mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: case refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
⊢ IsZero ((K ⊞ K).X i)

pp: case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
j : ℤ
hj : (ComplexShape.up ℤ).Rel i j
⊢ IsZero (K.X j)

Tactic:
· refine IsZero.of_iso ?_ ((HomologicalComplex.eval C (.up ℤ) i).mapBiprod _ _)
    simpa using K.isZero_of_isStrictlyGE n i

Goal state after:
case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
j : ℤ
hj : (ComplexShape.up ℤ).Rel i j
⊢ IsZero (K.X j)
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{v_1, u_1} C₁
inst✝³² : Category.{v_2, u_2} C₂
inst✝³¹ : Category.{v_3, u_5} C₃
inst✝³⁰ : Category.{v_4, u_6} C₄
inst✝²⁹ : Category.{v_5, u_3} C₁₂
inst✝²⁸ : Category.{v_6, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h : ¬c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
⊢ mapBifunctor₁₂.ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j ≫
      (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j).hom =
    ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
      mapBifunctor₂₃.ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j

Tactic:
rw [mapBifunctor₁₂.ιOrZero_eq_zero _ _ _ _ _ _ _ _ _ _ _ h,
      mapBifunctor₂₃.ιOrZero_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h,
      zero_comp, comp_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
X : C
⊢ t.IsGE X n₁ ↔ IsIso ((t.truncGTπ n₀).app X)

Tactic:
by
  rw [t.isGE_iff_isIso_truncGEπ_app n₁ X]
  exact (MorphismProperty.isomorphisms _).arrow_mk_iso_iff
    (Arrow.isoMk (Iso.refl _) ((t.truncGTIsoTruncGE _ _ hn₁).symm.app X))

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝¹ : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this : L₁.Additive
⊢ ((A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Faithful

Tactic:
have := Localization.functor_additive L₂ B.trW

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
⊢ ((A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Faithful
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (biprod.inr ≫
        { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
          (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr)) ≫
      biprod.snd =
    (biprod.inr ≫
        𝟙
          { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂,
              zero := ⋯ }.X₁) ≫
      biprod.snd

Tactic:
· simp [reassoc_of% σ₁.s_g, σ₂.f_r]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
this : L₂.op.IsLocalization B.op.trW
⊢ (B.inverseImage A.ι).trW.op ≤ (B.op.inverseImage A.op.ι).trW.isoClosure.inverseImage A.opEquivalence.symm.functor

Tactic:
rw [← trW_op, ← inverseImage_opEquivalence_inverse_trW_inverseImage_ι_op]
      intro _ _ f hf
      simp only [MorphismProperty.inverseImage_iff, Equivalence.symm_functor] at hf ⊢
      exact MorphismProperty.le_isoClosure _ _ hf

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ℕ ⥤ C
app : (n : ℕ) → F.obj n ⟶ G.obj n
naturality : ∀ (n : ℕ), F.map (homOfLE ⋯) ≫ app (n + 1) = app n ≫ G.map (homOfLE ⋯)
⊢ ∀ ⦃X Y : ℕ⦄ (f : X ⟶ Y), F.map f ≫ app Y = app X ≫ G.map f

Tactic:
intro i j φ

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ℕ ⥤ C
app : (n : ℕ) → F.obj n ⟶ G.obj n
naturality : ∀ (n : ℕ), F.map (homOfLE ⋯) ≫ app (n + 1) = app n ≫ G.map (homOfLE ⋯)
i j : ℕ
φ : i ⟶ j
⊢ F.map φ ≫ app j = app i ≫ G.map φ
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : ℕᵒᵖ ⥤ C
app : (n : ℕ) → F.obj (Opposite.op n) ⟶ G.obj (Opposite.op n)
naturality : ∀ (n : ℕ), F.map (homOfLE ⋯).op ≫ app n = app (n + 1) ≫ G.map (homOfLE ⋯).op
x✝¹ x✝ : ℕᵒᵖ
f : x✝¹ ⟶ x✝
⊢ F.map f ≫ app (Opposite.unop x✝) = app (Opposite.unop x✝¹) ≫ G.map f

