mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.DayConvolution

Goal state before:
pp: C✝ : Type u₁
inst✝¹⁶ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹⁵ : Category.{v₂, u₂} V✝
inst✝¹⁴ : MonoidalCategory C✝
inst✝¹³ : MonoidalCategory V✝
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
V : Type u₂
inst✝¹¹ : Category.{v₂, u₂} V
inst✝¹⁰ : MonoidalCategory C
inst✝⁹ : MonoidalCategory V
D : Type u₃
inst✝⁸ : Category.{v₃, u₃} D
inst✝⁷ : MonoidalCategoryStruct D
inst✝⁶ : LawfulDayConvolutionMonoidalCategoryStruct C V D
inst✝⁵ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁴ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝³ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝² :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C × C), Limits.PreservesColimitsOfShape (CostructuredArrow ((tensor C).prod (𝟭 C)) d) (tensorRight v)
x✝ y₁ y₂ : D
f : y₁ ⟶ y₂
x y : C
⊢ ((ι C V D).obj x✝).obj x ◁ ((ι C V D).map f).app y ≫ (convolutionExtensionUnit C V x✝ y₂).app (x, y) =
    (convolutionExtensionUnit C V x✝ y₁).app (x, y) ≫ ((ι C V D).map (x✝ ◁ f)).app (x ⊗ y)

Tactic:
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app

Goal state after:
C✝ : Type u₁
inst✝¹⁶ : Category.{v₁, u₁} C✝
V✝ : Type u₂
inst✝¹⁵ : Category.{v₂, u₂} V✝
inst✝¹⁴ : MonoidalCategory C✝
inst✝¹³ : MonoidalCategory V✝
C : Type u₁
inst✝¹² : Category.{v₁, u₁} C
V : Type u₂
inst✝¹¹ : Category.{v₂, u₂} V
inst✝¹⁰ : MonoidalCategory C
inst✝⁹ : MonoidalCategory V
D : Type u₃
inst✝⁸ : Category.{v₃, u₃} D
inst✝⁷ : MonoidalCategoryStruct D
inst✝⁶ : LawfulDayConvolutionMonoidalCategoryStruct C V D
inst✝⁵ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝⁴ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
inst✝³ :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorLeft v)
inst✝² :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (Functor.fromPUnit (𝟙_ C)) d) (tensorRight v)
inst✝¹ :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow ((𝟭 C).prod (Functor.fromPUnit (𝟙_ C))) d) (tensorRight v)
inst✝ :
  ∀ (v : V) (d : C × C), Limits.PreservesColimitsOfShape (CostructuredArrow ((tensor C).prod (𝟭 C)) d) (tensorRight v)
x✝ y₁ y₂ : D
f : y₁ ⟶ y₂
x y : C
⊢ ((ι C V D).obj x✝).obj x ◁ ((ι C V D).map f).app y ≫ (convolutionExtensionUnit C V x✝ y₂).app (x, y) =
    ((ι C V D).obj x✝).obj x ◁ ((ι C V D).map f).app y ≫ (convolutionExtensionUnit C V x✝ y₂).app (x, y)
Module: Mathlib.CategoryTheory.MorphismProperty.Ind

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X : C
⊢ P.ind.underObj = P.underObj.ind

Tactic:
ext f

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X : C
f : Under X
⊢ P.ind.underObj f ↔ P.underObj.ind f
Module: Mathlib.CategoryTheory.MorphismProperty.LocalEpi

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
inst✝ : Epi f
⊢ P.localEpi f

Tactic:
intro Z hZ u v huv
  rwa [← cancel_epi f]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
⊢ (M.tensorBimod N).whiskerLeft f =
    (M.associatorBimod N P).hom ≫ M.whiskerLeft (N.whiskerLeft f) ≫ (M.associatorBimod N P').inv

