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.Monoidal.Cartesian.CommGrp_ | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 19
} | [
{
"pp": "case h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\nX₁ X₂ : CommGrp C\nψ : X₁ ⟶ X₂\nY : (Grp C)ᵒᵖ\nf g : unop Y ⟶ X₁.toGrp\n⊢ ((f * g) ≫ ψ.hom).hom.hom = (f ≫ ψ.hom * g ≫ ψ.hom).hom.hom",
"usedConstants": [
"CategoryTheory.Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | {
"line": 139,
"column": 41
} | {
"line": 140,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝² : ChosenPullbacksAlong f\ninst✝¹ : ChosenPullbacksAlong g\ninst✝ : ChosenPullbacksAlong (f ≫ g)\n⊢ Functor.whiskerRight (pullbackComp f g).hom (map (f ≫ g)) ≫ (mapPullbackAdj (f ≫ g)).counit =\n (mapPullbackAdj (f ≫ g... | by
rw [pullbackComp, Adjunction.rightAdjointUniq_hom_counit] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | {
"line": 211,
"column": 66
} | {
"line": 211,
"column": 77
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nY Z X : C\nf : Y ⟶ X\ng : Z ⟶ X\ninst✝ : ChosenPullbacksAlong g\nW : C\nφ₁ φ₂ : W ⟶ pullbackObj f g\nh₁ : φ₁ ≫ fst f g = φ₂ ≫ fst f g\nh₂ : φ₁ ≫ snd f g = φ₂ ≫ snd f g\nadj : map g ⊣ pullback g := mapPullbackAdj g\nU : Over Z := Over.mk (φ₁ ≫ snd f g)\nφ₁' : U... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 15
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nY Z X : C\nf : Y ⟶ X\ng : Z ⟶ X\ninst✝ : ChosenPullbacksAlong g\nW : C\nφ₁ φ₂ : W ⟶ pullbackObj f g\nh₁ : φ₁ ≫ fst f g = φ₂ ≫ fst f g\nh₂ : φ₁ ≫ snd f g = φ₂ ≫ snd f g\nadj : map g ⊣ pullback g := ⋯\nU : Over Z := ⋯\nφ₁' : U ⟶ (pullback g).obj (Over.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nZ X : C\ng : Z ⟶ X\ninst✝ : ChosenPullbacksAlong g\nT : Over X\n⊢ Hom.left ((pullbackIsoOverPullback g).hom.app T) ≫ pullback.fst T.hom g = fst T.hom g",
"usedConstants": [
"CategoryTheory.ChosenPullbacksAlong.hasPullbackAlong",
"CategoryTheory... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Internal.Limits | {
"line": 92,
"column": 50
} | {
"line": 92,
"column": 61
} | [
{
"pp": "J : Type w\ninst✝² : Category.{v_1, w} J\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nF : J ⥤ Mon C\nc : Cone (F ⋙ forget C)\nhc : IsLimit c\ns : Cone F\nm : s.pt ⟶ (limitCone F c hc).pt\nw : ∀ (j : J), m ≫ (limitCone F c hc).π.app j = s.π.app j\nj : J\n⊢ m.hom ≫ c.π.app j = { h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Mod | {
"line": 393,
"column": 6
} | {
"line": 393,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\ninst✝⁸ : MonoidalCategory C\nD : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D\ninst✝⁶ : MonoidalLeftAction C D\nA B : C\ninst✝⁵ : MonObj A\ninst✝⁴ : MonObj B\nf : A ⟶ B\ninst✝³ : IsMonHom f\nM N : D\ninst✝² : ModObj B M\ninst✝¹ : ModObj B N\ng : M ⟶ N\ninst✝ : IsModH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Ring | {
"line": 106,
"column": 19
} | {
"line": 106,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : BraidedCategory C\nR : C\ninst✝ : RingObj R\nX✝ X : C\na : X ⟶ R\n⊢ 0 * a = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Ring | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\ninst✝¹ : BraidedCategory C\nR : C\ninst✝ : RingObj R\nX✝ X : C\na : X ⟶ R\n⊢ a * 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subterminal | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\nhA : IsSubterminal A\ninst✝ : HasBinaryProduct A A\n⊢ A ⨯ A ≅ A",
"usedConstants": [
"CategoryTheory.IsSubterminal.isIso_diag",
"CategoryTheory.Iso.symm",
"CategoryTheory.Limits.diag",
"CategoryTheory.Limits.prod",
"Cat... | letI := IsSubterminal.isIso_diag hA
apply (asIso (diag A)).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subterminal | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\nhA : IsSubterminal A\ninst✝ : HasBinaryProduct A A\n⊢ A ⨯ A ≅ A",
"usedConstants": [
"CategoryTheory.IsSubterminal.isIso_diag",
"CategoryTheory.Iso.symm",
"CategoryTheory.Limits.diag",
"CategoryTheory.Limits.prod",
"Cat... | letI := IsSubterminal.isIso_diag hA
apply (asIso (diag A)).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 42
} | [
{
