mathlib-initiative/mathlib-tactics huggingface dataset:
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
B : Type u_5
inst✝³ : AddCommMonoid B
inst✝² : HasShift C B
inst✝¹ : HasShift D B
inst✝ : F.CommShift B
X : C
a b : B
eq :
  F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (commShiftIso F (a + b)).hom.app X =
    (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫ (shiftFunctorAdd D a b).inv.app (F.obj X)
⊢ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ F.map ((shiftFunctorAdd' C b a (a + b) ⋯).hom.app X) =
    F.map ((shiftFunctorAdd C a b).inv.app X) ≫
      (commShiftIso F (a + b)).hom.app X ≫
        (shiftFunctorAdd' D b a (a + b) ⋯).hom.app (F.obj X) ≫
          (shiftFunctor D a).map ((commShiftIso F b).inv.app X) ≫ (commShiftIso F a).inv.app ((shiftFunctor C b).obj X)

Tactic:
simp only [NatTrans.congr_app (congr_arg Iso.hom (F.commShiftIso_add' (add_comm b a))) X,
    CommShift.isoAdd'_hom_app, Category.assoc, Iso.inv_hom_id_app_assoc,
    ← Functor.map_comp_assoc, Iso.hom_inv_id_app,
    Functor.map_id, Category.id_comp, comp_obj, Category.comp_id]

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

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₁ = (f + (g + h)).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₂ = (f + (g + h)).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₃ = (f + (g + h)).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₁ = (f + (g + h)).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₂ = (f + (g + h)).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g h : T₁ ⟶ T₂
⊢ (f + g + h).hom₃ = (f + (g + h)).hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₁ = f.hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₂ = f.hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₃ = f.hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₁ = f.hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₂ = f.hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (0 + f).hom₃ = f.hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₁ = f.hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₂ = f.hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₃ = f.hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₁ = f.hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₂ = f.hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (f + 0).hom₃ = f.hom₃
Module: Mathlib.CategoryTheory.Triangulated.Rotate

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
⊢ (rotate C).IsEquivalence

Tactic:
change (triangleRotation C).functor.IsEquivalence

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
⊢ (triangleRotation C).functor.IsEquivalence
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₁ = (n • f + f).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₂ = (n • f + f).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₃ = (n • f + f).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₁ = (n • f + f).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₂ = (n • f + f).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ ((n + 1) • f).hom₃ = (n • f + f).hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₁ = (f + -g).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₂ = (f + -g).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₃ = (f + -g).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₁ = (f + -g).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₂ = (f + -g).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f - g).hom₃ = (f + -g).hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₁ = (↑n • f + f).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₂ = (↑n • f + f).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₃ = (↑n • f + f).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₁ = (↑n • f + f).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₂ = (↑n • f + f).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (↑n.succ • f).hom₃ = (↑n • f + f).hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₁ = (-(↑n.succ • f)).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₂ = (-(↑n.succ • f)).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₃ = (-(↑n.succ • f)).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₁ = (-(↑n.succ • f)).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₂ = (-(↑n.succ • f)).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₃ = (-(↑n.succ • f)).hom₃
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₁ = TriangleMorphism.hom₁ 0

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₂ = TriangleMorphism.hom₂ 0

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₃ = TriangleMorphism.hom₃ 0

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₁ = TriangleMorphism.hom₁ 0

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₂ = TriangleMorphism.hom₂ 0

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f : T₁ ⟶ T₂
⊢ (-f + f).hom₃ = TriangleMorphism.hom₃ 0
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₁ = (g + f).hom₁

pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₂ = (g + f).hom₂

pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₃ = (g + f).hom₃

Tactic:
<;>

Goal state after:
case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₁ = (g + f).hom₁

case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₂ = (g + f).hom₂

case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
f g : T₁ ⟶ T₂
⊢ (f + g).hom₃ = (g + f).hom₃
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : F₁.CommShift A
inst✝² : F₂.CommShift A
inst✝¹ : IsIso τ
inst✝ : CommShift τ A
this : CommShift (asIso τ).hom A
⊢ CommShift (inv τ) A

Tactic:
change NatTrans.CommShift (asIso τ).inv A

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : F₁.CommShift A
inst✝² : F₂.CommShift A
inst✝¹ : IsIso τ
inst✝ : CommShift τ A
this : CommShift (asIso τ).hom A
⊢ CommShift (asIso τ).inv A
Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasShift C ℤ
J : Type u_1
T : J → Triangle C
inst✝⁴ : HasProduct fun j ↦ (T j).obj₁
inst✝³ : HasProduct fun j ↦ (T j).obj₂
inst✝² : HasProduct fun j ↦ (T j).obj₃
inst✝¹ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
inst✝ : HasZeroMorphisms C
h : ∀ (j : J), (T j).mor₃ ≫ (shiftFunctor C 1).map (T j).mor₁ = 0
⊢ (productTriangle T).mor₃ ≫ (shiftFunctor C 1).map (productTriangle T).mor₁ = 0

Tactic:
by
  have : HasProduct (fun j => (T j).obj₂⟦(1 : ℤ)⟧) :=
    ⟨_, isLimitFanMkObjOfIsLimit (shiftFunctor C (1 : ℤ)) _ _
      (productIsProduct (fun j => (T j).obj₂))⟩
  dsimp
  change _ ≫ (Pi.lift (fun j => Pi.π _ j ≫ (T j).mor₁))⟦(1 : ℤ)⟧' = 0
  rw [assoc, ← cancel_mono (piComparison _ _), zero_comp, assoc, assoc]
  ext j
  simp [h j]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
n : ℤ
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (n.negOnePow • (CategoryTheory.shiftFunctor C n).map X✝.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app X✝.obj₁) ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map f.hom₁) =
    (CategoryTheory.shiftFunctor C n).map f.hom₃ ≫
      (n.negOnePow • (CategoryTheory.shiftFunctor C n).map Y✝.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app Y✝.obj₁)

Tactic:
rw [Linear.units_smul_comp, Linear.comp_units_smul, ← Functor.map_comp_assoc, ← f.comm₃,
          Functor.map_comp, assoc, assoc, dsimp% (shiftFunctorComm C 1 n).hom.naturality]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₁ ≫
        (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₁))

Tactic:
NatTrans.naturality,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁ ≫
        (CategoryTheory.shiftFunctor C a ⋙ CategoryTheory.shiftFunctor C b).map T.mor₁ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₁))
Module: Mathlib.CategoryTheory.Triangulated.TriangleShift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₂ ≫
        (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₂))

Tactic:
NatTrans.naturality,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
        (CategoryTheory.shiftFunctor C a ⋙ CategoryTheory.shiftFunctor C b).map T.mor₂ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₂))
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr.left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂

pp: case mpr.right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Epi T.mor₂

Tactic:
<;>

Goal state after:
case mpr.left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂

case mpr.right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Epi T.mor₂
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
j : J
⊢ T'.mor₁ ≫ Pi.π (fun j ↦ (T j).obj₂) j = Pi.π (fun j ↦ (T j).obj₁) j ≫ (T j).mor₁

Tactic:
simp [f₁, T']

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

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝² : IsIso φ'.hom₁
this✝¹ : IsIso φ'.hom₂
this✝ : IsIso φ'.hom₃
this : IsIso φ'
⊢ productTriangle T ∈ distinguishedTriangles

Tactic:
exact isomorphic_distinguished _ hT' _ (asIso φ').symm

Goal state after: