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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
F G K : CochainComplex C ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
Φ : C ⥤ D
inst✝ : Φ.Additive
p q : ℤ
hpq : p + n₁₂ = q
⊢ ((z₁.comp z₂ h).map Φ).v p q hpq = ((z₁.map Φ).comp (z₂.map Φ) h).v p q hpq

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
F G K : CochainComplex C ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
Φ : C ⥤ D
inst✝ : Φ.Additive
p q : ℤ
hpq : p + n₁₂ = q
⊢ Φ.map ((z₁.comp z₂ h).v p q hpq) = ((z₁.map Φ).comp (z₂.map Φ) h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: case right
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n : ℤ
⊢ (H.map ((↑(fst φ)).v n (n + 1) ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) n ⋯ +
        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫
      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n (n + 1) ⋯ ≫
          ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ).f (n + 1) +
        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫
          ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj G).d n (n + 1)) =
    ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).d n (n + 1) ≫
      (H.map ((↑(fst φ)).v (n + 1) (n + 2) ⋯) ≫
            (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 2) (n + 1) ⋯ +
          H.map ((snd φ).v (n + 1) (n + 1) ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f (n + 1)) ≫
        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) (n + 1) ⋯

Tactic:
dsimp

Goal state after:
case right
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n : ℤ
⊢ (H.map ((↑(fst φ)).v n (n + 1) ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) n ⋯ +
        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫
      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n (n + 1) ⋯ ≫ H.map (φ.f (n + 1)) +
        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map (G.d n (n + 1))) =
    H.map ((mappingCone φ).d n (n + 1)) ≫
      (H.map ((↑(fst φ)).v (n + 1) (n + 2) ⋯) ≫
            (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 2) (n + 1) ⋯ +
          H.map ((snd φ).v (n + 1) (n + 1) ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f (n + 1)) ≫
        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) (n + 1) ⋯
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: case neg
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h✝ k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
h : ¬c.Rel (c.prev i) i
⊢ (f.f (c.prev i) - g.f (c.prev i) - K.d (c.prev i) i ≫ ho.hom i (c.prev i)) ≫ ((shortComplexFunctor C c i).obj L).f = 0

Tactic:
dsimp

Goal state after:
case neg
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h✝ k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
h : ¬c.Rel (c.prev i) i
⊢ (f.f (c.prev i) - g.f (c.prev i) - K.d (c.prev i) i ≫ ho.hom i (c.prev i)) ≫ L.d (c.prev i) i = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ (δ n m (z.map Φ)).v p q hpq = ((δ n m z).map Φ).v p q hpq

Tactic:
dsimp

Goal state after:
case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ (δ n m (z.map Φ)).v p q hpq = Φ.map ((δ n m z).v p q hpq)
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: case neg
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h✝ k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
h : ¬c.Rel i (c.next i)
⊢ ((shortComplexFunctor C c i).obj K).g ≫ (f.f (c.next i) - g.f (c.next i) - ho.hom (c.next i) i ≫ L.d i (c.next i)) = 0

Tactic:
dsimp

Goal state after:
case neg
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h✝ k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
h : ¬c.Rel i (c.next i)
⊢ K.d i (c.next i) ≫ (f.f (c.next i) - g.f (c.next i) - ho.hom (c.next i) i ≫ L.d i (c.next i)) = 0
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
⊢ ((shortComplexFunctor C c i).map f).τ₁ =
    (((shortComplexFunctor C c i).obj K).f ≫ ho.hom i (c.prev i) +
        if c.Rel (c.prev i) i then ho.hom (c.prev i) (c.prev (c.prev i)) ≫ L.d (c.prev (c.prev i)) (c.prev i)
        else f.f (c.prev i) - g.f (c.prev i) - K.d (c.prev i) i ≫ ho.hom i (c.prev i)) +
      ((shortComplexFunctor C c i).map g).τ₁