Tactic:
let φ : G.rightOp ⟶ F.rightOp := ofSequence (fun n ↦ (app n).op)
      (fun n ↦ Quiver.Hom.unop_inj (naturality n).symm)
    exact Quiver.Hom.op_inj (φ.naturality f.unop).symm

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Injective fun f ↦ f.postcomp g

Tactic:
intro f₀ f₁ h

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
f₀ f₁ : LeftHomotopyClass X Y
h : (fun f ↦ f.postcomp g) f₀ = (fun f ↦ f.postcomp g) f₁
⊢ f₀ = f₁
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Injective fun g ↦ g.precomp f

Tactic:
intro f₀ f₁ h

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
f₀ f₁ : RightHomotopyClass Y Z
h : (fun g ↦ g.precomp f) f₀ = (fun g ↦ g.precomp f) f₁
⊢ f₀ = f₁
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
⊢ w.GuitartExact ↔ ∀ {X₂ : C₂} {X₃ : C₃} (g : R.obj X₂ ⟶ B.obj X₃), IsConnected (w.CostructuredArrowDownwards g)

Tactic:
simp only [guitartExact_iff_isConnected_rightwards,
    isConnected_rightwards_iff_downwards]

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
H : Type u₃
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₃, u₃} H
F : C ⥤ H
W : MorphismProperty C
X Y : C
w : X ⟶ Y
hw : W w
⊢ W.Q.HasPointwiseLeftKanExtensionAt F (W.Q.obj X) ↔ W.Q.HasPointwiseLeftKanExtensionAt F (W.Q.obj Y)

Tactic:
exact hasPointwiseLeftKanExtensionAt_iff_of_iso W.Q F (Localization.isoOfHom W.Q W w hw)

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
H : Type u₃
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₃, u₃} H
F : C ⥤ H
W : MorphismProperty C
hF : W.IsInvertedBy F
⊢ F.HasPointwiseRightDerivedFunctor W

Tactic:
intro X
  rw [hasPointwiseRightDerivedFunctorAt_iff F W.Q W]
  exact (isPointwiseLeftKanExtensionOfIsoOfIsLocalization W
    (Localization.fac F hF W.Q).symm).hasPointwiseLeftKanExtension _

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X Y : DifferentialObject S C
f : X.obj ≅ Y.obj
hf : X.d ≫ (shiftFunctor C 1).map f.hom = f.hom ≫ Y.d
⊢ { f := f.hom, comm := hf } ≫ { f := f.inv, comm := ⋯ } = 𝟙 X

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X Y : DifferentialObject S C
f : X.obj ≅ Y.obj
hf : X.d ≫ (shiftFunctor C 1).map f.hom = f.hom ≫ Y.d
⊢ autoParam (({ f := f.hom, comm := hf } ≫ { f := f.inv, comm := ⋯ }).f = (𝟙 X).f) ext._auto_1
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X Y : DifferentialObject S C
f : X.obj ≅ Y.obj
hf : X.d ≫ (shiftFunctor C 1).map f.hom = f.hom ≫ Y.d
⊢ { f := f.inv, comm := ⋯ } ≫ { f := f.hom, comm := hf } = 𝟙 Y

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddMonoidWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
X Y : DifferentialObject S C
f : X.obj ≅ Y.obj
hf : X.d ≫ (shiftFunctor C 1).map f.hom = f.hom ≫ Y.d
⊢ autoParam (({ f := f.inv, comm := ⋯ } ≫ { f := f.hom, comm := hf }).f = (𝟙 Y).f) ext._auto_1
Module: Mathlib.Algebra.Homology.Double

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ i₁ : ι
hi₀₁ : c.Rel i₀ i₁
h : i₀ ≠ i₁
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
φ₁ : X₁ ⟶ K.X i₁
comm : φ₀ ≫ K.d i₀ i₁ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel i₁ k → φ₁ ≫ K.d i₁ k = 0
k : ι
hk₀ : ¬k = i₀
hk₁ : k = i₁
⊢ (double f hi₀₁).X k = (double f hi₀₁).X i₁

Tactic:
by rw [hk₁]

