mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Quotient.Preadditive

Goal state before:
pp: case mk.mk
C : Type ?u.2
inst✝² : Category.{v_1, ?u.2} C
inst✝¹ : Preadditive C
r : HomRel C
inst✝ : Congruence r
hr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)
P Q : Quotient r
iZ : Zero (P ⟶ Q) := { zero := Quot.mk (HomRel.CompClosure r) 0 }
iA : Add (P ⟶ Q) := { add := Preadditive.add r hr }
iN : Neg (P ⟶ Q) := { neg := Preadditive.neg r hr }
a✝² : P ⟶ Q
a✝¹ : P.as ⟶ Q.as
b✝ : P ⟶ Q
a✝ : P.as ⟶ Q.as
⊢ Quot.mk (HomRel.CompClosure r) a✝¹ + Quot.mk (HomRel.CompClosure r) a✝ =
    Quot.mk (HomRel.CompClosure r) a✝ + Quot.mk (HomRel.CompClosure r) a✝¹

Tactic:
;

Goal state after:
case mk.mk
C : Type ?u.2
inst✝² : Category.{v_1, ?u.2} C
inst✝¹ : Preadditive C
r : HomRel C
inst✝ : Congruence r
hr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)
P Q : Quotient r
iZ : Zero (P ⟶ Q) := { zero := Quot.mk (HomRel.CompClosure r) 0 }
iA : Add (P ⟶ Q) := { add := Preadditive.add r hr }
iN : Neg (P ⟶ Q) := { neg := Preadditive.neg r hr }
a✝² : P ⟶ Q
a✝¹ : P.as ⟶ Q.as
b✝ : P ⟶ Q
a✝ : P.as ⟶ Q.as
⊢ Quot.mk (HomRel.CompClosure r) a✝¹ + Quot.mk (HomRel.CompClosure r) a✝ =
    Quot.mk (HomRel.CompClosure r) a✝ + Quot.mk (HomRel.CompClosure r) a✝¹
Module: Mathlib.Algebra.Homology.HomotopyCategory

Goal state before:
pp: case mpr
ι : Type u_2
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C : HomologicalComplex V c
⊢ Nonempty (Homotopy (𝟙 C) 0) → 𝟙 ((quotient V c).obj C) = 0

Tactic:
rintro ⟨h⟩
    simpa using (eq_of_homotopy _ _ h)

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
n : ℤ
X✝ : CochainComplex C ℤ
⊢ { f := fun x ↦ (𝟙 X✝).f (x + n), comm' := ⋯ } =
    𝟙 { 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' := ⋯ }

Tactic:
;

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✝ : CochainComplex C ℤ
⊢ { f := fun x ↦ (𝟙 X✝).f (x + n), comm' := ⋯ } =
    𝟙 { 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' := ⋯ }
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✝ Z✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
⊢ { f := fun x ↦ (f✝ ≫ g✝).f (x + n), comm' := ⋯ } =
    { f := fun x ↦ f✝.f (x + n), comm' := ⋯ } ≫ { f := fun x ↦ g✝.f (x + n), comm' := ⋯ }

Tactic:
;

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✝ Z✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
⊢ { f := fun x ↦ (f✝ ≫ g✝).f (x + n), comm' := ⋯ } =
    { f := fun x ↦ f✝.f (x + n), comm' := ⋯ } ≫ { f := fun x ↦ g✝.f (x + n), comm' := ⋯ }
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
h : n = 0
X✝ Y✝ : CochainComplex C ℤ
x✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ ((shiftFunctor C n).map x✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n i i ⋯) ⋯).hom).f i✝ =
    ((Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n i i ⋯) ⋯).hom ≫ (𝟭 (CochainComplex C ℤ)).map x✝).f i✝

Tactic:
;

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
h : n = 0
X✝ Y✝ : CochainComplex C ℤ
x✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ ((shiftFunctor C n).map x✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n i i ⋯) ⋯).hom).f i✝ =
    ((Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n i i ⋯) ⋯).hom ≫ (𝟭 (CochainComplex C ℤ)).map x✝).f i✝
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 n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
X✝ Y✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
⊢ (shiftFunctor C n₁₂).map f✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom =
    (Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom ≫
      (shiftFunctor C n₁ ⋙ shiftFunctor C n₂).map f✝

Tactic:
;

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 n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
X✝ Y✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
⊢ (shiftFunctor C n₁₂).map f✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom =
    (Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom ≫
      (shiftFunctor C n₁ ⋙ shiftFunctor C n₂).map f✝
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 n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
X✝ Y✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ ((shiftFunctor C n₁₂).map f✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom).f
      i✝ =
    ((Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom ≫
          (shiftFunctor C n₁ ⋙ shiftFunctor C n₂).map f✝).f
      i✝

Tactic:
;

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 n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
X✝ Y✝ : CochainComplex C ℤ
f✝ : X✝ ⟶ Y✝
i✝ : ℤ
⊢ ((shiftFunctor C n₁₂).map f✝ ≫ (Hom.isoOfComponents (fun i ↦ Y✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom).f
      i✝ =
    ((Hom.isoOfComponents (fun i ↦ X✝.shiftFunctorObjXIso n₁₂ i (i + n₂ + n₁) ⋯) ⋯).hom ≫
          (shiftFunctor C n₁ ⋙ shiftFunctor C n₂).map f✝).f
      i✝
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 + b + a = n + (a + b)

