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
P Q : ObjectProperty.{v_1, u_1} C
X : C
⊢ P.isoClosure.extensionProduct Q X ≤ P.extensionProduct Q X
have this : ∀ (Y Z : C) (f : Y ⟶ X) (g : X ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h))
  (Y' : C) (hP : P Y') (i : Y ≅ Y') (hQ : Q Z), P.extensionProduct Q X
Tactic: ⟨Y, Z, f, g, h, hT, ⟨Y', hP, ⟨i⟩⟩, hQ⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
Y' : C
hP : P Y'
i : Y ≅ Y'
hQ : Q Z
⊢ P.extensionProduct Q X
have refine : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
  (Triangle.mk (i.inv ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) i.hom))
Tactic: refine ⟨Y', Z, i.inv ≫ f, g, h ≫ i.hom⟦1⟧', ?_, hP, hQ⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
⊢ P.extensionProduct Q.isoClosure = P.extensionProduct Q
have refine : P.extensionProduct Q.isoClosure ≤ P.extensionProduct Q
Tactic: refine le_antisymm ?_ (monotone_extensionProduct_right _ Q.le_isoClosure)
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X : C
⊢ P.extensionProduct Q.isoClosure X ≤ P.extensionProduct Q X
have this : ∀ (Y Z : C) (f : Y ⟶ X) (g : X ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h))
  (hP : P Y) (Z' : C) (hQ : Q Z') (i : Z ≅ Z'), P.extensionProduct Q X
Tactic: ⟨Y, Z, f, g, h, hT, hP, ⟨Z', hQ, ⟨i⟩⟩⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
Z' : C
hQ : Q Z'
i : Z ≅ Z'
⊢ P.extensionProduct Q X
have refine : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
  (Triangle.mk f (g ≫ i.hom) (i.inv ≫ h))
Tactic: refine ⟨Y, Z', f, g ≫ i.hom, i.inv ≫ h, ?_, hP, hQ⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.ContainsZero
X : C
hX : P X
⊢ P.extensionProduct Q X
have rw : P.extensionProduct Q.isoClosure X
Tactic: rw [← extensionProduct_isoClosure_right]
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.ContainsZero
X : C
hX : P X
⊢ P.extensionProduct Q.isoClosure X
have obtain : ∀ (Z : C) (hZ : IsZero.{v_1, u_1} Z) (hQ : Q Z), P.extensionProduct Q.isoClosure X
Tactic: obtain ⟨Z, hZ, hQ⟩ := Q.exists_prop_of_containsZero
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.ContainsZero
X : C
hX : P X
Z : C
hZ : IsZero.{v_1, u_1} Z
hQ : Q Z
⊢ P.extensionProduct Q.isoClosure X
have refine : Q.isoClosure 0
Tactic: refine ⟨_, _, _, _, _, contractible_distinguished X, hX, ?_⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X : C
hX : Q X
⊢ P.extensionProduct Q X
have rw : P.isoClosure.extensionProduct Q X
Tactic: rw [← extensionProduct_isoClosure_left]
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X : C
hX : Q X
⊢ P.isoClosure.extensionProduct Q X
have obtain : ∀ (Z : C) (hZ : IsZero.{v_1, u_1} Z) (hP : P Z), P.isoClosure.extensionProduct Q X
Tactic: obtain ⟨Z, hZ, hP⟩ := P.exists_prop_of_containsZero
-- 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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X : C
hX : Q X
Z : C
hZ : IsZero.{v_1, u_1} Z
hP : P Z
⊢ P.isoClosure.extensionProduct Q X
have refine : P.isoClosure
  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) (Triangle.obj₃.{v_1, u_1} (contractibleTriangle X)))
Tactic: refine ⟨_, _, _, _, _, inv_rot_of_distTriang _ (contractible_distinguished X), ?_, hX⟩
-- 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 Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X : C
⊢ P.extensionProduct Q X ≤ (P.extensionProduct Q).shift a X
have this : ∀ (Y Z : C) (f : Y ⟶ X) (g : X ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h))
  (hP : P Y) (hQ : Q Z), (P.extensionProduct Q).shift a X
Tactic: ⟨Y, Z, f, g, h, hT, hP, hQ⟩
-- 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 Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ (P.extensionProduct Q).shift a X
have refine_1 : P (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₁)
Tactic: refine ⟨_, _, _, _, _, Triangle.shift_distinguished _ hT a, ?_, ?_⟩
-- 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 Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
refine_1 : P (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₁)
⊢ (P.extensionProduct Q).shift a X
have refine_2 : Q (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₃)
Tactic: refine ⟨_, _, _, _, _, Triangle.shift_distinguished _ hT a, ?_, ?_⟩
-- 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 Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ P (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₁)
have refine_1 : P (Triangle.mk f g h).obj₁
Tactic: apply IsStableUnderShiftBy.le_shift
-- 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 Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ Q (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₃)
