫ (closedSie... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 34
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 85,
"column": 42
} | {
"line": 85,
"column": 93
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nr : Retract X Y\nhY : P.triangEnvelope Y\n⊢ P.triangEnvelope X",
"usedCon... | rw [prop_triangEnvelope_iff] at hY ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Opposite.Functor | {
"line": 132,
"column": 42
} | {
"line": 137,
"column": 6
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : HasShift C ℤ\ninst✝¹ : HasShift D ℤ\nF : C ⥤ D\ninst✝ : F.CommShift ℤ\nX : Cᵒᵖ\nn : ℤ\n⊢ F.map ((opShiftFunctorEquivalence C n).unitIso.inv.app X).unop =\n ((opShiftFunctorEquivalence D n).unitIso.in... | by
rw [← cancel_mono (F.map ((opShiftFunctorEquivalence C n).unitIso.hom.app X).unop),
← F.map_comp, ← unop_comp, Iso.hom_inv_id_app,
map_opShiftFunctorEquivalence_unitIso_hom_app_unop, assoc, assoc,
Iso.inv_hom_id_app_assoc, ← Functor.map_comp_assoc, ← unop_comp]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 116,
"column": 44
} | {
"line": 116,
"column": 88
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁶ : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Adjunction | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 40
} | [
{
"pp": "case right\nC : Type u_1\nD : Type u_2\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasZeroObject D\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : Preadditive D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : HasShift D ℤ\ninst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 119,
"column": 24
} | {
"line": 121,
"column": 48
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁴ : Pretriangulated C\ninst✝³ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunc... | by
rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]
exact eq_zero_of_hom_shift_pos hι _ (by lia) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 134,
"column": 24
} | {
"line": 136,
"column": 48
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁴ : Pretriangulated C\ninst✝³ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunc... | by
rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]
exact eq_zero_of_hom_shift_pos hι _ (by lia) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {
"line": 289,
"column": 6
} | {
"line": 289,
"column": 60
} | [
{
"pp": "case refine_2\nC : Type u_1\nD : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝¹³ : Category.{v_1, u_1} C\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : Category.{v_3, u_3} D₁\ninst✝¹⁰ : Category.{v_4, u_4} D₂\nA B : ObjectProperty C\ninst✝⁹ : HasZeroObject C\ninst✝⁸ : HasShift C ℤ\ninst✝⁷ : Preadditive C\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 260,
"column": 52
} | {
"line": 260,
"column": 67
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\na b : ℤ\nhn : a ≤ b + 1\n⊢ t.IsLE ((t.truncLT a).obj X) (a - 1)",
"usedConstants": [
... | dsimp [truncLT] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nh : a ≤ b\nX : C\n⊢ (t.natTransTruncLTOfLE a b h).app X ≫ (t.truncLTι b).app X = (t.tru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 358,
"column": 2
} | {
"line": 359,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nh : a ≤ b\nX : C\n⊢ (t.truncGEπ a).app X ≫ (t.natTransTruncGEOfLE a b h).app X = (t.tru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn₀ : ℤ\nX : C\nx✝ : t.IsLE X n₀\ne : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 495,
"column": 49
} | {
"line": 497,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn : ℤ\nX : C\n⊢ t.IsGE X n ↔ IsZero ((t.truncLT n).obj X)",
"usedConstants": [
"Eq.mpr... | by
rw [t.isGE_iff_isIso_truncGEπ_app n X]
exact (Triangle.isZero₁_iff_isIso₂ _ (t.triangleLTGE_distinguished n X)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 72
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftF... | exact Triangle.isoMk _ _ (-(Iso.refl _)) (Iso.refl _) (Iso.refl _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory | {
"line": 333,
"column": 60
} | {
"line": 333,
"column": 77
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁷ : Pretriangulated C\ninst✝⁶ : Category.{v_2, u_2} A\nι : A ⥤ C\nhι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 602,
"column": 35
} | {
"line": 602,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn₀ n₁ : ℤ\nh : n₀ + 1 = n₁\nX : C\nhX : ∀ (Y : C) (f : X ⟶ Y), t.IsGE Y n₁ → f = 0\n⊢ (t.truncGE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 608,
"column": 33
} | {
"line": 608,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn₀ n₁ : ℤ\nh : n₀ + 1 = n₁\nX : C\nhX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0\n⊢ 𝟙 ((t.tru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 50,
"column": 53
} | {
"line": 50,
"column": 64
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 91
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ns : Set α\ninst✝¹ : FunLike F α β\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (⇑f) (s * s)\nh : ThreeGPFree s\n⊢ ThreeGPFree (⇑f '' s)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMu... | rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ habc