Tactic:
;

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.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 + (a + b) = n + b + a

Tactic:
;

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 + b + a = n + ab

Tactic:
;

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 + ab = n + b + a

Tactic:
;

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
n : ℤ
⊢ n = n + 0

Tactic:
;

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 + 0 = n

Tactic:
;

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✝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'

Tactic:
;

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:
;

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
⊢ ((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✝
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
⊢ 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))

Tactic:
;

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))
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 L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i + n + 1 = i + 1 + n

Tactic:
;

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 L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i + n + 1 = i + 1 + 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
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i - 1 + n + 1 = i + n

Tactic:
;

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 L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i - 1 + n + 1 = i + n
Module: Mathlib.CategoryTheory.Shift.CommShift

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
A : Type u_4
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
x✝ : C
⊢ (isoZero' F A 0 ⋯).hom.app x✝ = (isoZero F A).hom.app x✝

Tactic:
;

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
A : Type u_4
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
x✝ : C
⊢ (isoZero' F A 0 ⋯).hom.app x✝ = (isoZero F A).hom.app 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
n : A
⊢ s n = 𝟭 D ⋙ s n

Tactic:
;

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
n : A
⊢ s n = 𝟭 D ⋙ s n
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
n : A
⊢ s n = s n ⋙ 𝟭 D

Tactic:
;

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
n : A
⊢ s n = s n ⋙ 𝟭 D
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
n : A
⊢ ∀ (X : D), (Induced.add F s i n 0).hom.app X = eqToHom ⋯ ≫ (Induced.zero F s i).inv.app ((s n).obj X)

Tactic:
suffices (Induced.add F s i n 0).hom =
            eqToHom (by rw [add_zero]; rfl) ≫ whiskerLeft (s n) (Induced.zero F s i).inv by
          intro X
          simpa using NatTrans.congr_app this X

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
n : A
⊢ (Induced.add F s i n 0).hom = eqToHom ⋯ ≫ (s n).whiskerLeft (Induced.zero F s i).inv
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

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 M : CochainComplex C ℤ
n✝ : ℤ
γ γ₁ γ₂ : Cochain K L n✝
n a : ℤ
a✝ b✝ : Cochain K L n
⊢ (a✝ + b✝).shift a = a✝.shift a + b✝.shift a

Tactic:
;

Goal state after:
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 M : CochainComplex C ℤ
n✝ : ℤ
γ γ₁ γ₂ : Cochain K L n✝
n a : ℤ
a✝ b✝ : Cochain K L n
⊢ (a✝ + b✝).shift a = a✝.shift a + b✝.shift a
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
⊢ (-γ).leftUnshift n hn = -γ.leftUnshift n hn

Tactic:
change (leftShiftAddEquiv K L n a n' hn).symm (-γ) = _
  apply map_neg

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
⊢ commShiftIso F 0 = CommShift.isoZero' F A 0 ⋯

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
⊢ commShiftIso F 0 = CommShift.isoZero' F A 0 ⋯
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

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 M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn' : n' + a = n
x : R
γ : Cochain K L n
⊢ (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn'

Tactic:
;

Goal state after:
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 M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn' : n' + a = n
x : R
γ : Cochain K L n
⊢ (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn'
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

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 M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn : n + a = n'
x : R
γ : Cochain K L n
⊢ (x • γ).leftShift a n' hn = x • γ.leftShift a n' hn

Tactic:
;

Goal state after:
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 M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn : n + a = n'
x : R
γ : Cochain K L n
⊢ (x • γ).leftShift a n' hn = x • γ.leftShift a n' hn
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

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 M : CochainComplex C ℤ
n✝ : ℤ
γ γ₁ γ₂ : Cochain K L n✝
n a : ℤ
x✝¹ : R
x✝ : Cochain K L n
⊢ (x✝¹ • x✝).shift a = x✝¹ • x✝.shift a

Tactic:
;

Goal state after:
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 M : CochainComplex C ℤ
n✝ : ℤ
γ γ₁ γ₂ : Cochain K L n✝
n a : ℤ
x✝¹ : R
x✝ : Cochain K L n
⊢ (x✝¹ • x✝).shift a = x✝¹ • x✝.shift a
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₁ = A.obj₁

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₁ = A.obj₁
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₁ = eqToHom ⋯

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₁ = eqToHom ⋯
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₂ = A.obj₂

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₂ = A.obj₂
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₂ = eqToHom ⋯

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₂ = eqToHom ⋯
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₃ = A.obj₃

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₃ = A.obj₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₃ = eqToHom ⋯

Tactic:
;

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A : Triangle C
⊢ (eqToHom ⋯).hom₃ = eqToHom ⋯
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := ofIso e A
a : A
x✝ : C
⊢ ((commShiftIso F a).hom ≫ whiskerRight e.hom (shiftFunctor D a)).app x✝ =
    ((shiftFunctor C a).whiskerLeft e.hom ≫ (commShiftIso G a).hom).app x✝

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := ofIso e A
a : A
x✝ : C
⊢ ((commShiftIso F a).hom ≫ whiskerRight e.hom (shiftFunctor D a)).app x✝ =
    ((shiftFunctor C a).whiskerLeft e.hom ≫ (commShiftIso G a).hom).app x✝