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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
q : ℕ
⊢ (-1) ^ q = -((-1) ^ q * (-1) ^ 1)

Tactic:
pow_one,

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
q : ℕ
⊢ (-1) ^ q = -((-1) ^ q * -1)
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
i₁ i₁' : I₁
hi₁ : c₁.Rel i₁ i₁'
i₂ i₂' : I₂
hi₂ : c₂.Rel i₂ i₂'
j : J
hj : c₁.π c₂ c (i₁', i₂) = j
hj' : c₁.π c₂ c (i₁, i₂') = j
⊢ -(c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
      (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
          (F.obj (L₁.X i₁)).map (L₂.d i₂ i₂') ≫
            (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).ιTotal c i₁ i₂' j hj' =
    -((c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
        (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (K₂.d i₂ i₂') ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂') ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂' j hj')

Tactic:
Units.neg_smul,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
i₁ i₁' : I₁
hi₁ : c₁.Rel i₁ i₁'
i₂ i₂' : I₂
hi₂ : c₂.Rel i₂ i₂'
j : J
hj : c₁.π c₂ c (i₁', i₂) = j
hj' : c₁.π c₂ c (i₁, i₂') = j
⊢ -((c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
        (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫
            (F.obj (L₁.X i₁)).map (L₂.d i₂ i₂') ≫
              (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).ιTotal c i₁ i₂' j hj') =
    -((c₁.ε₂ c₂ c (i₁', i₂) * c₁.ε₁ c₂ c (i₁, i₂')) •
        (F.map (h₁.hom i₁' i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (K₂.d i₂ i₂') ≫ (F.obj (L₁.X i₁)).map (f₂.f i₂') ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂' j hj')
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n n' : ℤ
x✝ : (up ℤ).Rel n n'
⊢ (K.totalShift₁XIso x n (n + x) ⋯).hom ≫ ((shiftFunctor (HomologicalComplex C (up ℤ)) x).obj (K.total (up ℤ))).d n n' =
    (((shiftFunctor₁ C x).obj K).total (up ℤ)).d n n' ≫ (K.totalShift₁XIso x n' (n' + x) ⋯).hom

Tactic:
dsimp
      simp only [total_d, Preadditive.add_comp, Preadditive.comp_add, smul_add,
        Linear.comp_units_smul, K.D₁_totalShift₁XIso_hom x n n' _ _ rfl rfl,
        K.D₂_totalShift₁XIso_hom x n n' _ _ rfl rfl]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₁.π c₂ c₁₂ (c₁.next i₁, i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = j'

Tactic:
← h₂,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₁.π c₂ c₁₂ (c₁.next i₁, i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃)
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (i₁, c₂.next i₂, i₃) = j'

Tactic:
← h₂,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : 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
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
h₂ : c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = c₁₂.next (c₁.π c₂ c₁₂ (i₁, i₂))
⊢ c₁.r c₂ c₃ c₁₂ c₄ (i₁, c₂.next i₂, i₃) = c₁₂.π c₃ c₄ (c₁₂.next i₁₂, i₃)
Module: Mathlib.Algebra.Homology.CochainComplexOpposite

Goal state before:
pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
H :
  ∀ (p q p' q' : ℤ) (hp : p = p') (hq : q = q'),
    h.hom p q = (HomologicalComplex.XIsoOfEq L ⋯).hom.op ≫ h.hom p' q' ≫ (HomologicalComplex.XIsoOfEq K ⋯).hom.op
n : ℤ
this : (f.f (-n)).op = (dNext n) h.hom + (prevD n) h.hom + (g.f (-n)).op
⊢ (dNext n) h.hom =
    ((prevD (-n)) fun p q ↦
        (HomologicalComplex.XIsoOfEq K ⋯).hom ≫ (h.hom (-q) (-p)).unop ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom).op

Tactic:
refine (dNext_eq _ (i' := n + 1) (by dsimp)).trans ?_
      rw [prevD_eq _ (j' := - (n + 1)) (by simp)]
      dsimp
      simp [H (- -(n + 1)) (- -n) (n + 1) n (by simp) (by simp), ← op_comp_assoc, ← op_comp]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q : ℤ
hp : p + -p = 0
hq : q + -q = 0
⊢ (homotopyUnop h).hom p q =
    (HomologicalComplex.XIsoOfEq K ⋯).hom ≫ (h.hom (-q) (-p)).unop ≫ (HomologicalComplex.XIsoOfEq L ⋯).hom