have refine_2 : Q (Triangle.mk f g h).obj₃
Tactic: apply IsStableUnderShiftBy.le_shift
-- 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
P Q R : ObjectProperty.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
⊢ (P.extensionProduct Q).extensionProduct R = P.extensionProduct (Q.extensionProduct R)
have h : ∀ (X : C), (P.extensionProduct Q).extensionProduct R X ↔ P.extensionProduct (Q.extensionProduct R) X
Tactic: ext X
-- 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
P Q R : ObjectProperty.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X : C
⊢ (P.extensionProduct Q).extensionProduct R X ↔ P.extensionProduct (Q.extensionProduct R) X
have h : (P.extensionProduct Q).extensionProduct R X → P.extensionProduct (Q.extensionProduct R) X
Tactic: constructor
-- 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
P Q R : ObjectProperty.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X : C
h : (P.extensionProduct Q).extensionProduct R X → P.extensionProduct (Q.extensionProduct R) X
⊢ (P.extensionProduct Q).extensionProduct R X ↔ P.extensionProduct (Q.extensionProduct R) X
have h₁ : P.extensionProduct (Q.extensionProduct R) X → (P.extensionProduct Q).extensionProduct R X
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X Y C : C_1
f : Y ⟶ X
g : X ⟶ C
h : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
A B : C_1
f' : A ⟶ Y
g' : Y ⟶ B
h' : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hP : P A
hQ : Q B
hR : R C
⊢ P.extensionProduct (Q.extensionProduct R) X
have h₁ : ∀ (Y_1 : C_1) (g'' : X ⟶ Y_1) (h'' : Y_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A)
  (hT'' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk (f' ≫ f) g'' h'')),
  P.extensionProduct (Q.extensionProduct R) X
Tactic: obtain ⟨Y, g'', h'', hT''⟩ := distinguished_cocone_triangle (f' ≫ f)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X A Z : C_1
f : A ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P A
B C : C_1
f' : B ⟶ Z
g' : Z ⟶ C
h' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) B
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hQ : Q B
hR : R C
⊢ (P.extensionProduct Q).extensionProduct R X
have h₁ : ∀ (Y : C_1) (f'' : Y ⟶ X) (h'' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y)
  (hT'' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f'' (g ≫ g') h'')),
  (P.extensionProduct Q).extensionProduct R X
Tactic: obtain ⟨Y, f'', h'', hT''⟩ := distinguished_cocone_triangle₁ (g ≫ g')
-- 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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i : C
⊢ P₁.extensionProduct P₂ i ≤ Q.isoClosure i
have this : ∀ (w w_1 : C) (w_2 : w ⟶ i) (w_3 : i ⟶ w_1) (w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk w_2 w_3 w_4))
  (hY : P₁ w) (hZ : P₂ w_1), Q.isoClosure i
Tactic: ⟨_, _, _, _, _, hT, hY, hZ⟩
-- 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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
inst_7 : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i : C
⊢ P₁.extensionProduct P₂ i ≤ Q i
have this : ∀ (w w_1 : C) (w_2 : w ⟶ i) (w_3 : i ⟶ w_1) (w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk w_2 w_3 w_4))
  (hY : P₁ w) (hZ : P₂ w_1), Q i
Tactic: ⟨_, _, _, _, _, hT, hY, hZ⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X : C
⊢ P.retractClosure.extensionProduct Q.retractClosure X ≤ (P.extensionProduct Q).retractClosure X
have this : ∀ (A B : C) (f₁ : A ⟶ X) (f₂ : X ⟶ B) (f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f₁ f₂ f₃))
  (A' : C) (hP : P A') (a₁ : A ⟶ A') (b₁ : A' ⟶ A) (h₁ : a₁ ≫ b₁ = 𝟙 A) (B' : C) (hQ : Q B') (a₃ : B ⟶ B') (b₃ : B' ⟶ B)
  (h₃ : a₃ ≫ b₃ = 𝟙 B), (P.extensionProduct Q).retractClosure X
Tactic: ⟨A, B, f₁, f₂, f₃, hT, ⟨A', hP, ⟨a₁, b₁, h₁⟩⟩, ⟨B', hQ, ⟨a₃, b₃, h₃⟩⟩⟩
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
⊢ (P.extensionProduct Q).retractClosure X
have obtain : ∀ (X' : C) (g₁ : A' ⟶ X') (g₂ : X' ⟶ B')
  (hT' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))),
  (P.extensionProduct Q).retractClosure X
Tactic: obtain ⟨X', g₁, g₂, hT'⟩ := distinguished_cocone_triangle₂ (b₃ ≫ f₃ ≫ a₁⟦(1 : ℤ)⟧')
-- 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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
⊢ (Triangle.mk f₁ f₂ f₃).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁ =
    a₃ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₃
have dsimp : f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁ = a₃ ≫ b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁
Tactic: dsimp