Tactic:
dsimp

Goal state after:
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
⊢ f.f (c.prev i) =
    (K.d (c.prev i) i ≫ ho.hom i (c.prev i) +
        if c.Rel (c.prev i) i then ho.hom (c.prev i) (c.prev (c.prev i)) ≫ L.d (c.prev (c.prev i)) (c.prev i)
        else f.f (c.prev i) - g.f (c.prev i) - K.d (c.prev i) i ≫ ho.hom i (c.prev i)) +
      g.f (c.prev i)
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
⊢ ((shortComplexFunctor C c i).map f).τ₃ =
    (if c.Rel i (c.next i) then K.d (c.next i) (c.next (c.next i)) ≫ ho.hom (c.next (c.next i)) (c.next i)
        else f.f (c.next i) - g.f (c.next i) - ho.hom (c.next i) i ≫ L.d i (c.next i)) +
        ho.hom (c.next i) i ≫ ((shortComplexFunctor C c i).obj L).g +
      ((shortComplexFunctor C c i).map g).τ₃

Tactic:
dsimp

Goal state after:
ι✝ : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c✝ : ComplexShape ι✝
C✝ D E : HomologicalComplex V c✝
f✝ g✝ : C✝ ⟶ D
h k : D ⟶ E
i✝ : ι✝
C : Type u_2
inst✝² : Category.{v_1, u_2} C
inst✝¹ : Preadditive C
ι : Type ?u.42
c : ComplexShape ι
inst✝ : DecidableRel c.Rel
K L : HomologicalComplex C c
f g : K ⟶ L
ho : Homotopy f g
i : ι
⊢ f.f (c.next i) =
    (if c.Rel i (c.next i) then K.d (c.next i) (c.next (c.next i)) ≫ ho.hom (c.next (c.next i)) (c.next i)
        else f.f (c.next i) - g.f (c.next i) - ho.hom (c.next i) i ≫ L.d i (c.next i)) +
        ho.hom (c.next i) i ≫ L.d i (c.next i) +
      g.f (c.next i)
Module: Mathlib.CategoryTheory.Quotient

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r : HomRel C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
F₁ F₂ : Quotient r ⥤ D
h✝ : functor r ⋙ F₁ = functor r ⋙ F₂
X Y : C
f g : X ⟶ Y
h : r f g
⊢ (functor r ⋙ F₂).map f = (functor r ⋙ F₂).map g

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r : HomRel C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
F₁ F₂ : Quotient r ⥤ D
h✝ : functor r ⋙ F₁ = functor r ⋙ F₂
X Y : C
f g : X ⟶ Y
h : r f g
⊢ F₂.map ((functor r).map f) = F₂.map ((functor r).map g)
Module: Mathlib.CategoryTheory.Quotient

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r : HomRel C
D : Type ?u.8
inst✝ : Category.{v_2, ?u.8} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ : functor r ⋙ F ≅ functor r ⋙ G
⊢ natTransLift r τ.inv ≫ natTransLift r τ.hom = 𝟙 G

Tactic:
by rw [comp_natTransLift, τ.inv_hom_id, natTransLift_id]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Quotient

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X : C
⊢ ((functor r).whiskerLeft (LiftCommShift.iso F r hF 0).hom).app X =
    ((functor r).whiskerLeft (Functor.CommShift.isoZero (lift r F hF) A).hom).app X

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X : C
⊢ (LiftCommShift.iso F r hF 0).hom.app ((functor r).obj X) =
    (Functor.CommShift.isoZero (lift r F hF) A).hom.app ((functor r).obj X)
Module: Mathlib.CategoryTheory.Shift.Induced

Goal state before:
pp: C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
m₁ m₂ m₃ : A
X : C
⊢ (((whiskeringLeft C D D).obj F).map
          ((Induced.add F s i (m₁ + m₂) m₃).hom ≫ whiskerRight (Induced.add F s i m₁ m₂).hom (s m₃))).app
      X =
    (((whiskeringLeft C D D).obj F).map
          (eqToHom ⋯ ≫ (Induced.add F s i m₁ (m₂ + m₃)).hom ≫ (s m₁).whiskerLeft (Induced.add F s i m₂ m₃).hom)).app
      X

