awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
rw : s ≫ f₁ - s ≫ f₂ = 0
rw₁ : s ≫ f₂ - s ≫ f₂ = 0
⊢ s ≫ (f₁ - f₂) = 0
have rw₂ : Eq.{v_1 + 1} (α := X' ⟶ Y) 0 0
Tactic: rw [comp_sub, hf₁, sub_self]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
⊢ ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
have hf₂ : s ≫ (f₁ - f₂) = 0
Tactic: have hf₂ : s ≫ (f₁ - f₂) = 0 := by rw [comp_sub, hf₁, sub_self]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
⊢ ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
have obtain : ∀ (q : (Triangle.mk s g h).obj₃ ⟶ Y) (hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q),
  ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
Tactic: obtain ⟨q, hq⟩ := Triangle.yoneda_exact₂ _ H _ hf₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
⊢ ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
have obtain : ∀ (Y' : C) (r : Y ⟶ Y') (t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃)
  (mem' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk q r t)),
  ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
Tactic: obtain ⟨Y', r, t, mem'⟩ := distinguished_cocone_triangle q
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
⊢ ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
have refine_1 : P.trW r
Tactic: refine ⟨Y', r, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
refine_1 : P.trW r
⊢ ∃ (Y' : C) (t : Y ⟶ Y') (_ : P.trW t), f₁ ≫ t = f₂ ≫ t
have refine_2 : f₁ ≫ r = f₂ ≫ r
Tactic: refine ⟨Y', r, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
⊢ f₁ ≫ r = f₂ ≫ r
have eq : (Triangle.mk q r t).mor₁ ≫ (Triangle.mk q r t).mor₂ = 0
Tactic: have eq := comp_distTriang_mor_zero₁₂ _ mem'
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2 : f₁ ≫ r - f₂ ≫ r = 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
refine_2 : f₁ ≫ r - f₂ ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2₁ : (f₁ - f₂) ≫ r = 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
refine_2 : f₁ ≫ r - f₂ ≫ r = 0
refine_2₁ : (f₁ - f₂) ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2₂ : ((Triangle.mk s g h).mor₂ ≫ q) ≫ r = 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
refine_2 : f₁ ≫ r - f₂ ≫ r = 0
refine_2₁ : (f₁ - f₂) ≫ r = 0
refine_2₂ : ((Triangle.mk s g h).mor₂ ≫ q) ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2₃ : (Triangle.mk s g h).mor₂ ≫ q ≫ r = 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
refine_2 : f₁ ≫ r - f₂ ≫ r = 0
refine_2₁ : (f₁ - f₂) ≫ r = 0
refine_2₂ : ((Triangle.mk s g h).mor₂ ≫ q) ≫ r = 0
refine_2₃ : (Triangle.mk s g h).mor₂ ≫ q ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2₄ : Eq (α := X ⟶ Y') ((Triangle.mk s g h).mor₂ ≫ 0) 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X'
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk s g h)
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk s g h).obj₃
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk q r t)
eq : q ≫ r = 0
refine_2 : f₁ ≫ r - f₂ ≫ r = 0
refine_2₁ : (f₁ - f₂) ≫ r = 0
refine_2₂ : ((Triangle.mk s g h).mor₂ ≫ q) ≫ r = 0
refine_2₃ : (Triangle.mk s g h).mor₂ ≫ q ≫ r = 0
refine_2₄ : Eq (α := X ⟶ Y') ((Triangle.mk s g h).mor₂ ≫ 0) 0
⊢ f₁ ≫ r = f₂ ≫ r
have refine_2₅ : Eq.{v_1 + 1} (α := X ⟶ Y') 0 0
Tactic: rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X Y : C
φ : P.trW.LeftFraction X Y
⊢ ∃ (ψ : P.trW.RightFraction X Y), ψ.s (W := P.trW) ≫ φ.f (W := P.trW) = ψ.f (W := P.trW) ≫ φ.s (W := P.trW)
have obtain : ∀ (Z : C) (f : φ.Y' (W := P.trW) ⟶ Z) (g : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y)
  (H :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk (φ.s (W := P.trW)) f g))
  (mem : P Z),
  ∃ (ψ : P.trW.RightFraction X Y), ψ.s (W := P.trW) ≫ φ.f (W := P.trW) = ψ.f (W := P.trW) ≫ φ.s (W := P.trW)
Tactic: obtain ⟨Z, f, g, H, mem⟩ := φ.hs
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
X Y : C
φ : P.trW.LeftFraction X Y
Z : C
f : φ.Y' (W := P.trW) ⟶ Z
g : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (φ.s (W := P.trW)) f g)
mem : P Z
⊢ ∃ (ψ : P.trW.RightFraction X Y), ψ.s (W := P.trW) ≫ φ.f (W := P.trW) = ψ.f (W := P.trW) ≫ φ.s (W := P.trW)
have obtain : ∀ (X' : C) (f' : X' ⟶ X) (h' : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X')
  (mem' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f' (φ.f (W := P.trW) ≫ f) h')),
  ∃ (ψ : P.trW.RightFraction X Y), ψ.s (W := P.trW) ≫ φ.f (W := P.trW) = ψ.f (W := P.trW) ≫ φ.s (W := P.trW)
Tactic: obtain ⟨X', f', h', mem'⟩ := distinguished_cocone_triangle₁ (φ.f ≫ f)