mathlib-initiative/mathlib-tactics huggingface dataset:
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 f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
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
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
⊢ (K₁.mapBifunctorFlipIso K₂ F c).inv ≫ mapBifunctorMap f₂' f₁ F.flip c ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom =
    mapBifunctorMap f₁ f₂' F c

Tactic:
by simp

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
⊢ ((K₁.mapBifunctor K₂ F₁₂ c₁₂).mapBifunctor K₃ G c₄).d j j' =
    D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j'

Tactic:
rw [mapBifunctor.d_eq]

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
⊢ mapBifunctor.D₁ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ j j' +
      mapBifunctor.D₂ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ j j' =
    D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j'
Module: Mathlib.Algebra.Homology.CochainComplexOpposite

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
x✝¹ x✝ : ℤ
⊢ (fun n ↦ -n) x✝¹ = (fun n ↦ -n) x✝ → x✝¹ = x✝

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
⊢ ∀ {i₁ i₂ : ℤ}, (up ℤ).Rel i₁ i₂ → (down ℤ).Rel (-i₁) (-i₂)

Tactic:
by simp

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✝ : HasZeroMorphisms C
K : ChainComplex C ℤ
n : ℤ
⊢ n = ComplexShape.embeddingUpIntDownInt.f (ComplexShape.embeddingDownIntUpInt.f n)

Tactic:
by simp

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✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
⊢ ComplexShape.embeddingDownIntUpInt.f (ComplexShape.embeddingUpIntDownInt.f n) = n

Tactic:
by simp

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 f g
n : ℤ
⊢ (ComplexShape.up ℤ).Rel (-(n + 1)) (-n)

Tactic:
by simp

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 f g
n : ℤ
⊢ (ComplexShape.up ℤ).Rel n (n + 1)

Tactic:
by simp

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 f g
n : ℤ
⊢ (ComplexShape.up ℤ).Rel (n - 1) n

Tactic:
by simp

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
⊢ p = ComplexShape.embeddingUpIntDownInt.f (-p)

Tactic:
by simp

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
⊢ ComplexShape.embeddingUpIntDownInt.f (-q) = q

Tactic:
by simp

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)
n : ℤ
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 = - -n

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
J : Type u_2
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
inst✝ : HasLimitsOfShape J C
K : CochainComplex C ℤ
p : (CochainComplex.plus C).LimitOfShape J K
n : J → ℤ
hn : ∀ (j : J), (p.diag.obj j).IsStrictlyGE (n j)
j : J
⊢ n j ∈ Finset.image n ⊤ ∪ {0}

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
J : Type u_2
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
inst✝ : HasColimitsOfShape J C
K : CochainComplex C ℤ
p : (CochainComplex.plus C).ColimitOfShape J K
n : J → ℤ
hn : ∀ (j : J), (p.diag.obj j).IsStrictlyGE (n j)
j : J
⊢ n j ∈ Finset.image n ⊤ ∪ {0}

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : F.PreservesZeroMorphisms
K : CochainComplex.Plus C
i : ℤ
hi : K.obj.IsStrictlyGE i
⊢ ((CochainComplex.Plus.ι C ⋙ F.mapHomologicalComplex (ComplexShape.up ℤ)).obj K).IsStrictlyGE i

Tactic:
dsimp [CochainComplex.Plus.ι]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : F.PreservesZeroMorphisms
K : CochainComplex.Plus C
i : ℤ
hi : K.obj.IsStrictlyGE i
⊢ ((F.mapHomologicalComplex (ComplexShape.up ℤ)).obj K.obj).IsStrictlyGE i
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)
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
⊢ (ComplexShape.up ℤ).Rel (-(n + 1)) (-n)

Tactic:
by simp

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)
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
⊢ - -(n + 1) = n + 1

Tactic:
by simp

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)
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
⊢ - -n = n

Tactic:
by simp

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 : CochainComplex C ℤ
m : ℤ
hK : HomologicalComplex.ExactAt K (-m)
hm : -m + m = 0
⊢ (ComplexShape.up ℤ).prev m = m - 1

Tactic:
by simp

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 : CochainComplex C ℤ
m : ℤ
hK : HomologicalComplex.ExactAt K (-m)
hm : -m + m = 0
⊢ (ComplexShape.up ℤ).next m = m + 1

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ n - 1 + 1 = n

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ δ (n - 1) n 0 = ↑0

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ ∀ {x : Cocycle K L n},
    x ∈ {α | ∃ m, ∃ (_ : m + 1 = n), ∃ β, δ m n β = ↑α} → -x ∈ {α | ∃ m, ∃ (_ : m + 1 = n), ∃ β, δ m n β = ↑α}