Goal state after:
Module: Mathlib.Algebra.Homology.Double

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ i₁ : ι
hi₀₁ : c.Rel i₀ i₁
h : i₀ ≠ i₁
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
φ₁ : X₁ ⟶ K.X i₁
comm : φ₀ ≫ K.d i₀ i₁ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel i₁ k → φ₁ ≫ K.d i₁ k = 0
k : ι
hk₀ : ¬k = i₀
hk₁ : k = i₁
⊢ K.X i₁ = K.X k

Tactic:
by rw [hk₁]

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ { f := (CategoryTheory.shiftFunctor C n).map (𝟙 X).f, comm := ⋯ } =
    𝟙
      { obj := (CategoryTheory.shiftFunctor C n).obj X.obj,
        d := (CategoryTheory.shiftFunctor C n).map X.d ≫ (shiftComm X.obj 1 n).hom, d_squared := ⋯ }

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ autoParam
    ({ f := (CategoryTheory.shiftFunctor C n).map (𝟙 X).f, comm := ⋯ }.f =
      (𝟙
          { obj := (CategoryTheory.shiftFunctor C n).obj X.obj,
            d := (CategoryTheory.shiftFunctor C n).map X.d ≫ (shiftComm X.obj 1 n).hom, d_squared := ⋯ }).f)
    ext._auto_1
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ Z✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { f := (CategoryTheory.shiftFunctor C n).map (f ≫ g).f, comm := ⋯ } =
    { f := (CategoryTheory.shiftFunctor C n).map f.f, comm := ⋯ } ≫
      { f := (CategoryTheory.shiftFunctor C n).map g.f, comm := ⋯ }

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ Z✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ autoParam
    ({ f := (CategoryTheory.shiftFunctor C n).map (f ≫ g).f, comm := ⋯ }.f =
      ({ f := (CategoryTheory.shiftFunctor C n).map f.f, comm := ⋯ } ≫
          { f := (CategoryTheory.shiftFunctor C n).map g.f, comm := ⋯ }).f)
    ext._auto_1
Module: Mathlib.Algebra.Homology.Double

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ : ι
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
k₀ k₁ : ι
hk : c.Rel k₀ k₁
h₀ : ¬k₀ = i₀
hi₀₁ : c.Rel i₀ k₀
h : i₀ ≠ k₀
φ₁ : X₁ ⟶ K.X k₀
comm : φ₀ ≫ K.d i₀ k₀ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel k₀ k → φ₁ ≫ K.d k₀ k = 0
⊢ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ K.d k₀ k₁ =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = i₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0

Tactic:
hφ k₁ hk,

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X₀ X₁ : C
f : X₀ ⟶ X₁
ι : Type u_2
c : ComplexShape ι
i₀ : ι
K : HomologicalComplex C c
φ₀ : X₀ ⟶ K.X i₀
k₀ k₁ : ι
hk : c.Rel k₀ k₁
h₀ : ¬k₀ = i₀
hi₀₁ : c.Rel i₀ k₀
h : i₀ ≠ k₀
φ₁ : X₁ ⟶ K.X k₀
comm : φ₀ ≫ K.d i₀ k₀ = f ≫ φ₁
hφ : ∀ (k : ι), c.Rel k₀ k → φ₁ ≫ K.d k₀ k = 0
⊢ (doubleXIso₁ f hi₀₁ h).hom ≫ 0 =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = i₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case pos.ofNat
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
n : ℕ
hnm : Int.ofNat n + 1 = m
⊢ h.d (Int.ofNat n) m ≫ h.d m p = 0

Tactic:
obtain rfl : m = .ofNat (n + 1) := by simp [← hnm]
    obtain rfl : p = .ofNat (n + 2) := by simp [← hmp]; lia
    simp only [Int.ofNat_eq_natCast, X_ofNat, d_ofNat, HomologicalComplex.d_comp_d]

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
m₁ m₂ m₃ : S
X : DifferentialObject S C
⊢ (shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X ≫ (shiftFunctor C m₃).map ((shiftFunctorAdd C m₁ m₂).hom.app X) =
    eqToHom ⋯ ≫
      (shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X)

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
m₁ m₂ m₃ : S
X : DifferentialObject S C
⊢ autoParam
    (((shiftFunctorAdd C (m₁ + m₂) m₃).hom.app X ≫ (shiftFunctor C m₃).map ((shiftFunctorAdd C m₁ m₂).hom.app X)).f =
      (eqToHom ⋯ ≫
          (shiftFunctorAdd C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X)).f)
    ext._auto_1
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ (shiftFunctorAdd C 0 n).hom.app X = eqToHom ⋯ ≫ (shiftFunctor C n).map ((shiftZero C).inv.app X)

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ autoParam (((shiftFunctorAdd C 0 n).hom.app X).f = (eqToHom ⋯ ≫ (shiftFunctor C n).map ((shiftZero C).inv.app X)).f)
    ext._auto_1
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ (shiftFunctorAdd C n 0).hom.app X = eqToHom ⋯ ≫ (shiftZero C).inv.app ((shiftFunctor C n).obj X)

Tactic:
ext1

Goal state after:
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ autoParam (((shiftFunctorAdd C n 0).hom.app X).f = (eqToHom ⋯ ≫ (shiftZero C).inv.app ((shiftFunctor C n).obj X)).f)
    ext._auto_1
Module: Mathlib.CategoryTheory.Generator.HomologicalComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
ι : Type w
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : (ObjectProperty.ofObj X).IsSeparating
⊢ (ObjectProperty.ofObj (separatingFamily c X)).IsSeparating

Tactic:
intro K L f g h

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
ι : Type w
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : (ObjectProperty.ofObj X).IsSeparating
K L : HomologicalComplex C c
f g : K ⟶ L
h : ∀ (G : HomologicalComplex C c), ObjectProperty.ofObj (separatingFamily c X) G → ∀ (h : G ⟶ K), h ≫ f = h ≫ g
⊢ f = g
Module: Mathlib.Algebra.Homology.ExactSequenceFour

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
n : ℕ
S : ComposableArrows C (n + 3)
hS : S.IsComplex
k : ℕ
hk : k ≤ n
cc : CokernelCofork (S.map' k (k + 1) ⋯ ⋯)
kf : KernelFork (S.map' (k + 2) (k + 3) ⋯ ⋯)
hcc : IsColimit cc
hkf : IsLimit kf
hS' : (S.sc hS k ⋯).Exact
⊢ Mono (hS.cokerToKer' k hk cc kf hcc hkf)

Tactic:
by
  have := hS'.hasZeroObject
  have := hS'.hasHomology
  let h := hS'.rightHomologyDataOfIsColimitCokernelCofork cc hcc
  have := h.exact_iff_mono_g'.1 hS'
  have fac : hS.cokerToKer' k hk cc kf hcc hkf ≫ kf.ι = h.g' := by
    rw [← cancel_epi h.p, h.p_g', ShortComplex.Exact.rightHomologyDataOfIsColimitCokernelCofork_p,
      cokerToKer'_fac]
  exact mono_of_mono_fac fac

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ CokernelCofork.ofπ (pOpcycles P.complex 0) ⋯ ≅ P.cokernelCofork

Tactic:
refine Cofork.ext (P.complex.isoHomologyι₀.symm ≪≫ isoOfQuasiIsoAt P.π 0 ≪≫
    singleObjHomologySelfIso _ _ _) ?_

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ Cofork.π (CokernelCofork.ofπ (pOpcycles P.complex 0) ⋯) ≫
      (P.complex.isoHomologyι₀.symm ≪≫
          isoOfQuasiIsoAt P.π 0 ≪≫ singleObjHomologySelfIso (ComplexShape.down ℕ) 0 Z).hom =
    Cofork.π P.cokernelCofork
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian D
C : Type u_2
inst✝² : Category.{v_2, u_2} C
inst✝¹ : Abelian C
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex).pOpcycles 0 ≫ P.fromLeftDerivedZero' F =
    F.map (P.π.f 0)

Tactic:
simp [fromLeftDerivedZero']

Goal state after: