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 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
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)
have obtain : ∀ (a : (Triangle.mk f' (φ.f (W := P.trW) ≫ f) h').obj₁ ⟶ (Triangle.mk (φ.s (W := P.trW)) f g).obj₁)
  (ha₁ :
    (Triangle.mk f' (φ.f (W := P.trW) ≫ f) h').mor₁ ≫ φ.f (W := P.trW) = a ≫ (Triangle.mk (φ.s (W := P.trW)) f g).mor₁),
  (Triangle.mk f' (φ.f (W := P.trW) ≫ f) h').mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a =
      𝟙 Z ≫ (Triangle.mk (φ.s (W := P.trW)) f g).mor₃ →
    ∃ (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 ⟨a, ⟨ha₁, _⟩⟩ := complete_distinguished_triangle_morphism₁ _ _
      mem' H φ.f (𝟙 Z) (by simp)
-- 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
Y Z Z' : C
f₁ f₂ : Y ⟶ Z
s : Z ⟶ Z'
hs : P.trW s
hf₁ : f₁ ≫ s = f₂ ≫ s
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have hs₁ : ∃ (X : C) (f : X ⟶ Z) (h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f s h)),
  P X
Tactic: rw [P.trW_iff'] at hs
Cleared hypotheses: hs, hf₁
-- 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
Y Z Z' : C
f₁ f₂ : Y ⟶ Z
s : Z ⟶ Z'
hs :
  ∃ (X : C) (f : X ⟶ Z) (h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f s h)),
    P X
hf₁ : f₁ ≫ s = f₂ ≫ s
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have obtain : ∀ (Z_1 : C) (g : Z_1 ⟶ Z) (h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z_1)
  (H :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk g s h))
  (mem : P Z_1), ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
Tactic: obtain ⟨Z, g, h, H, mem⟩ := hs
Cleared hypotheses: 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
⊢ (f₁ - f₂) ≫ s = 0
have rw : f₁ ≫ s - f₂ ≫ s = 0
Tactic: rw [sub_comp, 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
rw : f₁ ≫ s - f₂ ≫ s = 0
⊢ (f₁ - f₂) ≫ s = 0
have rw₁ : f₂ ≫ s - f₂ ≫ s = 0
Tactic: rw [sub_comp, 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
rw : f₁ ≫ s - f₂ ≫ s = 0
rw₁ : f₂ ≫ s - f₂ ≫ s = 0
⊢ (f₁ - f₂) ≫ s = 0
have rw₂ : Eq.{v_1 + 1} (α := Y ⟶ Z') 0 0
Tactic: rw [sub_comp, 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have hf₂ : (f₁ - f₂) ≫ s = 0
Tactic: have hf₂ : (f₁ - f₂) ≫ s = 0 := by rw [sub_comp, 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have obtain : ∀ (q : Y ⟶ (Triangle.mk g s h).obj₁) (hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁),
  ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
Tactic: obtain ⟨q, hq⟩ := Triangle.coyoneda_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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have obtain : ∀ (Y' : C) (r : Y' ⟶ Y) (t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y')
  (mem' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk r q t)),
  ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
Tactic: obtain ⟨Y', r, t, mem'⟩ := distinguished_cocone_triangle₁ 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
refine_1 : P.trW r
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂
have refine_2 : r ≫ f₁ = r ≫ f₂
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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ r ≫ f₁ = r ≫ f₂
have eq : (Triangle.mk r q t).mor₁ ≫ (Triangle.mk r q 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
⊢ r ≫ f₁ = r ≫ f₂
have refine_2 : r ≫ f₁ - r ≫ f₂ = 0
Tactic: rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]
-- 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
refine_2 : r ≫ f₁ - r ≫ f₂ = 0
⊢ r ≫ f₁ = r ≫ f₂
have refine_2₁ : r ≫ (f₁ - f₂) = 0
Tactic: rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]
-- 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
refine_2 : r ≫ f₁ - r ≫ f₂ = 0
refine_2₁ : r ≫ (f₁ - f₂) = 0
⊢ r ≫ f₁ = r ≫ f₂
have refine_2₂ : r ≫ q ≫ (Triangle.mk g s h).mor₁ = 0
Tactic: rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]
-- 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
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
refine_2 : r ≫ f₁ - r ≫ f₂ = 0
refine_2₁ : r ≫ (f₁ - f₂) = 0
refine_2₂ : r ≫ q ≫ (Triangle.mk g s h).mor₁ = 0
⊢ r ≫ f₁ = r ≫ f₂
have refine_2₃ : Eq (α := Y' ⟶ Z_1) (0 ≫ (Triangle.mk g s h).mor₁) 0
Tactic: rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]