awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A_1 A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
hg' : ∀ (j : J), (g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j) ≫ (T j).mor₂ = 0
hg'' : ∀ (j : J), ∃ (g_1 : A ⟶ (T j).obj₁), g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = g_1 ≫ (T j).mor₁
α : (j : J) → A ⟶ (T j).obj₁ :=
  fun (j : J) ↦ ⋯.choose (p := fun (g_1 : A ⟶ (T j).obj₁) ↦ g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = g_1 ≫ (T j).mor₁)
hα : ∀ (j : J), g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = α j ≫ (T j).mor₁
hg''' : g = Pi.lift α ≫ T'.mor₁
refine_1 : g ≫ T'.mor₂ = 0
refine_1₁ : (Pi.lift α ≫ T'.mor₁) ≫ T'.mor₂ = 0
refine_1₂ : Pi.lift α ≫ T'.mor₁ ≫ T'.mor₂ = 0
⊢ f = 0
have refine_1₃ : Eq (α := A ⟶ T'.obj₃) (CategoryStruct.comp (Y := piObj.{u_1, v, u} fun (b : J) ↦ (T b).obj₁) (Pi.lift α) 0) 0
Tactic: rw [hg, hg''', assoc, comp_distTriang_mor_zero₁₂ _ hT', comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A_1 A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
hg' : ∀ (j : J), (g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j) ≫ (T j).mor₂ = 0
hg'' : ∀ (j : J), ∃ (g_1 : A ⟶ (T j).obj₁), g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = g_1 ≫ (T j).mor₁
α : (j : J) → A ⟶ (T j).obj₁ :=
  fun (j : J) ↦ ⋯.choose (p := fun (g_1 : A ⟶ (T j).obj₁) ↦ g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = g_1 ≫ (T j).mor₁)
hα : ∀ (j : J), g ≫ Pi.π (fun (j : J) ↦ (T j).obj₂) j = α j ≫ (T j).mor₁
hg''' : g = Pi.lift α ≫ T'.mor₁
refine_1 : g ≫ T'.mor₂ = 0
refine_1₁ : (Pi.lift α ≫ T'.mor₁) ≫ T'.mor₂ = 0
refine_1₂ : Pi.lift α ≫ T'.mor₁ ≫ T'.mor₂ = 0
refine_1₃ :
  Eq (α := A ⟶ T'.obj₃) (CategoryStruct.comp (Y := piObj.{u_1, v, u} fun (b : J) ↦ (T b).obj₁) (Pi.lift α) 0) 0
⊢ f = 0
have refine_1₄ : Eq.{v + 1} (α := A ⟶ T'.obj₃) 0 0
Tactic: rw [hg, hg''', assoc, comp_distTriang_mor_zero₁₂ _ hT', comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
⊢ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
have h : ∀ (j : J), (T j).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (T j).mor₁ = 0
Tactic: rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
⊢ ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
have zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
Tactic: have zero : ((productTriangle T).mor₃) ≫ (shiftFunctor C 1).map T'.mor₁ = 0 := by
        rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
        intro j
        exact comp_distTriang_mor_zero₃₁ _ (hT j)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
⊢ (a ≫ (productTriangle T).mor₃) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
have rw : a ≫ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
Tactic: rw [assoc, zero, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
rw : a ≫ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
⊢ (a ≫ (productTriangle T).mor₃) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
have rw₁ : Eq (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₂) (a ≫ 0) 0
Tactic: rw [assoc, zero, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
rw : a ≫ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
rw₁ : Eq (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₂) (a ≫ 0) 0
⊢ (a ≫ (productTriangle T).mor₃) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
have rw₂ : Eq.{v + 1} (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₂) 0 0
Tactic: rw [assoc, zero, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
⊢ ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
have have : ∀ (g : A ⟶ T'.obj₃) (hg : a ≫ (productTriangle T).mor₃ = g ≫ T'.mor₃),
  ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
Tactic: have ⟨g, hg⟩ := T'.coyoneda_exact₁ hT' (a ≫ (productTriangle T).mor₃) (by
        rw [assoc, zero, comp_zero])
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
⊢ ∃ (a_1 : A ⟶ T'.obj₃), (fun (x : A ⟶ T'.obj₃) ↦ x ≫ φ'.hom₃) a_1 = a
have refine_2 : ∀ (a' : A ⟶ Z) (ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃),
  ∃ (a_1 : A ⟶ T'.obj₃), (fun (x : A ⟶ T'.obj₃) ↦ x ≫ φ'.hom₃) a_1 = a
Tactic: obtain ⟨a', ha'⟩ : ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃ := by
      have zero : ((productTriangle T).mor₃) ≫ (shiftFunctor C 1).map T'.mor₁ = 0 := by
        rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
        intro j
        exact comp_distTriang_mor_zero₃₁ _ (hT j)
      have ⟨g, hg⟩ := T'.coyoneda_exact₁ hT' (a ≫ (productTriangle T).mor₃) (by
        rw [assoc, zero, comp_zero])
      exact ⟨g, hg.symm⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
⊢ ((a - a' ≫ φ'.hom₃) ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j) ≫ (T j).mor₃ = 0
have simp : HSub.hSub (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁)
    (a ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃)
    (a' ≫ φ'.hom₃ ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃) =
  0
Tactic: simp only [sub_comp, assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
⊢ HSub.hSub (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁)
      (a ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃)
      (a' ≫ φ'.hom₃ ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃) =
    0
have erw : a ≫ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
    a' ≫
      φ'.hom₃ ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
  0
Tactic: erw [← (productTriangle.π T j).comm₃]