Tactic:
simp [homotopyUnop]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
inst✝ : ∀ (K₁ : CochainComplex C₁ ℤ) (K₂ : CochainComplex C₂ ℤ), K₁.HasMapBifunctor K₂ F
K₂ : CochainComplex C₂ ℤ
⊢ NatIso.ofComponents (fun K₁ ↦ K₁.mapBifunctorShift₁Iso K₂ F 0) ⋯ =
    CommShift.isoZero (F.map₂CochainComplex.flip.obj K₂) ℤ

Tactic:
ext K₂ n
    dsimp
    ext p q h
    simp [CochainComplex.ι_mapBifunctorShift₁Iso_hom_f _ _ F 0 p q n h p n
      (by lia) (by lia), CochainComplex.shiftFunctorZero_eq]

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
inst✝ : ∀ (K₁ : CochainComplex C₁ ℤ) (K₂ : CochainComplex C₂ ℤ), K₁.HasMapBifunctor K₂ F
K₂ L₂ : CochainComplex C₂ ℤ
g : K₂ ⟶ L₂
n : ℤ
⊢ (commShiftIso (F.map₂CochainComplex.flip.obj K₂) n).hom ≫
      whiskerRight (F.map₂CochainComplex.flip.map g) (shiftFunctor (CochainComplex D ℤ) n) =
    (shiftFunctor (CochainComplex C₁ ℤ) n).whiskerLeft (F.map₂CochainComplex.flip.map g) ≫
      (commShiftIso (F.map₂CochainComplex.flip.obj L₂) n).hom

Tactic:
ext K₁ d
    dsimp
    ext p q h
    simp [commShiftIso_map₂CochainComplex_flip_hom_app,
      CochainComplex.ι_mapBifunctorShift₁Iso_hom_f _ _ _ _ _ _ _ _ (p + n) (d + n) rfl rfl,
      CochainComplex.ι_mapBifunctorShift₁Iso_hom_f_assoc _ _ _ _ _ _ _ _ (p + n) (d + n) rfl rfl]

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
⊢ (MorphismProperty.isomorphisms (HomotopyCategory C (ComplexShape.up ℤ))).inverseImage
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)) f

Tactic:
MorphismProperty.inverseImage_iff,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
inst✝¹ : K.IsKInjective
inst✝ : L.IsKInjective
f : K ⟶ L
x✝ : QuasiIso f
⊢ MorphismProperty.isomorphisms (HomotopyCategory C (ComplexShape.up ℤ))
    ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f)
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case pos.h
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
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✝⁸ : 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✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = j'

Tactic:
← h₂,

Goal state after:
case pos.h
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
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✝⁸ : 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✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ c₁.r c₂ c₃ c₁₂ c₄ (c₁.next i₁, i₂, i₃) = c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃))
Module: Mathlib.Algebra.Homology.DerivedCategory.KInjective

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
n : ℤ
inst✝¹ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L
inst✝ : L.IsKInjective
⊢ Function.Bijective toSmallShiftedHom

Tactic:
by
  let := HasDerivedCategory.standard C
  rw [← Function.Bijective.of_comp_iff'
      (SmallShiftedHom.equiv _ DerivedCategory.Q).bijective,
    ← Function.Bijective.of_comp_iff' (Iso.homCongr ((quotientCompQhIso C).symm.app K)
      ((Q.commShiftIso n).symm.app L ≪≫ (quotientCompQhIso C).symm.app (L⟦n⟧))).bijective]
  convert! (CochainComplex.IsKInjective.Qh_map_bijective _ _).comp (toHom_bijective K L n)
  ext x
  obtain ⟨x, rfl⟩ := x.mk_surjective
  simp [toHom_mk, ShiftedHom.map]

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.RetractArgument

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝¹ : W₁.HasFactorization W₂
inst✝ : W₁.IsStableUnderRetracts
h₁ : W₁ ≤ W₂.llp
A B : C
i : A ⟶ B
hi : W₂.llp i
h : W₁.MapFactorizationData W₂ i
this : HasLiftingProperty i h.p
⊢ W₁ i

