mathlib-initiative/mathlib-tactics huggingface dataset:
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
hw : ∀ {X₂ : C₂} {X₃ : C₃} (g : R.obj X₂ ⟶ B.obj X₃), IsConnected (w.CostructuredArrowDownwards g)
X₂ : C₂
g : StructuredArrow (R.obj X₂) B
⊢ IsConnected (CostructuredArrow (w.structuredArrowDownwards X₂) g)

Tactic:
apply hw

Goal state after:
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
hw : ∀ (X₃ : C₃), (w.costructuredArrowRightwards X₃).Final
X₃ : C₃
⊢ (w.costructuredArrowRightwards X₃).Final

Tactic:
apply hw

Goal state after:
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
hw : ∀ (X₂ : C₂), (w.structuredArrowDownwards X₂).Initial
X₂ : C₂
⊢ (w.structuredArrowDownwards X₂).Initial

Tactic:
apply hw

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.KanExtension

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
D : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} C₃
inst✝³ : Category.{v₄, u₄} C₄
inst✝² : Category.{v₅, u₅} D
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
inst✝¹ : w.GuitartExact
inst✝ : B.EssSurj
F : C₂ ⥤ D
⊢ L.HasPointwiseLeftKanExtension (T ⋙ F) ↔ R.HasPointwiseLeftKanExtension F

Tactic:
by
  dsimp [Functor.HasPointwiseLeftKanExtension]
  simp only [hasPointwiseLeftKanExtensionAt_iff w]
  refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩
  rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]
  apply h

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

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ F'
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cocone (CostructuredArrow.proj L (L.obj Y) ⋙ F)
j : CostructuredArrow L (L.obj Y)
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj X₁ ⟶ L.obj Y
hφ :
  ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app (CostructuredArrow.mk φ) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk φ)
this : IsIso (L.map w)
⊢ IsIso (F.map w)

Tactic:
← NatIso.naturality_2 e w

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} H
F' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ F'
W : MorphismProperty C
G : D ⥤ H
e : F ≅ L ⋙ G
inst✝ : L.IsLocalization W
Y : C
s : Cocone (CostructuredArrow.proj L (L.obj Y) ⋙ F)
j : CostructuredArrow L (L.obj Y)
X₁ X₂ : C
w : X₁ ⟶ X₂
hw : W w
φ : L.obj X₁ ⟶ L.obj Y
hφ :
  ((LeftExtension.mk G e.hom).coconeAt (L.obj Y)).ι.app (CostructuredArrow.mk φ) ≫
      e.inv.app Y ≫ s.ι.app (CostructuredArrow.mk (𝟙 (L.obj Y))) =
    s.ι.app (CostructuredArrow.mk φ)
this : IsIso (L.map w)
⊢ IsIso (e.hom.app X₁ ≫ (L ⋙ G).map w ≫ e.inv.app X₂)
Module: Mathlib.CategoryTheory.GuitartExact.HorizontalComposition

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
T' : C₁ ⥤ D₁
B' : C₂ ⥤ D₂
inst✝ : w.GuitartExact
α : T ≅ T'
β : B ≅ B'
X₂ : C₂
⊢ ((w.whiskerHorizontal α.inv β.hom).costructuredArrowRightwards X₂).Final

Tactic:
let e : costructuredArrowRightwards (w.whiskerHorizontal α.inv β.hom) X₂ ≅
      w.costructuredArrowRightwards X₂ ⋙ (CostructuredArrow.mapIso (β.app X₂)).functor :=
    NatIso.ofComponents (fun f ↦ CostructuredArrow.isoMk (α.symm.app f.left))

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
T' : C₁ ⥤ D₁
B' : C₂ ⥤ D₂
inst✝ : w.GuitartExact
α : T ≅ T'
β : B ≅ B'
X₂ : C₂
e : (w.whiskerHorizontal α.inv β.hom).costructuredArrowRightwards X₂ ≅
  w.costructuredArrowRightwards X₂ ⋙ (CostructuredArrow.mapIso (β.app X₂)).functor :=
  NatIso.ofComponents (fun f ↦ CostructuredArrow.isoMk (α.symm.app f.left) ⋯) ⋯
⊢ ((w.whiskerHorizontal α.inv β.hom).costructuredArrowRightwards X₂).Final
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
d : ℤ
inst✝¹ : K.IsStrictlyLE d
inst✝ : ∀ (n : ℤ), Projective (K.X n)
L : CochainComplex Cᵒᵖ ℤ := (opEquivalence C).functor.obj (op K)
this✝ : ∀ (n : ℤ), Injective (L.X n)
this : L.IsStrictlyGE (-d)
⊢ K.IsKProjective

Tactic:
exact isKProjective_of_op (isKInjective_of_injective L (-d))

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
ι : Type u_2
c : ComplexShape ι
inst✝ : c.EulerCharSigns
X : GradedObject ι (ModuleCat R)
⊢ (Function.support fun i ↦ ↑(c.χ i) * ↑(Module.finrank R ↑(X i))) = finrankSupport X

Tactic:
by
  simp only [finrankSupport, Function.support_mul_of_ne_zero_left (fun i => Units.ne_zero (c.χ i))]
  ext i; simp [Function.mem_support]

Goal state after:
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
i₁ : ι
K : HomologicalComplex C c
φ₁ : X₁ ⟶ K.X i₁
hφ : ∀ (k : ι), c.Rel i₁ k → φ₁ ≫ K.d i₁ k = 0
k₀ k₁ : ι
hk : c.Rel k₀ k₁
hi₀₁ : c.Rel k₀ i₁
h : k₀ ≠ i₁
φ₀ : X₀ ⟶ K.X k₀
comm : φ₀ ≫ K.d k₀ i₁ = f ≫ φ₁
⊢ (if hk₀ : k₀ = k₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₀ = i₁ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0) ≫
      K.d k₀ k₁ =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = i₁ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0

Tactic:
rw [dif_pos rfl]

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
i₁ : ι
K : HomologicalComplex C c
φ₁ : X₁ ⟶ K.X i₁
hφ : ∀ (k : ι), c.Rel i₁ k → φ₁ ≫ K.d i₁ k = 0
k₀ k₁ : ι
hk : c.Rel k₀ k₁
hi₀₁ : c.Rel k₀ i₁
h : k₀ ≠ i₁
φ₀ : X₀ ⟶ K.X k₀
comm : φ₀ ≫ K.d k₀ i₁ = f ≫ φ₁
⊢ (eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯) ≫ K.d k₀ k₁ =
    (double f hi₀₁).d k₀ k₁ ≫
      if hk₀ : k₁ = k₀ then eqToHom ⋯ ≫ (doubleXIso₀ f hi₀₁).hom ≫ φ₀ ≫ eqToHom ⋯
      else if hk₁ : k₁ = i₁ then eqToHom ⋯ ≫ (doubleXIso₁ f hi₀₁ h).hom ≫ φ₁ ≫ eqToHom ⋯ else 0
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
⊢ P.π.f 1 ≫ ((single₀ C).obj Z).d 1 0 = 0

Tactic:
single_obj_d,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
P : ProjectiveResolution Z
⊢ P.π.f 1 ≫ 0 = 0
Module: Mathlib.CategoryTheory.Idempotents.FunctorExtension

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
G : Karoubi C ⥤ Karoubi D
P Q : Karoubi C
f : P ⟶ Q
⊢ ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).map f ≫ { f := (G.map Q.decompId_i).f, comm := ⋯ } =
    { f := (G.map P.decompId_i).f, comm := ⋯ } ≫
      (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).map f

Tactic:
by
              simpa only [hom_ext_iff, G.map_comp] using! G.congr_map (decompId_i_naturality f)

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X : C
⊢ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map
      (ProjectiveResolution.lift (𝟙 X) (projectiveResolution X) (projectiveResolution X)) =
    (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map (𝟙 (projectiveResolution X).complex)

Tactic:
apply HomotopyCategory.eq_of_homotopy

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X : C
⊢ Homotopy (ProjectiveResolution.lift (𝟙 X) (projectiveResolution X) (projectiveResolution X))
    (𝟙 (projectiveResolution X).complex)
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map
      (ProjectiveResolution.lift (f ≫ g) (projectiveResolution X✝) (projectiveResolution Z✝)) =
    (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map
      (ProjectiveResolution.lift f (projectiveResolution X✝) (projectiveResolution Y✝) ≫
        ProjectiveResolution.lift g (projectiveResolution Y✝) (projectiveResolution Z✝))