Tactic:
by
  dsimp [tensorHom, tensorBimod, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [TensorBimod.X, AssociatorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux, AssociatorBimod.inv]
  refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
  simp only [Functor.flip_obj_map, curriedTensor_map_app]
  slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_rhs 2 3 => rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]
  slice_rhs 3 4 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.invAux]
  slice_rhs 2 2 => rw [whiskerLeft_comp]
  slice_rhs 3 5 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_rhs 2 3 => rw [associator_inv_naturality_right]
  slice_rhs 1 3 => rw [Iso.hom_inv_id_assoc]
  slice_lhs 1 2 => rw [← whisker_exchange]
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.NoetherianObject

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y X✝ Y✝ : C
f : X✝ ⟶ Y✝
x✝ : Mono f
hY : isNoetherianObject Y✝
⊢ isNoetherianObject X✝

Tactic:
rw [← isNoetherianObject.is_iff] at hY ⊢
    exact isNoetherianObject_of_mono f

Goal state after:
Module: Mathlib.GroupTheory.EckmannHilton

Goal state before:
pp: X : Type u
m₁ m₂ : X → X → X
e₁ e₂ : X
h₁ : IsUnital m₁ e₁
h₂ : IsUnital m₂ e₂
distrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)
a b : X
⊢ m₂ a b = m₂ b a

Tactic:
by simpa [mul h₁ h₂ distrib, h₂.left_id, h₂.right_id] using distrib e₂ a b e₂

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Schur

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
⊢ ∃ c, c • 𝟙 X = f

Tactic:
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
c : 𝕜
nu : c ∈ spectrum 𝕜 (End.of f)
⊢ ∃ c, c • 𝟙 X = f
Module: Mathlib.CategoryTheory.Preadditive.Schur

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ Nonempty (X ≅ Y) → finrank 𝕜 (X ⟶ Y) = 1

Tactic:
rintro ⟨f⟩

Goal state after:
case mpr
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ≅ Y
⊢ finrank 𝕜 (X ⟶ Y) = 1
Module: Mathlib.CategoryTheory.Presentable.Directed

Goal state before:
pp: case refine_3
J : Type w
inst✝¹ : SmallCategory J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
j x✝² x✝¹ : J
x✝ : x✝² ⟶ x✝¹
h : __Diagram✝.W x✝
⊢ ∃ li lj, __Diagram✝.W li ∧ __Diagram✝.W lj ∧ x✝ ≫ lj = li

Tactic:
obtain ⟨⟨⟩⟩ := h

Goal state after:
case refine_3
J : Type w
inst✝¹ : SmallCategory J
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
j X Y : J
⊢ ∃ li lj, __Diagram✝.W li ∧ __Diagram✝.W lj ∧ 𝟙 j ≫ lj = li
Module: Mathlib.CategoryTheory.Presentable.Type

Goal state before:
pp: X : Type u
κ : Cardinal.{u}
inst✝ : Fact κ.IsRegular
⊢ IsCardinalPresentable X κ ↔ HasCardinalLT X κ

Tactic:
by
  refine ⟨fun _ ↦ ?_, fun hX ↦ hX.isCardinalPresentable⟩
  have := preservesColimitsOfShape_of_isCardinalPresentable X κ
  obtain ⟨⟨A, hA⟩, f, hf⟩ := Types.jointly_surjective_of_isColimit
    (isColimitOfPreserves (coyoneda.obj (op X))
      (HasCardinalLT.Set.isColimitCocone X κ
        (Cardinal.IsRegular.aleph0_le Fact.out))) (𝟙 X)
  obtain rfl : A = .univ := by
    ext x
    have := congr_hom hf x
    dsimp at this
    rw [← this]
    simp
  exact (hasCardinalLT_iff_of_equiv (Equiv.Set.univ X) _).1 hA

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.Basic

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Finite α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a ↦ pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a ↦ pullback.fst f (π a)
⊢ Presieve.ofArrows Z' π' ∈ {S | ∃ α, ∃ (x : Finite α), ∃ X π, S = Presieve.ofArrows X π ∧ IsIso (Sigma.desc π)}

pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Finite α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a ↦ pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a ↦ pullback.fst f (π a)
⊢ (Presieve.ofArrows Z' π').FactorsThruAlong S f