"pp": "case w.h.w.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\ninst✝ : MonoidalClosed V\nF G G' : C ⥤ V\nη : G ⟶ G'\nc c' : C\nf : c ⟶ c'\nj : Cᵒᵖ\nk : C\n⊢ (ihom (F.obj (unop j))).map (G.map (k ◁ f)) ≫ (ihom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 52
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalClosed V\nF G✝ H G : C ⥤ V\ninst✝ : DayConvolution F G\nℌ : DayConvolutionInternalHom F (F ⊛ G) H\nc c' : C\nf : c ⟶ c'\n⊢ G.map f ≫\n We... | apply Wedge.IsLimit.hom_ext <| ℌ.isLimitWedge c' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nhasDayConvolution : ∀ (F G : C ⥤ V), (tensor C).HasPointwiseLeftKanExtension (F ⊠ G)\nhasDayConvolutionUnit : (Functor.fromPUnit (𝟙_ C)).HasPointwiseLeftKanEx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 115,
"column": 10
} | {
"line": 115,
"column": 24
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : BraidedCategory C\nA B : C\ninst✝² : HopfObj A\ninst✝¹ : HopfObj B\nf : A ⟶ B\ninst✝ : IsBimonHom f\n| Δ ≫ A ◁ 𝒮 ≫ μ",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCa... | antipode_right | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 22
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (Δ ≫ A ◁ 𝒮 ≫ μ) ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"Ca... | antipode_right | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 929,
"column": 2
} | {
"line": 929,
"column": 34
} | [
{
"pp": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nV W X Y Z : Mon C\nM : Bimo... | dsimp only [AssociatorBimod.hom] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 381,
"column": 8
} | {
"line": 381,
"column": 22
} | [
{
"pp": "case a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| A ◁ (Δ ≫ A ◁ 𝒮 ≫ μ) ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"Ca... | antipode_right | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 22
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| (Δ ≫ A ◁ 𝒮 ≫ μ) ▷ A",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory... | antipode_right | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : C\ninst✝² : MonObj M\nN : C\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ (mopEquiv C).inverse.map (MonObj.one ≫ f.mop) = (mopEquiv C).inverse.map MonObj.one",
"usedConstants": [
"CategoryTheory.MonoidalOppo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : C\ninst✝² : MonObj M\nN : C\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ (mopEquiv C).inverse.map (MonObj.mul ≫ f.mop) = (mopEquiv C).inverse.map ((f.mop ⊗ₘ f.mop) ≫ MonObj.mul)",
"usedConstants": [
"Catego... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝² : MonObj M\nN : Cᴹᵒᵖ\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ (mopEquiv C).functor.map (MonObj.one ≫ f.unmop) = (mopEquiv C).functor.map MonObj.one",
"usedConstants": [
"CategoryTheory.Mono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Opposite.Mon | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : MonoidalCategory C\nM : Cᴹᵒᵖ\ninst✝² : MonObj M\nN : Cᴹᵒᵖ\ninst✝¹ : MonObj N\nf : M ⟶ N\ninst✝ : IsMonHom f\n⊢ (mopEquiv C).functor.map (MonObj.mul ≫ f.unmop) = (mopEquiv C).functor.map ((f.unmop ⊗ₘ f.unmop) ≫ MonObj.mul)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Rigid.Braided | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 17
} | [
{
"pp": "case e_g.e_g.e_f\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y : C\ninst : ExactPairing X Y\n⊢ ((((((α_ X Y X).inv ≫ (β_ X Y).hom ▷ X) ≫ inv (α_ Y X X).inv) ≫ inv (Y ◁ (β_ X X).inv)) ≫ inv (α_ Y X X).hom) ≫\n inv ((β_ Y X).inv ▷ X)) ≫\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Rigid.Braided | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 17
} | [
{
"pp": "case e_g.e_g.e_a\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX Y : C\ninst : ExactPairing X Y\n⊢ (((Y ◁ (β_ X Y).hom ≫ inv (α_ Y Y X).hom) ≫ inv ((β_ Y Y).inv ▷ X)) ≫ inv (α_ Y Y X).inv) ≫ inv (Y ◁ (β_ Y X).inv) =\n (((((inv (α_ Y X Y).hom ≫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Dense | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalAccessibleCategory C κ\nP : ObjectProperty C\nw✝ : ObjectProperty.EssentiallySmall.{w, v, u} P\nhP : P.IsCardinalFilteredGenerator κ\n⊢ P ≤ isCardinalPresentable C κ",
"usedConstants": [
"Ca... | exact hP.le_isCardinalPresentable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁵ : Category.{v_1, u_1} J✝\ninst✝⁴ : (j : J✝) → Category.{?u.49630, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝³ : SmallCategory J\ninst✝² : (j : J) → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 148,
"column": 8
} | {
"line": 148,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁵ : Category.{v_1, u_1} J✝\ninst✝⁴ : (j : J✝) → Category.{?u.49630, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝³ : SmallCategory J\ninst✝² : (j : J) → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : MorphismProperty C\nhp : P ≤ isFinitelyPresentable C\ninst✝ : LocallySmall.{w, v, u} C\nX Y : C\nf : X ⟶ Y\nhf : P.ind.ind f\nthis : P.underObj ≤ ObjectProperty.isFinitelyPresentable (Under X)\n⊢ P.ind f",
"usedConstants": [
"Eq.mpr",
"Categor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.ColimitPresentation | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ✝ : Type u_1\nI✝ : J✝ → Type u_2\ninst✝⁵ : Category.{v_1, u_1} J✝\ninst✝⁴ : (j : J✝) → Category.{?u.49630, u_2} (I✝ j)\nD✝ : J✝ ⥤ C\nP✝¹ : (j : J✝) → ColimitPresentation (I✝ j) (D✝.obj j)\nJ : Type w\nI : J → Type w\ninst✝³ : SmallCategory J\ninst✝² : (j : J) → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.ArtinianObject | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 11
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nh : ∀ (F : ℕ ⥤ (MonoOver X)ᵒᵖ), IsFiltered.IsEventuallyConstant F\nF : ℕ →o (Subobject X)ᵒᵈ\nn : ℕ\nhn : (⋯.functor ⋙ (orderDualEquivalence (Subobject X)).functor ⋙ Subobject.representative.op).IsEventuallyConstantFrom n\nm : ℕ\nhm : n ≤ m\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\nF G : C ⥤ V\ninst✝ : DayConvolution F G\nx x' y y' : C\nf : x ⟶ x'\ng : y ⟶ y'\n⊢ (F.map f ⊗ₘ G.map g) ≫ (unit F G).app (x', y') = (unit F G).app (x, y) ≫ (F ⊛... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.ArtinianObject | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y X✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Mono f\nhY : isArtinianObject.Is Y✝\n⊢ isArtinianObject.Is X✝",
"usedConstants": [
"CategoryTheory.isArtinianObject_of_mono"
]
}
] | exact isArtinianObject_of_mono f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\nF G : C ⥤ V\ninst✝ : DayConvolution F G\nx x' y : C\nf : x ⟶ x'\n⊢ F.map f ▷ G.obj y ≫ (unit F G).app (x', y) = (unit F G).app (x, y) ≫ (F ⊛ G).map (f ▷ y)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\nF G : C ⥤ V\ninst✝ : DayConvolution F G\nx y y' : C\ng : y ⟶ y'\n⊢ F.obj x ◁ G.map g ≫ (unit F G).app (x, y') = (unit F G).app (x, y) ≫ (F ⊛ G).map (x ◁ g)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 29
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁴ : Category.{v₂, u₂} V\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalCategory V\nF G : C ⥤ V\ninst✝¹ : DayConvolution F G\nF' G' : C ⥤ V\ninst✝ : DayConvolution F' G'\nf : F ⟶ F'\ng : G ⟶ G'\nx y : C\n⊢ (unit F G).app (x, y) ≫ (map f g).app... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 98,
"column": 4
} | {
"line": 99,
"column": 38
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nh : ∀ (F : ℕ ⥤ MonoOver X), IsFiltered.IsEventuallyConstant F\nF : ℕ →o Subobject X\nn : ℕ\nhn : (⋯.functor ⋙ Subobject.representative).IsEventuallyConstantFrom n\nm : ℕ\nhm : n ≤ m\n⊢ F n = F m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 178,
"column": 64
} | {
"line": 178,
"column": 75
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\nF G : C ⥤ V\ninst✝ : DayConvolution F G\nc : C\nv : V\nf g : (F ⊛ G).obj c ⟶ v\nh : ∀ {x y : C} (u : x ⊗ y ⟶ c), (unit F G).app (x, y) ≫ (F ⊛ G).map u ≫ f = (u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 439,
"column": 2
} | {
"line": 439,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝³ : Category.{v₂, u₂} V\ninst✝² : MonoidalCategory C\ninst✝¹ : MonoidalCategory V\nU : C ⥤ V\ninst✝ : DayConvolutionUnit U\nc : C\nv : V\ng h : U.obj c ⟶ v\ne : ∀ (f : 𝟙_ C ⟶ c), can ≫ U.map f ≫ g = can ≫ U.map f ≫ h\nj : CostructuredArrow (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 981,
"column": 8
} | {
"line": 981,
"column": 46
} | [
{