Tactic:
apply HomotopyCategory.eq_of_homotopy

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ Homotopy (ProjectiveResolution.lift (f ≫ g) (projectiveResolution X✝) (projectiveResolution Z✝))
    (ProjectiveResolution.lift f (projectiveResolution X✝) (projectiveResolution Y✝) ≫
      ProjectiveResolution.lift g (projectiveResolution Y✝) (projectiveResolution Z✝))
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ P.iso.inv ≫ (projectiveResolutions C).map f = (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ ≫ Q.iso.inv

Tactic:
apply HomotopyCategory.eq_of_homotopy

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ Homotopy (((projectiveResolution X).homotopyEquiv P).inv ≫ lift f (projectiveResolution X) (projectiveResolution Y))
    (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv)
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i₁ i₂ j : I
h₁ : c.Rel i₁ (c.next i₁)
h₂ : c.π c c (c.next i₁, i₂) = j
⊢ c.ε₁ c c (i₁, i₂) •
      ((curriedTensor C).map ((tensorUnit C c).d i₁ (c.next i₁))).app (K.X i₂) ≫
        (tensorUnit C c).ιMapBifunctor K (curriedTensor C) c (c.next i₁) i₂ j h₂ =
    0

Tactic:
single_obj_d,

Goal state after:
case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i₁ i₂ j : I
h₁ : c.Rel i₁ (c.next i₁)
h₂ : c.π c c (c.next i₁, i₂) = j
⊢ c.ε₁ c c (i₁, i₂) •
      ((curriedTensor C).map 0).app (K.X i₂) ≫
        (tensorUnit C c).ιMapBifunctor K (curriedTensor C) c (c.next i₁) i₂ j h₂ =
    0
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ c.ε₂ c c (i₁, i₂) •
      ((curriedTensor C).obj (K.X i₁)).map ((tensorUnit C c).d i₂ (c.next i₂)) ≫
        K.ιMapBifunctor (tensorUnit C c) (curriedTensor C) c i₁ (c.next i₂) j h₂ =
    0

Tactic:
single_obj_d,

Goal state after:
case pos
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ c.ε₂ c c (i₁, i₂) •
      ((curriedTensor C).obj (K.X i₁)).map 0 ≫
        K.ιMapBifunctor (tensorUnit C c) (curriedTensor C) c i₁ (c.next i₂) j h₂ =
    0
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
by
  by_cases h₁ : c.Rel i₂ (c.next i₂)
  · by_cases h₂ : ComplexShape.π c c c (i₁, c.next i₂) = j
    · rw [mapBifunctor.d₂_eq _ _ _ _ _ h₁ _ h₂, single_obj_d, Functor.map_zero,
        zero_comp, smul_zero]
    · rw [mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂]
  · rw [mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₁]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
inst✝ : PreservesFiniteColimits F
X : C
P : ProjectiveResolution X
⊢ IsIso (P.fromLeftDerivedZero' F)

Tactic:
dsimp [ProjectiveResolution.fromLeftDerivedZero']
  rw [ChainComplex.isIso_descOpcycles_iff, ShortComplex.exact_and_epi_g_iff_g_is_cokernel]
  exact ⟨CokernelCofork.mapIsColimit _ (P.isColimitCokernelCofork) F⟩

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : C ⥤ D ⥤ D
X : C
e : F.obj X ≅ 𝟭 D
inst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)
p : I × J → J
hp : ∀ (j : J), p (0, j) = j
Y Y' : GradedObject J D
φ : Y ⟶ Y'
j : J
s : Cofan ((((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).mapObjFun p j)
m : (mapBifunctorLeftUnitorCofan F X e p hp Y j).pt ⟶ s.pt
hm : ∀ (j_1 : ↑(p ⁻¹' {j})), Cofan.inj (mapBifunctorLeftUnitorCofan F X e p hp Y j) j_1 ≫ m = s.inj j_1
⊢ m = e.inv.app (Y j) ≫ (F.map (singleObjApplyIso 0 X).inv).app (Y j) ≫ s.inj ⟨(0, j), ⋯⟩

Tactic:
simp [← hm ⟨⟨0, j⟩, hp j⟩]

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

Goal state before:
pp: case e_a
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (if hi : 0 = 0 then GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 hi ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 hi).symm
      else { hom := 0, inv := 0, hom_inv_id := ⋯, inv_hom_id := ⋯ }).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Tactic:
rw [dif_pos rfl]

Goal state after:
case e_a
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 ⋯ ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 ⋯).symm).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: case e_a
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i : I
⊢ (if hi : 0 = 0 then GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 hi ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 hi).symm
      else { hom := 0, inv := 0, hom_inv_id := ⋯, inv_hom_id := ⋯ }).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Tactic:
rw [dif_pos rfl]

Goal state after:
case e_a
C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i : I
⊢ (GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 ⋯ ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 ⋯).symm).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv
Module: Mathlib.Algebra.Homology.ShortComplex.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ (a • φ₁).τ₁ = S₁.f ≫ (a • h.h₁) + a • h.h₀ + (a • φ₂).τ₁

Tactic:
dsimp
    rw [h.comm₁]
    simp only [smul_add, Linear.comp_smul]

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁ ≫ X.fromOpcycles f₁ f₂ f₁₂ h₁₂ n₁ = X.iCycles f₂ f₃ n₀ ≫ X.δ f₁₂ f₃ n₀ n₁ hn₁

Tactic:
by
  rw [← cancel_epi (X.toCycles f₂ f₃ _ rfl n₀),
    toCycles_Ψ_assoc .., p_fromOpcycles, toCycles_i_assoc]
  exact (X.δ_naturality _ _ _ _ _ _ _ _ rfl).symm

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.EpiMono

Goal state before:
pp: case h₂
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso ((X.H n₁).map (homMk₁ (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 1)) (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 2)) ⋯))

Tactic:
convert! (inferInstance : IsIso ((X.H n₁).map (𝟙 _)))

Goal state after:
case e'_5
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ homMk₁ (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 1)) (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 2)) ⋯ = 𝟙 (mk₁ f₄)
Module: Mathlib.Algebra.Homology.SpectralObject.Homology

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ (X.dKernelSequence f₁ f₂ f₃ f₄ f₅ f₂₃ h₂₃ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).Exact

Tactic:
rw [ShortComplex.exact_iff_exact_up_to_refinements]
  intro A x₂ hx₂
  dsimp at hx₂ ⊢
  obtain ⟨A₁, π₁, _, y₂, hy₂⟩ :=
    surjective_up_to_refinements_of_epi (X.πE f₃ f₄ f₅ n₀ n₁ n₂) x₂
  have hy₂' := hy₂ =≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ ≫ X.ιE ..
  simp only [assoc, reassoc_of% hx₂, zero_comp, comp_zero, πE_d_ιE] at hy₂'
  obtain ⟨A₂, π₂, _, y₁, hy₁⟩ :=
    ((X.sequenceΨ_exact f₂ f₃ f₄ f₂₃ h₂₃ _ rfl n₁ n₂).exact 0).exact_up_to_refinements y₂ hy₂'.symm
  refine ⟨A₂, π₂ ≫ π₁, inferInstance, y₁ ≫ X.πE f₂₃ f₄ f₅ n₀ n₁ n₂, ?_⟩
  simp [sequenceΨ, hy₂, reassoc_of% hy₁, X.πE_map f₂₃ f₄ f₅ f₃ f₄ f₅ (fourδ₁Toδ₀ f₂ f₃ f₄ f₅ f₂₃)
    (threeδ₁Toδ₀ f₂ f₃ f₄ f₂₃) n₀ n₁ n₂]

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.d (𝟙 i₀) f₁ (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ (X.EIsoH f₁ n₁ n₂ n₃ hn₂ hn₃).hom =
    (X.EIsoH f₂ n₀ n₁ n₂ hn₁ hn₂).hom ≫ X.δ f₁ f₂ n₁ n₂ hn₂

Tactic:
← cancel_epi (X.πE (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ hn₁ hn₂),

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.πE (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ hn₁ hn₂ ≫
      X.d (𝟙 i₀) f₁ (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ (X.EIsoH f₁ n₁ n₂ n₃ hn₂ hn₃).hom =
    X.πE (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ hn₁ hn₂ ≫ (X.EIsoH f₂ n₀ n₁ n₂ hn₁ hn₂).hom ≫ X.δ f₁ f₂ n₁ n₂ hn₂