Tactic:
· constructor
      exact ⟨hα, Z', π', ⟨by simp only,
        FinitaryPreExtensive.isIso_sigmaDesc_fst (fun x => π x) f h_iso⟩⟩

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Finite α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
Z' : α → C := fun a ↦ pullback f (π a)
π' : (a : α) → Z' a ⟶ Y := fun a ↦ pullback.fst f (π a)
⊢ (Presieve.ofArrows Z' π').FactorsThruAlong S f
Module: Mathlib.CategoryTheory.RegularCategory.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
m : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯
e : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)
k₁ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.fst m m
k₂ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := pullback.snd m m
d : pullback f f ⟶ pullback m m := pullback.lift (pullback.fst f f ≫ e) (pullback.snd f f ≫ e) ⋯
g₁ : pullback f f ⟶ pullback e k₁ := pullback.lift (pullback.fst f f) d ⋯
g₂ : pullback f f ⟶ pullback k₂ e := pullback.lift d (pullback.snd f f) ⋯
h : IsPullback g₁ g₂ (pullback.snd e k₁) (pullback.fst k₂ e)
⊢ IsRegularEpi g₁

Tactic:
apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback h.flip

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Regular C
X Y : C
f : X ⟶ Y
m : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := ⋯
e : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := ⋯
k₁ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := ⋯
k₂ : pullback m m ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := ⋯
d : pullback f f ⟶ pullback m m := ⋯
g₁ : pullback f f ⟶ pullback e k₁ := ⋯
g₂ : pullback f f ⟶ pullback k₂ e := ⋯
h : IsPullback g₁ g₂ (pullback.snd e k₁) (pullback.fst k₂ e)
⊢ MorphismProperty.regularEpi C (pullback.fst k₂ e)
Module: Mathlib.CategoryTheory.Sites.Coherent.Comparison

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝ : Finite α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ X₁
hg : g' ≫ Sigma.desc π₁ = g ≫ f
X₂ : α → C := fun a ↦ Limits.pullback g' (Sigma.ι X₁ a)
π₂ : (a : α) → Limits.pullback g' (Sigma.ι X₁ a) ⟶ B₂ := fun a ↦ pullback.fst g' (Sigma.ι X₁ a) ≫ g
π' : (a : α) → Limits.pullback g' (Sigma.ι X₁ a) ⟶ Y := fun a ↦ pullback.fst g' (Sigma.ι X₁ a)
⊢ ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ i ι, ∀ (b : α), ι b ≫ π₁ (i b) = π₂ b ≫ f

Tactic:
have _ := FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X₁ a) g' inferInstance

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : Preregular C
B₁ B₂ : C
f : B₂ ⟶ B₁
α : Type
x✝¹ : Finite α
X₁ : α → C
π₁ : (a : α) → X₁ a ⟶ B₁
h : EffectiveEpiFamily X₁ π₁
Y : C
g : Y ⟶ B₂
w✝ : EffectiveEpi g
g' : Y ⟶ ∐ X₁
hg : g' ≫ Sigma.desc π₁ = g ≫ f
X₂ : α → C := fun a ↦ Limits.pullback g' (Sigma.ι X₁ a)
π₂ : (a : α) → Limits.pullback g' (Sigma.ι X₁ a) ⟶ B₂ := fun a ↦ pullback.fst g' (Sigma.ι X₁ a) ≫ g
π' : (a : α) → Limits.pullback g' (Sigma.ι X₁ a) ⟶ Y := fun a ↦ pullback.fst g' (Sigma.ι X₁ a)
x✝ : IsIso (Sigma.desc fun x ↦ pullback.fst g' (Sigma.ι X₁ x))
⊢ ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ i ι, ∀ (b : α), ι b ≫ π₁ (i b) = π₂ b ≫ f
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits

Goal state before:
pp: A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
⊢ Presheaf.IsSheaf (extensiveTopology C) (pointwiseCocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)).pt

Tactic:
rw [Presheaf.isSheaf_iff_preservesFiniteProducts]