"pp": "case a\nC✝ : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹⁵ : Category.{v₂, u₂} V✝\ninst✝¹⁴ : MonoidalCategory C✝\ninst✝¹³ : MonoidalCategory V✝\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nV : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} V\ninst✝¹⁰ : MonoidalCategory C\ninst✝⁹ : MonoidalCatego... | ι_map_leftUnitor_hom_eq_leftUnitor_hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 15
} | [
{
"pp": "case h.e'_2.h.e'_7.h\nC : Type u\ninst✝³ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\no : HomOrthogonal s\nα : Type\ninst✝ : Finite α\nf : α → ι\nb a : α\nj_property✝ : a ∈ f ⁻¹' {f b}\nj_property : f a = f b\nh : ¬b = a\n⊢ biproduct.ι (fun a ↦ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.EckmannHilton | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 69
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\n⊢ e₁ = e₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.EckmannHilton | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 73
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b : X\n⊢ m₂ a b = m₂ b a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.EckmannHilton | {
"line": 83,
"column": 19
} | {
"line": 83,
"column": 75
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b c : X\n⊢ m₂ (m₂ a b) c = m₂ a (m₂ b c)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Pi.Monoidal | {
"line": 249,
"column": 20
} | {
"line": 249,
"column": 31
} | [
{
"pp": "case w\nI : Type w₁\nC : I → Type u₁\ninst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)\ninst✝⁵ : (i : I) → MonoidalCategory (C i)\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : MonoidalCategory D\nF G : D ⥤ ((i : I) → C i)\ninst✝² : F.LaxMonoidal\ninst✝¹ : G.LaxMonoidal\nτ : (i : I) → F ⋙ eval C i ⟶... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Pi.Monoidal | {
"line": 250,
"column": 26
} | {
"line": 250,
"column": 37
} | [
{
"pp": "case w\nI : Type w₁\nC : I → Type u₁\ninst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)\ninst✝⁵ : (i : I) → MonoidalCategory (C i)\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : MonoidalCategory D\nF G : D ⥤ ((i : I) → C i)\ninst✝² : F.LaxMonoidal\ninst✝¹ : G.LaxMonoidal\nτ : (i : I) → F ⋙ eval C i ⟶... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Adjunction | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 65
} | [
{
"pp": "C : Type u\nD : Type u'\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\nP : ObjectProperty C\nhP : P.IsCardinalFilteredGenerator κ\ninst✝² : G.IsCardinalAccessible κ\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nY : D\nt... | obtain ⟨J, _, _, ⟨hY⟩⟩ := hP.exists_colimitsOfShape (G.obj Y) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 166,
"column": 17
} | {
"line": 166,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_2\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 81
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_2\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : ¬Nonempty (X ≅ Y)... | · exact (finrank_hom_simple_simple_eq_zero_iff 𝕜 X Y).2 (not_nonempty_iff.mp h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 59,
"column": 23
} | {
"line": 59,
"column": 38
} | [
{
"pp": "X : Type u\nκ : Cardinal.{u}\nhX : HasCardinalLT X κ\ninst✝ : Fact κ.IsRegular\nJ : Type u\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ\nF : J ⥤ Type u\nc : Cocone F\nhc : IsColimit c\nthis : IsFiltered J\nj : J\nf g : X ⟶ F.obj j\nh : f ≫ c.ι.app j = g ≫ c.ι.app j\nk : ToType X → J\na : (x : To... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 303,
"column": 30
} | {
"line": 303,
"column": 41
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\nhg : f ≫ g₁ = f ≫ g₂\n⊢ g₁ ≫ toSucc W Z = g₂ ≫ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 285,
"column": 4
} | {
"line": 286,
"column": 65
} | [
{
"pp": "case refine_1.h.h₀.h\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nT : C\nhT : W.isLocal T\nφ₁ φ₂ : succ W Z ⟶ T\nh : toStep W Z ≫ fromSte... | · apply (hT d.1.1.hom d.1.2).1
simp only [← D₁.ι_comp_t_assoc, pushout.condition_assoc, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 300,
"column": 6
} | {
"line": 300,
"column": 46
} | [
{
"pp": "case refine_1.refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nx✝ : IsIso (toSucc W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 315,