rw [h ha hb hc (hf (mul_mem_mul ha hc) (mul_mem_mul hb hb) <| by rwa [map_mul, map_mul])] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 91
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ns : Set α\ninst✝¹ : FunLike F α β\ninst✝ : MulHomClass F α β\nf : F\nhf : InjOn (⇑f) (s * s)\nh : ThreeGPFree s\n⊢ ThreeGPFree (⇑f '' s)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMu... | rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ habc
rw [h ha hb hc (hf (mul_mem_mul ha hc) (mul_mem_mul hb hb) <| by rwa [map_mul, map_mul])] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\nhs : ThreeGPFree s\na : α\nha : a ∈ s\nc : α\nhc : c ∈ s\nhb : a ∈ s\nhabc : a * c = a * a\n⊢ a = c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 192,
"column": 47
} | {
"line": 192,
"column": 83
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Set α\na : α\nhs : ThreeGPFree s\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 87
} | [
{
"pp": "α : Type u_2\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : ThreeGPFree s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x ↦ a • x) b * (fun x ↦ a • x) d = (fun x ↦ a • x) c * (fun x ↦ a • x) c\n⊢ b * d = c * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanHom 2 (↑A) (↑B) f\nhf' : Set.BijOn f ↑A ↑B\ns : Finset β\nhsB : s ⊆ B\nhcard : #s = mulRothNumber B\nhs : ThreeGPFree ↑s\nhsA ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 15
} | [
{
"pp": "case refine_1\nk n : ℕ\nhkn : k ≤ n\n⊢ Set.MapsTo Nat.cast ↑(range k) ↑(Iio ↑k)",
"usedConstants": [
"Eq.mpr",
"Finset.coe_range",
"congrArg",
"Fin.instCommRing",
"Finset",
"PartialOrder.toPreorder",
"AddGroupWithOne.toAddMonoidWithOne",
"Finset.Iio",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 858,
"column": 52
} | {
"line": 858,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na b : ℤ\nX : C\nh : a ≤ b\nu₁₂ : (t.truncLT a).obj X ⟶ (t.truncLT b).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 95,
"column": 53
} | {
"line": 95,
"column": 64
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\na b : ℤ\nf : WithBotTop.coe a ⟶ WithBotTop.coe b\n⊢ a ≤ b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CovBySMul | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 40
} | [
{
"pp": "M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA₁ A₂ B : Set X\nhA : A₁ ⊆ A₂\nhAB : CovBySMul M K A₂ B\n⊢ CovBySMul M K A₁ B",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.CovBySMul | {
"line": 73,
"column": 29
} | {
"line": 73,
"column": 40
} | [
{
"pp": "M : Type u_1\nX : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nK : ℝ\nA B₁ B₂ : Set X\nhB : B₁ ⊆ B₂\nhAB : CovBySMul M K A B₁\n⊢ CovBySMul M K A B₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.RuzsaCovering | {
"line": 69,
"column": 64
} | {
"line": 69,
"column": 94
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nK : ℝ\nA B : Finset G\nhB₀ : (↑B).Nonempty\nhK : ↑(Nat.card ↑(↑A * ↑B)) ≤ K * ↑(Nat.card ↑↑B)\n⊢ ↑(?m.82 * B).card ≤ ?m.80 * ↑B.card",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 216,
"column": 44
} | {
"line": 216,
"column": 55
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nr : α → β → Prop\nh₁ : LeftTotal r\nh₂ : LeftUnique r\n⊢ ∀ a ∈ univ, ∃ b ∈ univ, r a b",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"instInhabitedTrue",
"Membership.mem",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 220,
"column": 45
} | {
"line": 220,
"column": 56
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nr : α → β → Prop\nh₁ : RightTotal r\nh₂ : RightUnique r\n⊢ ∀ b ∈ univ, ∃ a ∈ univ, r a b",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"instInhabitedTrue",
"Membership.mem",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 486,
"column": 4
} | {
"line": 486,
"column": 15
} | [
{
"pp": "case inr.inr\nN : ℕ\nhN : N > 0\nh₁ : N < 4096\n⊢ 1 ≤ ↑(rothNumberNat N)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Preorder.toLE",
"id",
"roth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 19
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\ng : β → α\nhg₁ : MapsTo g B A\nhg₂ : RightInvOn g f B\nhf : IsMulFreimanIso n A B f\ns t : Multiset β\nhsB : ∀ ⦃x : β⦄, x ∈ s → x ∈ B\nhtB : ∀ ⦃x : β⦄, x ∈ t → x ∈ B\nhs : s.card = n\nht : t... | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 12
} | [
{
"pp": "case coe\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nX : C\nn : ℤ\n⊢ (t.eTriangleLTGE.obj (WithBotTop.coe n)).obj X ∈ distinguishedTr... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : CommMonoid β\nA : Set α\nB : Set β\nf : α → β\nn : ℕ\nhf : IsMulFreimanHom n A B f\ns t : Finset α\nhsA : ↑s ⊆ A\nhtA : ↑t ⊆ A\nhs : s.card = n\nht : t.card = n\n⊢ ∏ i ∈ s, i = ∏ i ∈ t, i → ∏ i ∈ s, f i = ∏ i ∈ t, f i",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 256,
"column": 33
} | {
"line": 256,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nn : ℤ\nX : C\ni : ℤ\nh : WithBotTop.coe n ≤ WithBotTop.coe i\n⊢ n ≤ ?m.80",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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