Goal state after:
A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
⊢ PreservesFiniteProducts (pointwiseCocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)).pt
Module: Mathlib.CategoryTheory.Sites.EpiMono

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ IsLocallyInjective
    ((presheafToSheaf J A).map
        ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom)) ≫
      (sheafificationAdjunction J A).counit.app x✝.right)

Tactic:
← isLocallyInjective_sheafToPresheaf_map_iff,

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallyInjective J
    ((sheafToPresheaf J A).map
      ((presheafToSheaf J A).map
          ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom)) ≫
        (sheafificationAdjunction J A).counit.app x✝.right))
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j : ι'
a : Z ⟶ X' j
fac : a ≫ f' j = q ≫ f i
Z' : C
r : Z' ⟶ Z
r' : Z' ⟶ X i
a' : Z' ⟶ X' j
hr' : r ≫ q = r'
ha' : r ≫ a = a'
⊢ pullHom (D₁.hom (q ≫ f i) q (a ≫ p' j) ⋯ ⋯) r r' (r ≫ a ≫ p' j) ⋯ ⋯ ≫
      pullHom (φ.hom j) a' (r ≫ a ≫ p' j) (r ≫ a ≫ p' j) ⋯ ⋯ ≫ D₂.hom (r ≫ q ≫ f i) (r ≫ a ≫ p' j) r' ⋯ ⋯ =
    pullHom
      (D₁.hom (q ≫ f i) q (a ≫ p' j) ⋯ ⋯ ≫
        pullHom (φ.hom j) a (a ≫ p' j) (a ≫ p' j) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a ≫ p' j) q ⋯ ⋯)
      r r' r' ⋯ ⋯

Tactic:
← D₂.pullHom_hom r (q ≫ f i) _ rfl (a ≫ p' j) q (by cat_disch) rfl _ r' rfl hr'

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j : ι'
a : Z ⟶ X' j
fac : a ≫ f' j = q ≫ f i
Z' : C
r : Z' ⟶ Z
r' : Z' ⟶ X i
a' : Z' ⟶ X' j
hr' : r ≫ q = r'
ha' : r ≫ a = a'
⊢ pullHom (D₁.hom (q ≫ f i) q (a ≫ p' j) ⋯ ⋯) r r' (r ≫ a ≫ p' j) ⋯ ⋯ ≫
      pullHom (φ.hom j) a' (r ≫ a ≫ p' j) (r ≫ a ≫ p' j) ⋯ ⋯ ≫
        pullHom (D₂.hom (q ≫ f i) (a ≫ p' j) q ⋯ ⋯) r (r ≫ a ≫ p' j) r' ⋯ ⋯ =
    pullHom
      (D₁.hom (q ≫ f i) q (a ≫ p' j) ⋯ ⋯ ≫
        pullHom (φ.hom j) a (a ≫ p' j) (a ≫ p' j) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a ≫ p' j) q ⋯ ⋯)
      r r' r' ⋯ ⋯
Module: Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
b₀✝ b₁✝ b₂✝ b₃✝ : Cᵒᵖ
f : b₀✝ ⟶ b₁✝
g : b₁✝ ⟶ b₂✝
h : b₂✝ ⟶ b₃✝
⊢ ((Cat.Hom.isoMk (J.overMapPullbackComp A h.unop (f ≫ g).unop).symm).hom ≫
        Bicategory.whiskerRight (Cat.Hom.isoMk (J.overMapPullbackComp A g.unop f.unop).symm).hom
            (J.overMapPullback A h.unop).toCatHom ≫
          (Bicategory.associator (J.overMapPullback A f.unop).toCatHom (J.overMapPullback A g.unop).toCatHom
                (J.overMapPullback A h.unop).toCatHom).hom ≫
            Bicategory.whiskerLeft (J.overMapPullback A f.unop).toCatHom
                (Cat.Hom.isoMk (J.overMapPullbackComp A h.unop g.unop).symm).inv ≫
              (Cat.Hom.isoMk (J.overMapPullbackComp A (g ≫ h).unop f.unop).symm).inv).toNatTrans =
    (eqToHom ⋯).toNatTrans

