awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
refine_3 :
  w₁₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃) ≫
        Functor.map (shiftFunctor C (A := ℤ) 1)
          (Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
            NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₂₃) =
    w₂₃
⊢ w₁₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1)
        (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
            NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₂₃) =
    w₂₃
have refine_3₁ : w₁₃ ≫
    Functor.map (shiftFunctor C (A := ℤ) 1) (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃) ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃) ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₂₃) =
  w₂₃
Tactic: rw [Functor.map_comp, Functor.map_comp, shift_shiftFunctorCompIsoId_hom_app,
      shift_shiftFunctorCompIsoId_inv_app, shiftFunctorCompIsoId_naturality_1, eq₃]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ m₃ ≫ v₂₃ = v₁₃ ≫ u₁₂
have refine_4 : m₃ ≫
    CategoryStruct.comp (Y :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1 ⋙ shiftFunctor C (A := ℤ) (-1)) Z₂₃)
      (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₂₃)
      (Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂) =
  v₁₃ ≫ u₁₂
Tactic: rw [← shiftFunctorCompIsoId_naturality_1 v₂₃ 1 (-1) (Int.add_right_neg 1)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ m₃ ≫
      CategoryStruct.comp (Y :=
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1 ⋙ shiftFunctor C (A := ℤ) (-1)) Z₂₃)
        (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₂₃)
        (Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
          NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂) =
    v₁₃ ≫ u₁₂
have refine_4 : (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₂₃) ≫
    CategoryStruct.comp (Y :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃))
      (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₂₃)
      (Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂) =
  v₁₃ ≫ u₁₂
Tactic: dsimp [m₃]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
          NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₂₃) ≫
      CategoryStruct.comp (Y :=
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
          (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃))
        (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₂₃)
        (Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
          NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂) =
    v₁₃ ≫ u₁₂
have refine_4 : NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃ ≫
    Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂ =
  v₁₃ ≫ u₁₂
Tactic: rw [assoc, assoc, Iso.hom_inv_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
          NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂ =
    v₁₃ ≫ u₁₂
have refine_4 : NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃ ≫
    (Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃)) ≫
      NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂ =
  v₁₃ ≫ u₁₂
Tactic: nth_rw 2 [← assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
      (Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃)) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂ =
    v₁₃ ≫ u₁₂
have refine_4 : NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃ ≫
    Functor.map (shiftFunctor C (A := ℤ) (-1)) (o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
      NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂ =
  v₁₃ ≫ u₁₂
Tactic: rw [← Functor.map_comp, ← eq₄, ← Functor.map_comp, shiftFunctorCompIsoId_naturality_1]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
refine_4 :
  NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) (-1)) (o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂ =
    v₁₃ ≫ u₁₂
⊢ NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) ⋯)) Z₁₃ ≫
      (Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) (-1)) (Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃)) ≫
        NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) ⋯)) X₂ =
    v₁₃ ≫ u₁₂
have refine_4₁ : CategoryStruct.comp (Y :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃))
    (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) Z₁₃)
    (Functor.map (shiftFunctor C (A := ℤ) (-1))
        (Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂) ≫
      NatTrans.app (Iso.hom (shiftFunctorCompIsoId C 1 (-1) sorry)) X₂) =
  v₁₃ ≫ u₁₂
Tactic: rw [← Functor.map_comp, ← eq₄, ← Functor.map_comp, shiftFunctorCompIsoId_naturality_1]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk m₁ m₃ (v₂₃ ≫ w₁₂))
have refine_5 : Triangle.mk m₁ m₃ (v₂₃ ≫ w₁₂) ≅
  Functor.obj (shiftFunctor (Triangle.{v_1, u_1} C) (A := ℤ) (-1))
    (Triangle.mk o.m₁ o.m₃
      ((-Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₂₃ u₂₃ w₂₃).mor₁) ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₁₂ u₁₂ w₁₂).mor₃))
Tactic: apply isomorphic_distinguished _ ((Triangle.shift_distinguished_iff _ (-1 : ℤ)).mpr o.mem)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Triangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₂
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₂ u₁₂ w₁₂)
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₂₃
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₂₃ u₂₃ w₂₃)
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z₁₃
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk v₁₃ u₁₃ w₁₃)
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ :=
  Iso.inv (shiftShiftNeg Z₁₂ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₁ ≫ Iso.hom (shiftShiftNeg Z₁₃ 1)
m₃ : Z₁₃ ⟶ Z₂₃ :=
  Iso.inv (shiftShiftNeg Z₁₃ 1) ≫ Functor.map (shiftFunctor C (A := ℤ) (-1)) o.m₃ ≫ Iso.hom (shiftShiftNeg Z₂₃ 1)
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ = Functor.map (shiftFunctor C (A := ℤ) 1) v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ :
  Functor.map (shiftFunctor C (A := ℤ) 1) v₁₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) u₁₂ =
    o.m₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) v₂₃
⊢ Triangle.mk m₁ m₃ (v₂₃ ≫ w₁₂) ≅
    Functor.obj (shiftFunctor (Triangle.{v_1, u_1} C) (A := ℤ) (-1))
      (Triangle.mk o.m₁ o.m₃
        ((-Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₂₃ u₂₃ w₂₃).mor₁) ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₁₂ u₁₂ w₁₂).mor₃))
have refine_5 : (Triangle.mk m₁ m₃ (v₂₃ ≫ w₁₂)).mor₃ ≫
    Functor.map (shiftFunctor C (A := ℤ) 1) (Iso.hom (Iso.symm (shiftShiftNeg Z₁₂ 1))) =
  Iso.hom (Iso.symm (shiftShiftNeg Z₂₃ 1)) ≫
    (Functor.obj (shiftFunctor (Triangle.{v_1, u_1} C) (A := ℤ) (-1))
        (Triangle.mk o.m₁ o.m₃
          ((-Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₂₃ u₂₃ w₂₃).mor₁) ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk v₁₂ u₁₂ w₁₂).mor₃))).mor₃
Tactic: refine Triangle.isoMk _ _ (shiftShiftNeg Z₁₂ (1 : ℤ)).symm
      (-(shiftShiftNeg Z₁₃ (1 : ℤ)).symm) (shiftShiftNeg Z₂₃ (1 : ℤ)).symm (comm₃ := ?_)