"column": 27
} | {
"line": 315,
"column": 53
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 88
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | exact ⟨c.pt, fun i ↦ u i ≫ c.π ⟨⟩,
fun i ↦ ⟨fun hi ↦ (hm₀ i).false (t₁ i ≫ c.π ⟨⟩ ≫ hi)⟩,
fun i₁ i₂ j h₁ h₂ ↦ by simpa [index, shape] using c.condition ⟨⟨i₁, i₂, j⟩, h₁, h₂⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 313,
"column": 6
} | {
"line": 318,
"column": 47
} | [
{
"pp": "case refine_2.h.h₀.h\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nhZ : W.isLocal Z\nf : succ W Z ⟶ Z\nhf : toSucc W Z ≫ f = 𝟙 Z\nd : D₁ ... | simp only [Category.assoc] at hf
simp only [Category.comp_id, ← Category.assoc]
refine D₂.condition _ d.1.2 ?_
rw [Category.assoc, Category.assoc, Category.assoc,
← D₁.ι_comp_t_assoc, pushout.condition_assoc, reassoc_of% hf,
← D₁.ι_comp_t_assoc, pushout.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 313,
"column": 6
} | {
"line": 318,
"column": 47
} | [
{
"pp": "case refine_2.h.h₀.h\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nhZ : W.isLocal Z\nf : succ W Z ⟶ Z\nhf : toSucc W Z ≫ f = 𝟙 Z\nd : D₁ ... | simp only [Category.assoc] at hf
simp only [Category.comp_id, ← Category.assoc]
refine D₂.condition _ d.1.2 ?_
rw [Category.assoc, Category.assoc, Category.assoc,
← D₁.ι_comp_t_assoc, pushout.condition_assoc, reassoc_of% hf,
← D₁.ι_comp_t_assoc, pushout.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 33
} | [
{
"pp": "case inr\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\nhf : ∃ i, Arrow.mk f = Arrow.mk ((D i.fst).isTerminal.lift ⋯ ... | obtain ⟨⟨i, j, hj⟩, hi⟩ := hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 425,
"column": 44
} | {
"line": 425,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.ToType\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RegularCategory.Basic | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 70
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nX Y : C\nf : X ⟶ Y\nm : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯\ne : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)\nk₁ : pullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 13
} | [
{
"pp": "case h.a\nC : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.ToType\ninst✝¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 464,
"column": 37
} | {
"line": 464,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsSmall.{w, v, u} W\ninst✝¹ : LocallySmall.{w, v, u} C\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\ninst✝ : HasColimitsOfSize.{w, w, v, u}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 32
} | [
{
"pp": "case h.e'_5.w.h.h.toFun.h\nC : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nR : F.RepresentableBy X\nx✝¹ : Cᵒᵖ\nx✝ : (uliftYoneda.{w, v, u}.obj X).obj x✝¹\n⊢ (ConcreteCategory.hom ((uliftYonedaEquiv.symm { down := R.homEquiv (𝟙 X) }).app x✝¹)).toFun x✝ =\n (ConcreteCategory.hom\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 109,
"column": 14
} | {
"line": 109,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nF' : Cᵒᵖ ⥤ Type w\ne : F ≅ F'\nh : F'.IsRepresentedBy ((ConcreteCategory.hom (e.hom.app (op X))) x)\n⊢ F.IsRepresentedBy x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 17
} | [
{
"pp": "case e_a.e_a.e_m.h.h.w\nC : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₁\ninst✝⁴ : Category.{v₁, u₁} D\ninst✝³ : Preadditive D\ninst✝² : HasFiniteBiproducts D\nF : C ⥤ D\ninst✝¹ : F.Additive\nL : Mat_ C ⥤ D\ninst✝ : L.Additive\nα : embedding C ⋙ L ≅ F\nX✝ Y✝ : Mat_ C\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RegularCategory.Basic | {
"line": 198,
"column": 2
} | {
"line": 199,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nA B : C\nf : A ⟶ B\nA' : Subobject A\nB' : Subobject B\n⊢ IsPullback (frobeniusMorphism f A' B' ≫ ((«exists» f).obj A' ⊓ B').ofLE B' ⋯)\n ((A' ⊓ (Subobject.pullback f).obj B').ofLE A' ⋯) B'.arrow\n ((imageFactorisation f A').F.e ≫ ((«exis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 463,
"column": 9
} | {
"line": 463,
"column": 20
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD : D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 39