Tactic:
simpa using of_retract (RetractArrow.ofLeftLiftingProperty h.fac) h.hi

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.RetractArgument

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝¹ : W₁.HasFactorization W₂
inst✝ : W₂.IsStableUnderRetracts
h₂ : W₂ ≤ W₁.rlp
X Y : C
p : X ⟶ Y
hp : W₁.rlp p
⊢ W₂ p

Tactic:
have h := factorizationData W₁ W₂ p

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝¹ : W₁.HasFactorization W₂
inst✝ : W₂.IsStableUnderRetracts
h₂ : W₂ ≤ W₁.rlp
X Y : C
p : X ⟶ Y
hp : W₁.rlp p
h : W₁.MapFactorizationData W₂ p
⊢ W₂ p
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
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✝⁹ : 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✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
h₃ : c₂.Rel i₂ (c₂.next i₂)
⊢ c₁.r c₂ c₃ c₁₂ c₄ (i₁, c₂.next i₂, i₃) = j'

Tactic:
dsimp [ComplexShape.r]
              rw [← h₂, ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄,
                ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
A : C
P✝ : Cylinder A
inst✝² : IsCofibrant A
P P' : Cylinder A
inst✝¹ : P.IsGood
inst✝ : P'.IsGood
⊢ Cofibration (P.trans P').i

Tactic:
let ψ : P.I ⨿ A ⟶ (P.trans P').I := coprod.desc (pushout.inl _ _) (P'.i₁ ≫ pushout.inr _ _)
    rw [show (P.trans P').i = coprod.map P.i₀ (𝟙 A) ≫ ψ by simp [Precylinder.i, ψ]]
    have fac : coprod.map P.i₁ (𝟙 A) ≫ ψ = P'.i ≫ pushout.inr _ _ := by
      ext
      · simp [ψ, pushout.condition]
      · simp [ψ]
    have sq : IsPushout P.i₁ (coprod.inl ≫ P'.i) (coprod.inl ≫ ψ) (pushout.inr _ _) := by
      simpa [ψ] using IsPushout.of_hasPushout P.i₁ P'.i₀
    have : Cofibration ψ := by
      rw [cofibration_iff]
      exact (cofibrations C).of_isPushout
        (IsPushout.of_top sq fac (IsPushout.of_coprod_inl_with_id P.i₁ A).flip)
        (by rw [← cofibration_iff]; infer_instance)
    infer_instance

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h : LeftHomotopyRel f g
⊢ ∃ P, P.IsVeryGood ∧ Nonempty (P.LeftHomotopy f g)

Tactic:
by
  obtain ⟨P, _, ⟨h⟩⟩ := h.exists_good_cylinder
  let fac := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C) P.π
  let P' : Cylinder X :=
    { I := fac.Z
      i₀ := P.i₀ ≫ fac.i
      i₁ := P.i₁ ≫ fac.i
      π := fac.p
      weakEquivalence_π := weakEquivalence_of_precomp_of_fac fac.fac }
  have : Cofibration P'.i := by
    rw [show P'.i = P.i ≫ fac.i by cat_disch]
    infer_instance
  have sq : CommSq h.h fac.i (terminal.from _) (terminal.from _) := { }
  exact ⟨P', { }, ⟨{ h := sq.lift }⟩ ⟩

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Plus

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
hK : CochainComplex.plus C K
⊢ CochainComplex.plus C (pathObject K)

Tactic:
by
  obtain ⟨n, hn⟩ := hK
  refine ⟨n - 1, ?_⟩
  rw [CochainComplex.isStrictlyGE_iff]
  intro i hi
  refine pathObject.isZero_X _ _ (K.isZero_of_isStrictlyGE n i)
    (fun j hj ↦ ?_)
  simp only [ComplexShape.up_Rel] at hj
  exact K.isZero_of_isStrictlyGE n j

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Plus

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X : Plus C
n : ℤ
inst✝ : X.IsGE n
⊢ (ι.obj X).IsGE n

Tactic:
rw [isGE_ι_obj_iff]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X : Plus C
n : ℤ
inst✝ : X.IsGE n
⊢ X.IsGE n