Tactic:
dsimp

Goal state after:
C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
m₁ m₂ m₃ : A
X : C
⊢ (Induced.add F s i (m₁ + m₂) m₃).hom.app (F.obj X) ≫ (s m₃).map ((Induced.add F s i m₁ m₂).hom.app (F.obj X)) =
    (eqToHom ⋯).app (F.obj X) ≫
      (Induced.add F s i m₁ (m₂ + m₃)).hom.app (F.obj X) ≫ (Induced.add F s i m₂ m₃).hom.app ((s m₁).obj (F.obj X))
Module: Mathlib.CategoryTheory.Shift.Quotient

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b : A
X : C
⊢ ((functor r).whiskerLeft (LiftCommShift.iso F r hF (a + b)).hom).app X =
    ((functor r).whiskerLeft
          (Functor.CommShift.isoAdd (LiftCommShift.iso F r hF a) (LiftCommShift.iso F r hF b)).hom).app
      X

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b : A
X : C
⊢ (LiftCommShift.iso F r hF (a + b)).hom.app ((functor r).obj X) =
    (Functor.CommShift.isoAdd (LiftCommShift.iso F r hF a) (LiftCommShift.iso F r hF b)).hom.app ((functor r).obj X)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
X✝ Y✝ : CochainComplex C ℤ
φ : X✝ ⟶ Y✝
i✝ j✝ : ℤ
a✝ : (ComplexShape.up ℤ).Rel i✝ j✝
⊢ φ.f (i✝ + n) ≫
      { X := fun i ↦ Y✝.X (i + n), d := fun x x_1 ↦ n.negOnePow • Y✝.d (x + n) (x_1 + n), shape := ⋯, d_comp_d' := ⋯ }.d
        i✝ j✝ =
    { X := fun i ↦ X✝.X (i + n), d := fun x x_1 ↦ n.negOnePow • X✝.d (x + n) (x_1 + n), shape := ⋯, d_comp_d' := ⋯ }.d
        i✝ j✝ ≫
      φ.f (j✝ + n)

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
X✝ Y✝ : CochainComplex C ℤ
φ : X✝ ⟶ Y✝
i✝ j✝ : ℤ
a✝ : (ComplexShape.up ℤ).Rel i✝ j✝
⊢ φ.f (i✝ + n) ≫ (n.negOnePow • Y✝.d (i✝ + n) (j✝ + n)) = (n.negOnePow • X✝.d (i✝ + n) (j✝ + n)) ≫ φ.f (j✝ + n)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n₁ n₂ : ℤ
K : CochainComplex C ℤ
x✝² x✝¹ : ℤ
x✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹
⊢ (K.shiftFunctorObjXIso (n₁ + n₂) x✝² (x✝² + n₂ + n₁) ⋯).hom ≫
      ((shiftFunctor C n₁ ⋙ shiftFunctor C n₂).obj K).d x✝² x✝¹ =
    ((shiftFunctor C (n₁ + n₂)).obj K).d x✝² x✝¹ ≫ (K.shiftFunctorObjXIso (n₁ + n₂) x✝¹ (x✝¹ + n₂ + n₁) ⋯).hom

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n₁ n₂ : ℤ
K : CochainComplex C ℤ
x✝² x✝¹ : ℤ
x✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹
⊢ (XIsoOfEq K ⋯).hom ≫ (n₂.negOnePow • n₁.negOnePow • K.d (x✝² + n₂ + n₁) (x✝¹ + n₂ + n₁)) =
    ((n₁ + n₂).negOnePow • K.d (x✝² + (n₁ + n₂)) (x✝¹ + (n₁ + n₂))) ≫ (XIsoOfEq K ⋯).hom
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b n : ℤ
⊢ n + { as := b }.as + { as := a }.as = n + { as := a + b }.as