Tactic:
simpa [overMapPullbackCongr_eq_eqToIso] using!
      J.overMapPullback_assoc A h.unop g.unop f.unop

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Types

Goal state before:
pp: α : Type u
S : Sieve α
hs : S ∈ (Sheaf.finestTopologySingle (yoneda.obj (ULift.{u, 0} Bool))) α
x : α
hsx : ¬S.arrows (↾fun x_1 ↦ x)
β : Type u
f : β ⟶ PUnit.{u + 1}
hf : (Sieve.pullback (↾fun x_1 ↦ x) S).arrows f
⊢ (f ≫ ↾fun x ↦ { down := true }) = f ≫ ↾fun x ↦ { down := false }

Tactic:
ext y

Goal state after:
α : Type u
S : Sieve α
hs : S ∈ (Sheaf.finestTopologySingle (yoneda.obj (ULift.{u, 0} Bool))) α
x : α
hsx : ¬S.arrows (↾fun x_1 ↦ x)
β : Type u
f : β ⟶ PUnit.{u + 1}
hf : (Sieve.pullback (↾fun x_1 ↦ x) S).arrows f
y : β
⊢ ((ConcreteCategory.hom (f ≫ ↾fun x ↦ { down := true })).toFun y).down =
    ((ConcreteCategory.hom (f ≫ ↾fun x ↦ { down := false })).toFun y).down
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
k : B ⟶ K
hk : ((shiftFunctor C 1).map (ι.map k) ≫ α) ≫ f₃ = 0
l : (shiftFunctor C 1).obj (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : (shiftFunctor C 1).map (ι.map k) ≫ α = 0
m : (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj B)) ⟶ (Triangle.mk α β γ).obj₃
hm : (shiftFunctor C 1).map ((shiftFunctor C 1).map (ι.map k)) = m ≫ (Triangle.mk α β γ).mor₃
⊢ m = 0

Tactic:
← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _),

Goal state after:
C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
k : B ⟶ K
hk : ((shiftFunctor C 1).map (ι.map k) ≫ α) ≫ f₃ = 0
l : (shiftFunctor C 1).obj (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : (shiftFunctor C 1).map (ι.map k) ≫ α = 0
m : (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj B)) ⟶ (Triangle.mk α β γ).obj₃
hm : (shiftFunctor C 1).map ((shiftFunctor C 1).map (ι.map k)) = m ≫ (Triangle.mk α β γ).mor₃
⊢ (shiftFunctorAdd' C 1 1 2 mono_ιK._proof_1_4).hom.app (ι.obj B) ≫ m =
    (shiftFunctorAdd' C 1 1 2 mono_ιK._proof_1_4).hom.app (ι.obj B) ≫ 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
k : Q ⟶ B
hk : f₂ ≫ β ≫ ι.map k = 0
l : (shiftFunctor C 1).obj (Triangle.mk (ι.map f₁) f₂ f₃).obj₁ ⟶ ι.obj B
hl : β ≫ ι.map k = 0
m : (shiftFunctor C 1).obj (Triangle.mk α β γ).obj₁ ⟶ ι.obj B
hm : ι.map k = (Triangle.mk α β γ).mor₃ ≫ m
⊢ m = 0

Tactic:
← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _),

Goal state after:
C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
k : Q ⟶ B
hk : f₂ ≫ β ≫ ι.map k = 0
l : (shiftFunctor C 1).obj (Triangle.mk (ι.map f₁) f₂ f₃).obj₁ ⟶ ι.obj B
hl : β ≫ ι.map k = 0
m : (shiftFunctor C 1).obj (Triangle.mk α β γ).obj₁ ⟶ ι.obj B
hm : ι.map k = (Triangle.mk α β γ).mor₃ ≫ m
⊢ (shiftFunctorAdd' C 1 1 2 epi_πQ._proof_1_2).hom.app (ι.obj K) ≫ m =
    (shiftFunctorAdd' C 1 1 2 epi_πQ._proof_1_2).hom.app (ι.obj K) ≫ 0