} | [
{
"pp": "case mp.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\nY : C\nT : Presieve Y\nhS : T ∈ (coherentCoverage C).coverings Y\n⊢ ∃ α, ∃ (_ : Finite α), ∃ Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a)",
"usedConstants": []
}
] | obtain ⟨a, h, Y', π, h', _⟩ := hS | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 15
} | [
{
"pp": "case h\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\ninst✝ : HasFiniteCoproducts C\nX Y Z : C\nf : X ⟶ Y\ng : Z ⟶ Y\nx✝ : EffectiveEpi g\nhp : EffectiveEpi g → ∃ β, ∃ (_ : Finite β), ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f\nβ : Type\nw✝ : Finite β... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 82,
"column": 44
} | {
"line": 82,
"column": 55
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS✝ : Sieve X\nY : C\nR S : Sieve Y\na✝¹ : (regularCoverage C).Saturate Y R\na✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → (regularCoverage C).Saturate Y_1 (Sieve.pullback f S)\nb : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → ∃ Y_2 π,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 67
} | [
{
"pp": "case mk\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY✝ : C\nI : Type\nw✝ : Finite I\nX : I → C\nf : (a : I) → X a ⟶ Y✝\nhT : EffectiveEpiFamily X f\nR Y : C\ni✝¹ : Unit\nψ : R ⟶ ∐ fun i ↦ X i\nQ : C\ni✝ : I\ne : Q ⟶ X i✝\n⊢ P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 32
} | [
{
"pp": "case refine_1.a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nhP :\n ∀ (y : P.obj (op X)),\n (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →\n ∃! x, (C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 49
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nhP :\n ∀ (y : P.obj (op X)),\n (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →\n ∃! x, (Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 49
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (Concre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 49
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (Concre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 32
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology | {
"line": 41,
"column": 59
} | {
"line": 41,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : FinitaryPreExtensive C\nX✝ : C\nS✝ : Sieve X✝\nX : C\nS : Presieve X\nα : Type\nw✝ : Finite α\nY : α → C\nπ : (a : α) → Y a ⟶ X\nh : S = Presieve.ofArrows Y π\nh' : IsIso (Sigma.desc π)\n⊢ IsIso (Sigma.desc (Cofan.mk X π).inj)",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 32
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\n⊢ (parallelPair (ObjectProperty.homMk (Over.homMk c.fst ⋯)).op (ObjectProperty.homMk (Over.homMk c.snd ⋯)).op).Initial",
"usedConstants": [
"Unit.unit",
"CategoryTheory.Over",
"O... | apply Limits.parallelPair_initial_mk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.EpiMono | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 18
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁷ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝⁵ : ConcreteCategory A FA\ninst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A\n... | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 17
} | [
{
"pp": "case zero\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nP : Cᵒᵖ ⥤ D\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\nS : Presieve B := (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows\nX' : S.category := ... | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 15
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX : C\ninst✝¹ : Projective X\nF : Cᵒᵖ ⥤ Type u_4\nY : C\nf : Y ⟶ X\nhf : EffectiveEpi f\ninst✝ : (ofArrows (fun x ↦ Y) fun x ↦ f).regular\nx : Unit → F.obj (op Y)\nhx : Arrows.Compatible F (fun x ↦ f) x\nx✝ : Unit\n⊢ (ConcreteCategory.hom (F.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 17
} | [
{
"pp": "case refine_2.refine_1\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies\ninst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Preco... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 75