Tactic:
rintro α ⟨m, hm, β, hβ⟩
    exact ⟨m, hm, -β, by aesop⟩

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ mk 0 = 0

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ ∀ (x y : Cocycle K L n), mk (x + y) = mk x + mk y

Tactic:
by simp

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

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x : Cocycle K L n
h : toHom (mk x) = 0
⊢ x ∈ coboundaries K L n

Tactic:
simp only [coboundaries, exists_prop, AddSubgroup.mem_mk, AddSubmonoid.mem_mk,
      AddSubsemigroup.mem_mk, Set.mem_ofPred_eq]

Goal state after:
case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x : Cocycle K L n
h : toHom (mk x) = 0
⊢ ∃ m, m + 1 = n ∧ ∃ β, δ m n β = ↑x
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x : Cocycle K L n
h✝ : Nonempty (Homotopy (Cocycle.equivHomShift.symm x) 0)
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)
h : (↑x).rightShift n 0 ⋯ = δ (-1) 0 γ
⊢ n - 1 + 1 = n

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ (ComplexShape.up ℤ).prev n + 1 = n

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
⊢ n + 1 = (ComplexShape.up ℤ).next n

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : P.rightOrthogonal T.obj₁
h₃ : P.rightOrthogonal T.obj₃
X : C
f : X ⟶ T.obj₂
hX : P X
⊢ f = 0

Tactic:
obtain ⟨g, rfl⟩ := Pretriangulated.Triangle.coyoneda_exact₂ T hT f (h₃ _ hX)
    simp [h₁ g hX]

Goal state after:
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✝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
i₂₃ : ι₂₃
j : ι₄
h₂₃ : c₂.π c₃ c₂₃ (i₂, i₃) = i₂₃
h : c₁.π c₂₃ c₄ (i₁, i₂₃) = j
⊢ ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j ⋯ =
    (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ i₃ i₂₃ h₂₃) ≫
      K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ i₂₃ j h

Tactic:
subst h₂₃
  rfl

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
Y : C
hY : P.trW.isLocal Y
X : C
f : X ⟶ Y
hX : P X
⊢ (Triangle.mk 0 (𝟙 X) 0).mor₁ ≫ f = (Triangle.mk 0 (𝟙 X) 0).mor₁ ≫ 0

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : P.IsTriangulated
X : C
hX : P.trW.isColocal X
Y : C
f : X ⟶ Y
hY : P Y
⊢ f ≫ (Triangle.mk 0 0 (𝟙 ((shiftFunctor C 1).obj Y))).mor₁ = 0 ≫ (Triangle.mk 0 0 (𝟙 ((shiftFunctor C 1).obj Y))).mor₁

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ f = f ≫ 𝟙 L₂

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ f ≫ e.inv ≫ e.hom = (f ≫ e.inv) ≫ e.hom

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
L₁ L₂ : CochainComplex C ℤ
e : HomotopyEquiv L₁ L₂
inst✝ : L₁.IsKInjective
K : CochainComplex C ℤ
f : K ⟶ L₂
hK : HomologicalComplex.Acyclic K
⊢ 0 ≫ e.hom = 0

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
α : Cochain K L (-1)
n : ℤ
hα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n
hK : HomologicalComplex.ExactAt K (n + 1)
inst✝ : Injective (L.X (n + 1))
u : K.X (n + 1) ⟶ L.X (n + 1) :=
  f.f (n + 1) - α.v (n + 1) n ⋯ ≫ L.d n (n + 1) - K.d (n + 1) (n + 2) ≫ α.v (n + 2) (n + 1) ⋯
hu : K.d n (n + 1) ≫ u = 0
⊢ (ComplexShape.up ℤ).prev (n + 1) = n

Tactic:
by simp

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

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
H : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n
H_zero : H 0
n : ℤ
X : C
hX : t.ge (n + ↑1) X
⊢ n + 1 = n + ↑1

Tactic:
by simp

Goal state after:
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
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
⊢ ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j h ≫ D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' =
    d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
by
  simp [D₂]

Goal state after:
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
⊢ (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄).d j j' =
    D₁ F G₂₃ K₁ K₂ K₃ c₂₃ c₄ j j' + D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' + D₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j'

Tactic:
rw [mapBifunctor.d_eq]

Goal state after:
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
⊢ mapBifunctor.D₁ K₁ (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ j j' +
      mapBifunctor.D₂ K₁ (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ j j' =
    D₁ F G₂₃ K₁ K₂ K₃ c₂₃ c₄ j j' + D₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' + D₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j'