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b n : ℤ
⊢ n + b + a = n + (a + b)
Module: Mathlib.CategoryTheory.Shift.Quotient

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a : A
X : C
⊢ ((Functor.commShiftIso (functor r ⋙ lift r F hF) a).hom ≫
          Functor.whiskerRight (lift.isLift r F hF).hom (shiftFunctor D a)).app
      X =
    ((shiftFunctor C a).whiskerLeft (lift.isLift r F hF).hom ≫ (Functor.commShiftIso F a).hom).app X

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a : A
X : C
⊢ (Functor.commShiftIso (functor r ⋙ lift r F hF) a).hom.app X ≫ (shiftFunctor D a).map (𝟙 (F.obj X)) =
    𝟙 (F.obj ((shiftFunctor C a).obj X)) ≫ (Functor.commShiftIso F a).hom.app X
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b n : ℤ
⊢ n + { as := a + b }.as = n + { as := b }.as + { as := a }.as

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b n : ℤ
⊢ n + (a + b) = n + b + a
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + { as := b }.as + { as := a }.as = n + { as := ab }.as

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + b + a = n + ab
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + { as := b }.as + { as := a }.as = n + { as := ab }.as

Tactic:
by dsimp; rw [← h, add_assoc, add_comm a]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + { as := ab }.as = n + { as := b }.as + { as := a }.as

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + ab = n + b + a
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
a b ab : ℤ
h : a + b = ab
n : ℤ
⊢ n + { as := ab }.as = n + { as := b }.as + { as := a }.as

Tactic:
by dsimp; rw [← h, add_assoc, add_comm a]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
n : ℤ
⊢ n = n + { as := 0 }.as

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
n : ℤ
⊢ n = n + 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
n : ℤ
⊢ n + { as := 0 }.as = n

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K : CochainComplex C ℤ
n : ℤ
⊢ n + 0 = n
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
a b p : ℤ
⊢ (((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b (a + b) ⋯).symm ≪≫
                CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) b a (a + b) ⋯).hom.app
          K).f
      p =
    (XIsoOfEq K ⋯).hom

Tactic:
dsimp

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
a b p : ℤ
⊢ ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b (a + b) ⋯).inv.app K).f p ≫
      ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) b a (a + b) ⋯).hom.app K).f p =
    (XIsoOfEq K ⋯).hom
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n i i' : ℤ
hi : n + i = i'
K : CochainComplex C ℤ
⊢ i + { as := n }.as = i'

Tactic:
dsimp

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n i i' : ℤ
hi : n + i = i'
K : CochainComplex C ℤ
⊢ i + n = i'
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
n : ℤ
X✝ Y✝ : HomologicalComplex C (ComplexShape.up ℤ)
x✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ ((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n ⋙ F.mapHomologicalComplex (ComplexShape.up ℤ)).map x✝ ≫
          (HomologicalComplex.Hom.isoOfComponents
              (fun x ↦
                Iso.refl
                  (((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n ⋙
                            F.mapHomologicalComplex (ComplexShape.up ℤ)).obj
                        Y✝).X
                    x))
              ⋯).hom).f
      i✝ =
    ((HomologicalComplex.Hom.isoOfComponents
              (fun x ↦
                Iso.refl
                  (((shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n ⋙
                            F.mapHomologicalComplex (ComplexShape.up ℤ)).obj
                        X✝).X
                    x))
              ⋯).hom ≫
          (F.mapHomologicalComplex (ComplexShape.up ℤ) ⋙ shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) n).map
            x✝).f
      i✝

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
n : ℤ
X✝ Y✝ : HomologicalComplex C (ComplexShape.up ℤ)
x✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ F.map (x✝.f (i✝ + n)) ≫ 𝟙 (F.obj (Y✝.X (i✝ + n))) = 𝟙 (F.obj (X✝.X (i✝ + n))) ≫ F.map (x✝.f (i✝ + n))