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nA : Type u₃\ninst✝⁴ : Category.{v₃, u₃} A\nF : Cᵒᵖ ⥤ A\nB : Type u₄\ninst✝³ : Category.{v₄, u₄} B\ns : A ⥤ B\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nh : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f\ninst✝ : ReflectsFiniteLimits s\... | refine ⟨⟨fun n ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩, fun _ _ π _ c hc ↦ ⟨?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit | {
"line": 102,
"column": 19
} | {
"line": 102,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nF : ℕᵒᵖ ⥤ Sheaf (coherentTopology C) (Type v)\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Sheaf.IsLocallySurjective (F.map (homOfLE ⋯).op)\ninst✝ : HasLimitsOfShape ℕᵒᵖ C\nh : ∀ (G : ℕᵒᵖ ⥤ C), (∀ (n : ℕ), Effe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.IsStack | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsStack J\nS : C\nR : Presieve S\nhR : Sieve.generate R ∈ J S\n⊢ F.IsStackFor R",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 63
} | [
{
"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": 69,
"column": 32
} | {
"line": 69,
"column": 43
} | [
{
"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": 104,
"column": 2
} | {
"line": 104,
"column": 59
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\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\nhf' : Sieve.ofArrows X' f' ∈ J S\ni : ι\n⊢ sieve f f' i ∈ (J.over (X i)) (Over.mk (𝟙 (X i)))",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 160,
"column": 10
} | {
"line": 160,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD : F.DescentDataAsCoalgebra fun x ↦ f\ni₁ i₂ : ι\n⊢ ((𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).obj D).hom i₁ i₂ ≫\n (F.map f.op.toLoc).l.toFunctor.map ((F.map f.op.toLoc... | obtain rfl := Subsingleton.elim i₁ default | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 163,
"column": 6
} | {
"line": 165,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD₁ D₂ : F.DescentDataAsCoalgebra fun x ↦ f\nα : D₁ ⟶ D₂\n⊢ (𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).map α ≫ ((fun D ↦ isoMk (fun i ↦ eqToIso ⋯) ⋯) D₂).hom =\n ((fun D ↦ is... | ext i
obtain rfl := Subsingleton.elim i default
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 163,
"column": 6
} | {
"line": 165,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD₁ D₂ : F.DescentDataAsCoalgebra fun x ↦ f\nα : D₁ ⟶ D₂\n⊢ (𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).map α ≫ ((fun D ↦ isoMk (fun i ↦ eqToIso ⋯) ⋯) D₂).hom =\n ((fun D ↦ is... | ext i
obtain rfl := Subsingleton.elim i default
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\nι : Type u_1\ninst✝ : Finite ι\nY : ι → C\nf : (i : ι) → Y i ⟶ X\n⊢ ofArrows Y f ∈ (finite C).coverings X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Presieve",
"c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 61,
"column": 29
} | {
"line": 61,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nu : Y ⟶ X\ns : Presieve X\nhs : s ∈ (Precoverage.finite C).coverings X\n⊢ pullbackArrows u s ∈ (Precoverage.finite C).coverings Y",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 62,
"column": 31
} | {
"line": 62,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX : C\ns : Presieve X\nt : ⦃Y : C⦄ → (f : Y ⟶ X) → s f → Presieve Y\nhs : s ∈ (Precoverage.finite C).coverings X\nht : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (H : s f), t f H ∈ (Precoverage.finite C).coverings Y\n⊢ s.bind t ∈ (Precoverage.finite C).coverin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.GlobalSections | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\ninst✝¹ : HasWeakSheafify J A\ninst✝ : HasGlobalSectionsFunctor J A\nF : Sheaf J A\nc : Cone F.obj\nf : c.pt ⟶ F.coneΓ.pt\nhf : (Functor.const Cᵒᵖ).map f ≫ F.coneΓ.π = c.π\n⊢ f = ΓHomEquiv c.π"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 17
} | [
{
"pp": "case h.e'_6.h.h\nC : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 213,
"column": 44
} | {
"line": 213,
"column": 55
} | [
{
"pp": "case h\nC : 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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 221,
"column": 49
} | {
"line": 221,
"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 |
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